Modern Algebra A Logical Approach, Book One [1, 1 ed.]

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Book One

Modern Algebra A Logical Approach By HELEN R. PEARSON Head of the Mathematics Department Arlington High School INDIANAPOLIS , INDIANA

and FRANK B. ALLEN Chairman, Department of Mathematics Lyons Township High School and Junior College LA GRANGE , ILLINOIS

GINN AND COMPANY BOSTON

NEW YORK DALLAS

CHICAGO

PALO ALTO

ATLANTA

TORONTO

© COPYRIGHT, 1964, BY GINN AND COMPANY ALL RIGHTS RESERVED

PICTURE SOURCES

Page 14 : Bell Telephone Laboratories Page 28 : National Aeronautics and Space Administration Photographs Page 94: International Business Machines, Inc. Page 352: The Carpenter Steel Company Page 596: American Airlines

Cover Design- J oseph Loughman Typographical Design-J ohn Rechel

ii

Preface

This text constitutes a one-year course fo r students beginning the study of algebra . It is adapt able t o both average and above-average classes. It is a t ext that encourages thought as opposed t o mere mechan ical juggling of algebraic symbols. The stu dent sees how a syste m of algebra is developed and leqrns how to justify hi s mathematical state ments by logical argument. In its early pages the book explains the meanin g of "If p, then q" when p and q a re statements a nd leads students to see that if we accept p and the st at ement " p implies q," then we conclude that we must accept q. T his idea, symbolized by p and } -+ q, is emphasized as the basic building

p-+q

block in drawing inferences . T he text uses this idea t o explain the various rules of algebra. It shows th at by accepting the well-known properties of the numbers of arithmetic, we can build a mathe matical structure called algebra . Students a re encouraged to build their own proofs. The first five chapte rs of the t ext are devoted to developing the algebra of the numbers of arithmeti c. In them the stude nt gains a vocabulary of algebraic t erms; mee ts such concepts as variable, equation, inequality, solution set, and graph; and learns to recognize the properties of equality and inequality and the properties of operations with numbers while working within a set of numbers already fam iliar to him. He also learns that the fam iliar set of the numbers of a rithmetic has various subsets- the set of natural numbers , the set of whole numbers , the set of rational numbers of arithmetic, a nd the set of irrational numbers of arithmetic- each having some properties that are not possessed by other sets. In Chapters 6- 11 these ideas are gradually extended to all real numbers . Chapte r 12 introduces the student to the concepts of relation and function. The t ext is not a book on theo ry alone, however. Every bit of theory is explained by exa mples and followed by exercises . These exercises a re constructed to ge nera te thought as well as to develop manipulative skill. iii

iv

Preface

The text is carefully constructed and complete. Definitions are well stated and , once stated, are adhered to meticulously in the succeeding developments. Organization has been planned to reveal the stru cture of algebra gradually and with simplicity. The basic ideas appear again and again so that students come to recognize them for what they are, the foundation blocks of algebra . The book is a modern text . It is a synthesis of the experiences and conclusions of th e authors, gained from participation in and careful analysis of present-day cooperative programs to improve mathematics teaching. The spirit, the subj ect matter, and the symbolism of the book are up-to-date . . Moreover, the text contains those time-tested features desired by most teachers. At the end of each chapter there is a list of essentials that students should have learned from study of the chapter, a list of algebraic terms whose meaning and spelling should have been learned, a list of review exercises covering topics discussed in the chapter, and a chapter test. The book also contains four cumulative reviews and four cumulative tests. Individual differences are provided for by the organization of the text and by the groupin g of exercises. Most theorems in the book are accompanied by a discussion of the situation which calls for the theorem and usually by one or more examples. All proofs included are designed to enhance clarity of exposition , and their inclusion does not imply any attempt to achi eve an ultra-rigorous development. Many proofs may be omitted, however, without impairing the continuity of the text. Exercises are divided into A and B groups. Those in group A are the easier exercises and also the more basic; those in group B are a little more difficult and often lead pupils to think ahead of the text. Pictures and between-chapter articles a re used to add interest and to provoke thought on subj ects related to the topics discussed in the book. The tables needed by the stud ent are included . The authors take this opportunity to acknowledge the assistance of Mr. Darwin M. Newton and the fine editorial staff at Ginn and Company. They likewise wish to thank Mr. Roger B. Hooper for his assistance in preparing the notes and answe rs for the annotated edition, and Mr. J. Glenn Maxwell and Reverend Stanl ey Bezuszka, S.J. , for their careful reading of part or all of the manuscript and for their many hel~ful suggestions. H elen R . P earson Frank B. Allen

Contents

PAGE

1

1

Sets Meaning of Set, 1 · Defining a Set, 1 · Naming a Set, 2 · Belonging to a Set, 2 · Equal Sets, 2 · Number of Elements in a Set, 4 · Subsets, 5 · Universal Set, 6 · Venn Diagrams, 9 · Operations with Sets, 10 · Closure, 15 · Closure and the Numbers of Arithmetic, 15 • Equal and Unequal Expressions, 16 • Numerals, 18 · Variables, 18 · Essentials, 22 · Chapter Review, 23 · Chapter Test, 25

Systematic Counting

2

26

29

The Number Line The Numbers of Arithmetic, 29 · The Whole Numbers and the Number Line, 29 • The Natural Numbers and the Number Line, 31 · The Rational Numbers of Arithmetic and the Number Line, 32 · Betweenness on the Number Line, 35 • Order on the Number Line, 37 · Graphs of Sets, 39 · Compari son Symbols, 42 • Essentials, 46 · Chapter Review, 46 · Chapter Test, 48

Systems of Numeration

3

50

52

Expressions and Sentences Algebraic Expressions, 52 · Order of Operations, 53 · Indicating Multiplication with Parentheses, 55 · Evaluation, 55 · Simple Numerical Sentences, 57 • Open Sentences, 60 • Solution Sets, 61 • The Set Builder, 64 • Graphs of Truth Sets, 66 • Compound Sentences, 68 · Compound Sentences with the Connective "and," 68 · Compound Sentences with the Connective "or, " 70 · Graphs of Truth Sets of Compound Sentences, 72 · Essentials, 74 · Chapter R eview, 75 • Chapter Test, 77

Reaching Logical Conclusions

78 y

PAGE

4

80

Logic Statements, 80 • Equivalent Statements, 81 • Contradictions, 83 • More about Contradictory Statements and Equivalent Statements, 85 · A Special Notation for "Contradiction," 87 · Negation, 87 · Disjunction, 89 • Conjunction, 91 · Contradicting Disjunctions and Conjunctions, 95 • Conditional Sentences- Implications, 97 · Variation in Stating an Implica tion, 99 • More about Implications, 100 · Contradicting an Implication, 101 · Proof, 103 · Form for Writing a Syllogism, 105 · Transitive Property of Implication, 107 · Forms of Proof, 109 · Forms of Simple Implications, 11 2 • Proof by Using the Contrapositive, 114 · Indirect Proof, 116 · Other Ways to State Implications and Equ ivalences, 118 · Need for Assumptions and Definitions, 121 · Essentials, 122 · Chapter Review, 123 · Chapter Test, 125 · Cumulative Review, 126 · Cumulative T est, 130

Logic and Switching Circuits

5

132

Operations with the Numbers of Arithmetic

134

The Numbers of Arithmetic, 134 • Closure and the Numbers of Arithmetic, 135 · Binary Operations, 137 · The Properties of Equality, 139 · Properties of Addition and Multiplication, 142 · Evaluating Open Expressions, 145 · Inverse Operations, 148 · Solving Equations, 150 · Equations Whose Solutions R equire More than One Operation, 154 • Using Equations to Solve Problems, 157 · Equivalent Equations and Equation Solving, 160 · Identity Elements, 162 · The Multiplicative Inverse, 162 • The Distributive Property, 166 · Equi valent Expressions, 169 · Using the Distributive Property to Write Equivalent Expressions, 170 · Using the Distributive Property in Equation Solving, 174 · Multiplication Property of Zero, 178 • E xpressing Division by the Multiplicative Inverse, 181 · Fractions, 182 • Multiplication of Fractions, 183 · Simplifying Fractions, 186 · Addition of Fractions, 189 · Equations Involving Fractions, 190 · Essentials, 196 • Chapter Review, 197 · Chapter Test, 200

202

Boolean Algebra

6 The Real Numbers

204

The Real Numbers, 204 · Integers and Rational Numbers, 206 · The R eal Numbers and the Number Line, 207 • Order on the Real Number Line, 208 • The Properties of the Real Numbers, 209 • Uniqueness of the Additive Inverse, 212 · Addition of Real Numbers, 214 · Absolute Value, 217 · The Rules for Adding Real Numbers, 219 · Addition on the Number Line, 221 • Subtraction of R eal Numbers, 222 · Multiplication of Real Numbers, 227 · Signs of a Real Number and its Multiplicati ve Inverse, 230 • Some Theorems Concerning the Multiplication of Real Numbers, 23 1 • Equivalent Exvi

PAGE

pressions Containing Parentheses, 235 · Polynomials, 240 • Product of Two Monomials, 242 · Product of a Monomial and a Polynomial of Two or More Terms, 243 · Product of Two Binomials, 244 • Factoring a Polynomial, 247 · Problem Solving, 25 1 · Addition Property of Inequality, 256 • Multiplication Properties of Inequality, 259 • Essentials, 263 • Chapter Review, 264 · Chapter Test, 267

Repeating Decimals

7

268

270

Division of the Real Numbers Definition of Division of Real Numbers, 270 · Zero, Not a Divisor, 271 • Rules for Division of Real Numbers, 274 • The Signs of a Fraction, 276 • Rational Numbers as Quotients of Integers, 278 • Applications of Theorems on Fractions, 279 · Adding Fractions in the Real Number System, 282 • Ratio and Proportion, 285 · Complex Fractions, 288 • Equations Involving Fractions, 289 · Fractional Equations, 293 • Problems Whose Equations Involve Fractions, 296 · Formulas Involving Fractions, 301 • Inequalities Involving Fractions, 306 · Sentences Involving Absolute Value, 308 • Essentials, 309 · Chapter Review, 310 • Chapter Test , 313 • Cumulative Review, 314 · Cumulative Test, 317

Triangular Numbers

8

319

320

Factors and Exponents Fundamen tal Operations wi th Polynomials over the Integers, 320 · Factoring Integers, 322 • An Important Theorem, 325 • Even and Odd Integers, 326 · Tests fo r Divisibility, 327 · Prime Numbers, 329 · Complete Factorization, 331 · Least Common Multiple, 332 · Greatest Common Factor, 333 · Adding Fractions Having Integral Denominators, 334 • Some Theorems about Factors and Products of Integers, 337 · Exponents, 340 · Laws of Exponents- Positive Integers as Exponents, 342 • Laws of ExponentsZero and Negati ve Integers as E xponents, 347 • Scientific Notation, 350 · Monomials Involving Variables, 353 • Adding Fractions Having Monomial Denominators, 355 · Essentials, 356 · Chapter Review, 357 Chapter Test, 359

Theory of Numbers

9

360

Polynomials and Rational Expressions Degree of a Polynomial, 362 · Adding Polynomials, 363 • The Meaning of " Factorable," 365 · Multiplying and Factoring, 366 • More about Multiplication of Polynomials, 368 · More about Equations and Inequalities Involving Products, 370 · Grouping before Factoring, 373 · Difference of Two Squares, 37 5 · Perfect Squares, 378 • Factoring after Completing the Square, vii

362

PAGE

381 · Quadratic Polynomials, 382 · Products Which Can be Expressed as Quadratic Polynomials, 383 · Factoring Quadratic Trinomials, 385 • Factoring ax 2 + bx + c; a~ 0, a~ 1, 387 · Roots of an Equation , 389 • Solving Quadratic Equations Using Factoring, 389 • Summary of Factoring Meth ods, 392 · More about Solving Equations Using Factoring, 393 • Poly nomials over the Rational Numbers, 395 · Rational Expressions, 397 • Simplification of Rational Expressions, 399 · Multiplication and Di vision of Rational Expressions, 401 · Least Common Multiple of P olynomials, 403 • Addition and Subtraction of Rational Expressions, 403 • More about Fractional Equations and Inequalities, 407 · Meaning of Division of One P olynom ial by Another Polynomial , 410 · H ow to Divide a Polynomial by a Polynomial, 412 · Essentials, 415 · Chapter Rev iew, 416 • Chapter T est, 419 • Cumulati ve Review, 420 · Cumulative T est, 423 More about Theory of Numbers

10

426

The Real Number Plane

428

Ordered Pairs of Numbers, -128 · Open Sentences in Two Variables , 429 · Cartesian Products, 43 1. · Graphs of Sets of Ordered Pairs of Numbers, 432 · The Graph of IX I , 437 · The Graphs of Q X Q and RX R, 44 1 · Distance betwee n T wo P oints, 443 · Directed Distance, 446 · Graphs of Open Sent ences in Two Variables, 448 · Straight Lines, 452 · y-Intercepts, 453 · Slope of a Straight Line, 454 · Graphing Equations by the Slope-Intercept Method, 457 · D etermining the Equation of a Line by the Slope-Intercept Method, 458 · Parallel Lines and Perpendicular Lines, 460 · Disjun cti ve Sentences Whose Clauses are Equations, 464 · Systems of Equations, 465 · Solving Systems of Equations by the Addition Method, 467 · Solving Systems of Equations by the Substitution Method, 477 • Summary of Ways to Solve Systems of Equations, 478 · Types of Systems of Equations, 480 · P roblems Involving Systems of Equations, 482 • Graphs of Open Sentences Involving Inequalities, 487 · Graphs of Sentences Involving Absolute Value, 489 · Systems Involving Inequalities, 490 · Essentials, 492 · Chapter Review, -193 · Chapter T est , 496

498

Linear Programing

11

500

Radicals R oots of Real Numbers, 501 · Square Roots of Real Numbers, 502 · Cube R oots and Other Roots, 504 · Irrational Numbers, 506 · Products of Radicals, 508 • Quotients of Radicals, 511 · Sums Involving Radicals, 515 · Multiplication and Division of Binomials Containing Radicals, 516 · Square Root , 519 • Rational Numbers as Exponents, 526 · More about Rational viii

PAGE

Exponents, 529 · Equations Containing Radicals, 530 • More about the Pythagorean Theorem, 532 • Essentials, 536 Chapter Review, 537 Chapter T est, 539 More about the Irrationality of

12

Y2

540

542

Functions and Other Relations R elations, 542 · D omain and Range of a R elation, 5-lc5 · Functions, 548 · Function Notation, 554 · Graph s of Fun ctions, 556 · Linear Functions, 559 · Direct Variation and Proportion, 561 · Inve rse Variation, 566 · The D egree Measure of a n Angle, 570 · The Tangent Ratio, 570 · Applications of the Tange nt Ratio, 575 · T he Cotangent Ratio, 578 · T he Sine and Cosine Ratios, 582 · R eview- Functions Associated with Angles, 586 · Quad ra tic Functions, 588 · Grap hing a Quadratic Function, 588 · The Graph of y ax 2 + bx + c, 59 1 · Solvi ng Quadratic Equations Using Factoring, 592 · More about Solvi ng Quadratic Equations, 594 · Essentials, 598 · Chapter Review, 599 · Chapter Test, 602 · Cumulative R eview, 603 · Cumulat ive T est , 607

=

Table I. Square Roots, Squares Table II. Natural Functions: Sine, Tangent, Cotangent, Cosine

608 609

Index

610

NOTE TO TEACHERS. An annotated edition and other aids are available with this text. ix

List of Symbols C

w F A

Q N I R

A '{; B ACB

0

u At\B AUB

A' X EA

A= {a, b, c, · · ·} A= {a, b, c, · · ·, n} {x I }

=

{}, [], () , p or q p and q p~q p+-+q

,..._,p

Ir I AB /ABI AB

AXB J(x)

Set of counting numbers Set of whole numbers Set of rational numbers of arithmetic Set of numbers of arithmetic Set of rational real numbers Set of negative numbers Set of integers Set of real numbers A is a subset of B. (A r;J,. B A is not a subset of B. ) A is a proper subset of B. The empty set, or null set The universal set The intersection of set A and set B The union of set A and set B The complement of set A x is an element of set A. (x i A x is not an element of set A.) A is the set of elements a, b, c, and so on. A is the set of elements a, b, c, and so on to, and including, n . Set-builder notation Is equal to (,e Is not equal to) Is less than ( Is not greater than) Is less than or equal to ( i Is not less than or equal to) Is greater than or equal to ( ;t Is no t greater than or equal to) Is approximately equal to Symbols of grouping Disjunction of p and q Conjunction of p and q If p, then q. pis equivalent to q. Contradiction of p, read " not p" Absolute value of r Line segment AB Measure of line segment A B Directed distance from A to B Cartesian product of set A and set B Function notation Therefore X

Chapt,r

1 Sets In this chapter you will learn to work with sets of numbers

Meaning of Set

We often find it useful to talk about collections or groups of objects. For example, we might want to talk about the collection of coins owned by Mr. Jones, the collection of cars in the school parking lot, the group of children in the family next door, or the collection of all jet planes owned by the navy. Each of these is an example of a set of objects. We use the word "object" in a very broad sense to mean any person, thing, action, or idea. Any object in a particular set is called a member or element of the set. For example, in a set of tires each tire is a member of the set. Defining a Set

Sometimes we describe a set by listing its members. Thus if Miss Thomas says, "Consider the set of letters a, e, i, o, u," you know that she wants you to consider these letters and no others. Braces, { } , are often used to enclose the members of a set. If we write {a, e, i, o, u} we mean "the set whose members are a, e, i, o, and u." Listing the members of a set within braces is frequently referred to as the roster method of defining a set. Sometimes we describe a set by stating the conditions under which a given object belongs to the set. This is called the rule method. Thus Miss Thomas might indicate the set above by saying, "Consider the standard set of vowels in the English alphabet not including w and y." Again you would know that a, e, i, o, and u are members of the set and that no other letters of the alphabet are members. When we make clear which objects belong to a set and which do not, we have defined the set. Whether Miss Thomas chooses to write {a, e, i, o, u} or to

speak of "the standard set of vowels in the English alphabet," she has defined the set she wants you to consider.

2

CHAPTER

1

Naming a Set

We use a capital letter to name a set. Thus we might find it convenient to speak of the set of vowels as the set V. We may write: Vis {a, e, i, o, u} or V= {a, e, i, o, u}. Belonging to a Set

We use the symbol e to indicate that an object belongs to a particular set and the symbol t to indicate that an object does not belong to the set . Thus to write, "The letter a is an element of set V," or "The letter a belongs to the set V, " we may write a e V. To write, "The letter h is not an element of set V, " or "The letter h does not belong to set V," we may write ht V. Equal Sets

Equal sets are sets that have exactly the same members. The order in which the elements of a set are listed does not matter. Thus if V = {a, e, i, o, u} and W = {i, o, u, e, a}, then V and Ware equal sets. We write V = W. If A = {1, 2, 3, 4} and B = {1, 2, 3, S}, sets A and Bare not equal because they do not have the same members. Which members are unlike? To write, "A is not equal to B," we write A -,e B. ORAL EXERCISES

1. What are two ways of defining a set? 2. How do we name a set? 3. What is the meaning of the symbol e?

4. What are equal sets?

0

EXERCIS~S

1. By listing the members between braces, indicate each of the sets described below. a. Your teachers this semester

b. The months of the year c. The states of the United States that are completely surrounded by water d. All counting numbers less than 10 (Zero is not one of the counting numbers.) e. All odd counting numbers greater than 5 but less than 15

f. All counting numbers less than 20 that are exactly divisible by 2

3

Sets

g. The counting numbers less than 20 that are exactly divisible by 4

h. The two-digit numbers each having its tens digit and units digit alike i. The two-digit numbers each having its tens digit twice its units digit 2. Write a word description of each of the sets described below. Be sure to make your description complete enough to exclude objects that do not belong to the set. a. {red, white, blue}

b. {Monday, Tuesday, Wednesday, Thursday, Friday, Saturday, Sunday} c. {2, 4, 6, 8}

d. {21, 23, 25, 27, 29} e. {George Washington, John Adams, Thomas Jefferson, James Madison, James Monroe, John Quincy Adams}

f. {January, June, July} g. {10, 20, 30, 40, 50, 60, 70, 80, 90} 3. If P is the set of presidents of the United States, indicate which of the following statements are true.

a. George Washington e P.

d. Christopher Columbus t P.

b. Thomas A. Edison e P.

e. Franklin D . Roosevelt t P.

c. Abraham Lincoln t P.

f. Theodore Roosevelt e P.

4. Insert either the symbol e or the symbol t to make each of the following statements true.

a. If A = {1, 3, 5, 7, 9}, then 1 _?_ A, 2 _?_ A, and 7 _?_ A.

b. If S = the set of states bordering the Pacific Ocean, then Illinois _? _ S, Ohio _? _ S, and California _? _ S. c. If L is the set of letters appearing exactly twice in the word Kokomo, k _? _ L, o _? _ L, and m _? _ L.

5. In each of the following statements insert either the symbol = or the symbol r5- to make the statement true. a. If A= {2, 4, 6, 8} and B

= {1, 3, 5, 7}, A

_? _ B.

b. If A= {2, 4, 6, 8} and C= {2, 4, 7, 8}, A_?_ C.

= d. If A = c. If A

= and E =

{2, 4, 6, 8} and D

{8, 4, 2, 6}, A _? _ D.

{2, 4, 6, 8}

{2, 4, 6, 8, 9}, A _? _ E.

4

CHAPTER

1

Number of Elements in a Set

A set m ay have any number of elements. Let us consider the sets S

= {14}

B = {the planet M ars , the junior prom, the Golden Rule, Socrat es} N = {l , 2, 3, 4, 5, 6, 7, 8, 9} D = the set of all people who live in Chicago E = the set of vowels which are successors to vowels when all of the letters of the English alphabet are arranged in st andard order 0 = the set of all odd counting numbers W = the set of all whole numbers

It is easy t o count the number of elements in sets S, B, and N. S has one element, B has four elements, and N has nine elements. While it would be possible to count all the elements of set D, to do so would require an enorm ous amount of time and energy. Set E is remarkable in that it contains no elements. (There is no vowel that is a successor to another vowel when the letters of the alphabet are arranged in standard order.) A set having no elements is called the empty set or the null set. Sin ce all empty set s have exactly the same members, that is, no members, they are all equal. This is why we refer to the empty set or the null set. Other examples of the null set are the set of authentic coins bearing the minting date 200 B.C . and the set of counting numbers greater than 11 but less than 12. To represent the null set we often use the symbol 0· We do not use braces around 0. Observe that 0 is not the same as {0}. The set represented by {0} is not an empty set because it has the one m ember, 0. We shall say that a set is a.finite set if the elements of the set can be counted with the counting coming to an end or if the set is the null set. All other set s will be called infinite sets. There is no end to the counting of elements in an infinite set . Set s S, B, N, D , and E above are finite sets. Sets O and W are infinite set s. We cannot list all the elements of an infinite set, but we can sometimes describe the set by listing the first three or four members and by following these with three dots . The three dots mean, "and so on." For example, {l , 2, 3, · · ·} represents the set of all counting numbers beginning 1, 2, 3 and continuing in the same pattern without end. Similarly, {O, 1, 2, 3, · · ·} rep. resents the set of whole numbers. When this procedure fails to describe an infinite set clearly, we state the conditions for membership in the set . We also use three dots to repres~nt the words, "and so on to, and including." For exampl e, {l , 3, 5, 7, · · ·, 99} indicates " the set 1, 3, 5, 7, and so on to , and including, 99."

5

S ets

ORAL EXERCISES

1. What is a finite set? an infinite set? 2. What is the difference between 0 and {O}? 3. What do the three dots mean in each of the sets {l, 2, 3, · · ·} and {l , 2, 3, · · ·, 100}?

0

EXERCISES

1. C= {l , 2, 3, 4, · · ·}. Which of the following statements are correct? a. 10 f C b. 0 f C c. 784,698 f C 2. T = {2, 4, 6, · · ·, 98}. Which of the following statements are correct? a. 8 E T b. 700 E T c. 59 f T

3. Which of the following sets are finite and which infinite? a. The set of all blades of grass growing in Kentucky

b. The set of all counting numbers greater than 4 c. The set of all counting numbers less than 1,000,000

d. The set of all counting numbers great er than 21 but less than 22 e. The set of all whole numbers that when multiplied by 4, give 20

f. The set of all the numbers studied in arithmetic 4. Describe the following sets by listing their members. a. The set of all one-digit counting numbers greater than 9

b. The set of numbers obtained by adding one t o each of the numbers in {0, 1, 2, 3, · · ·}

c. The set of sums for the addition exercises in A= {l + 1, 2 + 2, 3 + 3, 4 + 4, · · ·} Subsets If every element of set A is also an element of set B, then set A is a subset of set B; that is, set A is included in set B. For example, the set of words on this page is a subset -of the set of words in this book. Similarly, the set of lakes in Minnesot a is a subset of the set of all lakes in the United States. To write, "Set A is a subset of set B," or "Set A is included in set B," we write A ~ B . If A is included in B then, of course, B includes A. To write, "B includes A," we write B ;;;2 A. We write A r;J,. B or B '12. A to mean, "Set A is not a subset of set B." This means that set A contains at least one element which is not an element of set B . For example, the set of students in your algebra class is a subset of

6

CHAPTER

1

the set of all students in your school, but the set of people in the room during an algebra recitation is not a subset of all the students in school because the set contains at least one person, the teacher, who is not a student . If B is the set of all students in a certain four-year high school, B will contain , among others, the following subsets: 1. All members of the high school varsity basketball team (set V). Thus V~B. 2. All members of the high school freshman class (set F ) . Thus F ~ B . 3. All students in the high school who are over sixteen years of age (set 0). Thus O ~ B. 4. All students in the high school enrolled in foreign language courses (set L). Thus L ~ B. 5. All the students in the high school. Thus B ~ B. We must accept the last statement as true because it meets the requirements of our definition for subset. It is certainly true that any element of Bis an element of B. Thus we must conclude: Any set is a subset of itself. We agree that the null set or empty set is a subset of every set. In reaching this agreement our reasoning is as follows . If A is a subset of B, then every element of A is an element of B. This means that A can be obtained from B by taking away none, some, or all of the elements of B. The subset of Bobtained by taking away all of its elements is the empty set. If every element of A is also an element of B and there is at least one element in B which is not in A, we say that A is a proper subset of B. To write "A is a proper subset of B," or "A is properly included in B," we write A C B. Of course, if A is properly included in B, then B properly includes A, and we write B:) A. For example, if A= {2, 4} and B = {l, 2, 3, 4}, then A is a proper subset of B because B contains two elements, 1 and 3, that are not elements of A. If C = {1, 2, 3, 4}, then C is not a proper subset of B because B has no elements that are not in C. You recall that we have said that two sets are equal if they have exactly the same members. We now have another way to determine that two sets are equal. Two sets, A and B, are equal if both of the following statements are true. 1. Every member of A is a member of B (A ~ B). 2. Every member of Bis a member of A (B ~ A). Universal Set

If, in a certain discussion, all the sets under consideration are subsets of set A, we may designate A as the universal set for this discussion. For example, if all of the sets we are discussing at a particular time are subsets of

7

Sets

the set of whole numbers, W = {O, 1, 2, 3, ·· ·}, then W can be chosen as the universal set for the discussion. In other discussions our universal set might be the set of all counting numbers, the set of all fractions, the set of all people who voted in the last election, or the set of mines located in North America. In any discussion it is a good idea to describe the universal set from which the elements of all the other sets are to be chosen. We usually denote the universal set by U . For example, if we are considering subsets of the set of whole numbers, we might write U = {O, 1, 2, · · ·}. ORAL EXERCISES

1. What is a subset of a given set? 2. How do we know whether a subset is a proper subset of a given set? 3. What is a universal set? 4. What can you say about the subsets of the empty set?

0

EXERCISES

1. Write eight possible subset s of R if R = {umbrella, raincoat, overshoes} . How many of these subsets are proper subsets of R? 2. Given D = {1, 2, 3, 4, 5, 6, 7, 8, 9}, write the following subsets of D:

a. All even numbers in D

b. All even numbers in D that are greater than 0 c. All members in D that are exactly divisible by 3

d. All members in D. that are less than 1 e. All odd numbers in D that are greater than 3 but less than 7

3. From S = {O, 1, 2, 3, 4, 5} select the following subsets.

(When more than one subse t is possible choose the one having the most members. )

a. The subset such that 3 times each of its elements is in S

b. The subset such that one more than twice any one of its elements is in S c. The subset such that the sum of any two of its elements is in S

4. If U = {2, 4, 6, 8, 10}, M = {2, 3, 4}, N = {4, 6, 8, 10}, and O = {4, 8}, which of the following statements are true? a. Mc U

b. N

~

U

c. NC U

d. 0 CU e. 0 ~ U f. 0 ~ U

g. 0 CU

h. UC U i. UC U

8

CHAPTER

1

5. If Sis the set of students listed below, together with their current grades, write the subsets of S indicated in a, b, c, and d. GRADES Name

Algebra

English

Foreign Language

Science

Ben Alton David Evans Dick Grove Don Leslie Tom White

B

B A B

B

A

A

B

A B

C F

C F

A C

A B A

A B B

B

~

A

B A D

Girls

Mary Adams Nancy Bray Sally Thorn

A C

B

C

A

a. Suppose that only boys are eligible to try for the track team and only those boys who have a passing grade in at least three of the subjects listed. If F is a failing grade and A, B, C, and D are passing grades, write the subset of students who are eligible to try for the track team.

b. Suppose that to be eligible for consideration as a reporter for the school paper a student (either a boy or a girl) must have a grade of A in English and a grade of C or above in each of the other subjects listed. Write the subset af students who are eligible for consideration as a reporter. Grades of A and B are higher than a grade of C. c. Suppose that to be eligible for the school honor roll a student (either a boy or a girl) must have at least 30 honor points. If 8 honor points are given for an A, 6 for a B, 4 for a C, 2 for a D, and none for an F, write the subset of students who are eligible for the honor roll .

d. To be eligible for a position on the student council a student (either a boy or a girl) must have an average of at least 5 honor points in the subjects listed. Write the subset of students eligible for a position on the student council. Use the honor point evaluation scale of part c, above. 6. Given that A = {a, b, c, d} and B = {b, d, a, c}, answer the following.

a. Is A a subset of B?

h. Is B a subset of A? c. How does this tell you that A= B?

0

EXERCISE

7. Can you suggest a short cut for counting the number of subsets in any given finite set? Venn Diagrams

We sometimes use Venn diagrams to picture relationships between sets. The figure at the right is a Venn diagram showing that the set of juniors (set J) in a certain school U is a subset of the set of all pupils in: the school (set U). We may think of the rectangle as a line enclosing points which represent the members of set U, and of the closed curve within the rectangle as a line enclosing points which represent the members of set J. Since the curve is entirely inside the rectangle, set] is pictured as a subset of set U. Actually the outlines above might have been of any shape. We usually use a rectangle to picture a universal set, however. In the figure at the right the closed curve u labeled G indicates the set G of girls who are juniors. Since the curve G is entirely within the closed curve J, it pictures the fact that the set of all girls in the junior class is a subset of the set of all juniors in the school. In the third of these figures at the right the u closed curve labeled M indicates the set of all junior girls· who are studying mathematics. Is this set pictured as a subset of the set of junior girls?

0

EXERCISES

1. Draw a Venn diagram to illustrate each of the following statements: a. The set B of all boys on the basketball team is a proper subset of the set U of all boys in school. b. The set of counting numbers, C = {l, 2, 3, • • •}, is a proper subset of the set of all whole numbers, W = {O, 1, 2, 3, •··},which is a proper subset of U, the set of numbers of arithmetic. c. The set B 1 (read B sub one) of all books in your school library is a proper subset of the set B 2 of all books in your state, which is a proper subset of the set B 3 of all of the books in the United States, which is a proper subset of the set U of all of the books in the world.

10

CHAPTER

1

d.R C U

e. R C U and S C R

f. R C U, S C R, and T C S

2. Figures 1- 5 are Venn diagrams illustrating five

u ~

possible relationships among sets E, M, and U, where E is the set of all seniors studying English in a certain high school, Mis the set of all seniors studying mathematics, and U is the set of all seniors in the school. Statements a- f describe conditions relating these sets. Try matching each statement with the appropriate diagrams. Be careful because sometimes more than one diagram belongs with a particular statement.

(__/LJ

a. Some, but not all, of the seniors who study English also study mathematics.

Fig. 2

Fig. 1 U ~

~

u

~

b. Some, but not all, of the seniors who study mathematics study English . c. All of the seniors who study mathematics study English, but some who study English do not study mathematics.

d. Some of the seniors study mathematics but do

Fig. 3

u

not study English, some study English but do not study mathematics, and some study both. e. No senior who studies mathematics studies English and no senior who studies English studies mathematics.

Fig. 4 U

f. Not all seniors study mathematics, but every senior studying mathematics studies English. Operations with Sets

Fig. 5

You know that if we are given two numbers we can add them, multiply them, and perhaps divide them or subtract them. You know that each of these operations produces a number. For example, if we add 2 and 3, we obtain the number S; if we add O and 7, we obtain the number 7. You will not be surprised, therefore, to learn that if we are given two sets, we can perform certain operations with them to produce a third set, which may be one of the given sets. Two very important operations are finding the intersection of two given sets and finding the union of two given sets.

11

Sets

The intersection of two sets A and B is the set of elements that belong to both A and B. To study the meaning of this definition let us consider the two sets

and

A= {2, 4, 6, 8, 10, 12, 14, 16, 18} B = {3, 6, 9, 12, 15, 18}.

We see that elements 6, 12, and 18 belong to both A and B . Therefore, by the definition, {6, 12, 18} is called the intersection of A and B. To write , "the intersection of A and B," we write A n B. We sometimes read A n B as "A intersection B," or " A cap B." In the example above An B= {6, 12 , 18}. In the Venn diagram at the right the closed curve labeled A encloses the points representing the members of A and the U closed curve labeled B encloses the points representing the members of B. The portion of the figure shaded in red is common to both regions and c~nsequently represents A n B. If we consider the sets

and

A B

= =

{3.l2, 4.12, 5.12, · · ·'J {3 4 5 .. ·}

' ' '

we see that there are no elements which belong to both A and B. In this case A n B = 0. When two sets A and B have no u elements in common we say that they are disjoint or mutually exclusive sets. In a Venn diagram two disjoint sets appear as two completely separate regions. In the figure at the right A and B are disjoint . An B=0 The union of two sets A and B is the set of elements which belong to A, to B, or to both A and B. To write, "the union of A and B," we write A U B. We sometimes read A U Bas "A union B," or "A cup B." The symbol U suggests the letter u of union and prevents us from confusing the symbols U and n. Let us consider the following example of the uni0n of two sets. Suppose that there are two ways for a competitor to qualify for the state track finals in the 100-yard dash: 1. Win first place in one of the six district meets 2. Run the distance in less than 10 seconds.

12

CHAPTER

1

Under this arrangement there can be two sets of qualifiers: 1. Set W consisting of the six district winners 2. Set T consisting of those who ran the distance in less than 10 seconds. Suppose that when the district results are compiled, members of sets W and Tare identified as follows (a, b, c, · · ·, i representing the qualifiers): W

= {a, b, d, e,f, i} and T = {b, c,f, g, h}.

Any competitor whose name appears in either of these lists is qualified to compete in the state finals. If Q is the set of qualifiers, then

Q = {a, b, c, d, e, f, g, h, i}. Thus Q is the union of Wand T, or Q =WU T. Observe that though qualifiers b and f appeared in both sets, they are listed only once. Let us consider the universe here to be the set of all finalists in the state track meet. A Venn diagram showing the relau tionship of sets W, T, and WU Tis at the right. In this case all of the area included within either the closed curve W or the closed curve T is shaded in red. The shaded area represents WU T, the union of Wand T. WUT

Another important operation is complementation. Unlike the operations producing intersection and union, complementation is an operation involving one set within a universe. If Xis a subset of a universal set U, the members of U that are not in . - U - - - - - - - - - - - , X form a set known as the complement of X. We shall denote the complement of X by X' . Thus if U = {l, 2, 3, 4, 5} and X = {2, 4}, then X' = {l, 3, 5}. In the Venn diagram at the right the portion X' shaded in red is the complement of X.

0

EXERCISES

1. If U= {l, 2, 3, 4, 5, 6, 7, 8, 9, 10}, A= {l, 2, 3, 4, 5, 6, 7}, and B= {2, 4, 6, 8}, find:

a. A

n

B

b. A

u

B

c. A I

d. B'

2. Make a Venn diagram co illustrate each part of Ex. 1. Shade the indicated area gray and write the name of each element in the proper location.

13

Sets

3. If U = {l, 2, 3, 4, · · ·}, C = {l , 3, 5, ···},and E = {2, 4, 6, 8, · ··}, find: a. en E b. CUE 4. If U= {l, 2, 3, 4, · · ·, 24}, E= {2, 4, 6, 8, · · ·, 24}, and T= {3, 6, 9, 12 , · · ·, 24}, find:

a. E

n

T

b. E

u

T

c. E'

5. Complete the following statement:

If M is the set of all people in Pleasantville who read the morning paper and E is the set of all people in Pleasantville who read the evening paper, then the intersection of M and E is the set of all people in Pleasantville who _ _ ? __.

6. If the members of sets A, B, and Care those shown in the Venn diagram at the right, state the members of:

a. A UB

f.B nC

b.AnB

g. (An B) n C

c.A UC

d. A nc e.BU C

n B) n C means the intersection n B and the set C.

Note . (A

set A

of the

h. (AU B) UC Note . (A U B) U C means the umon of the set A U B and the set C.

7. If in Ex. 6 U = {l, 2, 3, · · ·, 15}, find : a. The complement of A U B b. The complement of (A U B) U C c. The complement of (An B) n C

8. If A is any set whatsoever, find 0 n A. Find 0 U A. 9. Both boys and girls attend J efferson High School. Some, but not all, students in the school take swimming. Let U be the set of all boys and girls in the school, G the set of all girls in the school, B the set of all boys in the school, and S the set of all students in the school who t ake swimming. Draw Venn diagrams each showing one of the possible interrelationships among sets U, G, B, and S.

10. A, B , and Care subsets of universal set U. If A= {2, 4, 6, 8, 10, 12, 20}, B = {3, 6, 9, 12, 15}, C = {5, 10, 15 , 20}, and U is the set of all whole numbers, draw a Venn diagram showing the relation of U, A, B, and C. Write the names of the elements of each set in the proper areas in your diagram.

T~ ~ 9 3 I I 5 5 5 2 3

I:,

S

••••••••••••••• ::::, eee ooo ee ooo oc,

12 J

4ss oooo eee oo e ooo o 1a9eooooooooo e ooo D00000000000000 , 41 oo eeooooooooo o 2s a oooo•••• oo e ooo Js9ee oooooo ee oooo s10 P ee ooooooooo e o

•••••••••••••••

Were you to insert the card bearing John Jones's telephone number, 311-555-2368, into a card dialer telephone, the phone would be connected automatically with Mr. Jones's phone. The connection would be the result of the phone's reaction to holes punched in the card. These holes, appearing as black dots in the picture, signify Mr. Jones's number. To see that this is true we notice that: 1. Mr. Jones's number is written across the top of the card in such a way that each digit is above a column of circles which may later become punched holes. 2. Each of the first seven rows of circles is labeled with one of the seven sets of dig its {1, 2, 3} , {4, 5, 6} , {7, 8, 9}, {O} , {1, 4, 7 }, {2, 5, 8} , or {3, 6, 9} . 3. To indicate that the first digit in Mr. Jones's number is 3, a hole is punched in the column headed " 3 " and in each of the rows labeled with a set of digits containing 3. We notice that there are two of these, the first and the seventh. Similarly, to designate that the second digit in Mr. Jones's number is 1, a hole is punched in the first column headed "1" and in each of the rows labeled with sets containing 1. There are two of these, the first and the fifth . In like manner holes are punched for each of the other digits in the number. 4. Now we notice that each digit in the number is represented by two holes. This is because two complete sets of the digits 1 through 9 are used, one set in the first, second, and third rows and one set in the fifth, sixth, and seventh rows. Moreover, we notice that the sets are cleverly arranged so that the intersection of any set of three digits in the first, second, or third row with any set of three digits in the fifth, sixth, or seventh row is one and only one digit. For example, the intersection of {1, 2, 3} and {1, 4, 7} is {1} and the intersection of {1, 2, 3} and {3, 6, 9} is {3}. Mr. Jones's number did not contain 0, but had that digit appeared, it would have been rep resented by only one hole because O appears in only the fourth row.

15

Sets

Closure

If our numbers were only those in {l , 2, 3, 4}, we could not say, "When any two numbers in our set are added, their sum is also a member of our set ." We could find 1 1 = 2, 1 2 = 3, and 1 3 = 4, but we could not find 1 4 because there would be no number to express the sum. Would there be a number to exp ress 2 2? 2 3? 2 4? 3 3? Likewi se, we could not say, "When any two numbers in our set are multiplied, their product is also a member of our set ." Whi ch products would be members of our set? Which would not be? To express the thought that the sum of any two numbers in a set is always a member of the set, we say that the set is closed und er the operation of addition . Likewise , to express the thought that the product of any t wo numbers in a set is always a member of the set, we say th at the set is closed under the operation of multipli cation . When we consider the se t of even numbers, E = {O, 2, 4, 6, · · ·}, we see that , no matter which two numbers we choose to add, there is always a number in the set to express the su m. We say that set E is closed under the operation of addition . Likewise, we see that no matter which two numbers of E we choose to multiply, there is always a number in the set to express the product. Set E is closed under th e operation of multiplication . Since we cannot find sums for some pairs of numbers in {1, 2, 3, 4}, considered in the preceding paragraph , this set is not closed under addition. Likewise, the set is not closed und er multipli cation . If the result of performing a certain operation on any two elements in set Sis always an element in S , we say tha t set S is closed under that operation.

+

+

+

+

+

+

+

+

Closure and the Numbers of Arithmetic

We have learned from experience that the sum of any two of the numbers of arithmetic is a number of a rithm eti c; that is, the se t of numbers of arithm etic, A, is closed under addition. For example, we know tha t O + 7 = 7, 4.3 + 7.45 = 11.75 , f + ir = and so on. We have learned from ex peri ence tha t th e product of any two of the numbers of arithmetic is a number of arithmetic; that is, the se t of numbers of arithmetic is closed under multipli ca tion . For example, we know that 3 X 11 = 33, 4.8 x 7.9= 37.92, Js3 X ¾= J,jf, and so on. We know that so long as we ba r division by zero the set of the numbers of a rithmetic is closed under divi sion. Thus we know that ii= 3, 37.92 + 7.9 = 4 .8, Jl07 + ¾ = Js3, and so on. However, the set of the numbers of arithmetic is not closed under subtraction because the results of such subtra ctions as 11 - 13 and 48 - 73 are not numbers of arithmetic.

~t

16

CHAPTER

1

ORAL EXERCISES

1. What do we mean when we say that a se t of numbers is closed under the operation of addition? of multiplication? 2. The set of the numbers of arithmetic is closed under which of the following operations : addition, subtraction, multiplication, division ?

0

EXERCISES

1. Which of the following sets are closed under addition? If you find a set that is not closed, give one example to show that it is not closed .

a. The set of all counting numbers

b. The se t of all counting numbers less than 1,000,000 c. The set of all even counting numbers

d. The set of all odd counting numbers e. The se t of all counting numbers that are multiples of 3 (A multiple of 3 is a number which is the product of 3 and another whole number.)

f. The se t of all counting numbers which are multiples of 5 g. T he set of all counting numbers which end in 0

h.

The set of all counting numbers which are multiples of 7

2. Which of the sets of Ex . 1 are closed under multipli cation?

3. Which of the following sets are closed under division? a. The set of counting numbers

b. The set of whole numbers, {0, 1, 2, 3, · · ·} c. T he set of the numbers of arithmetic if zero is excluded as a divisor

4. Under which of the operations, addition, subtraction, multiplication, and division, are the following sets closed?

a. {1, 2, 3}

b. {l}

c. {O, l }

d. {O}

Equal and Unequal Expressions

In arithmetic you learned that there are many names for the same number. For example, each of the expressions V25, J,f, 27 - 22, 3 + 2, 500%, -1,l-f, and 22 + 1 is a name for the number with the common name 5. Thus we may write: V25 = 5, -V = 5, 27 - 22 = 5, 22 1 = 5, and so on. For the present we are using th e word "expression" to mean any symbol or combina tion of symbols which se rves to name a number.

+

17

Sets

When we write the equality sign(=) between two expressions, we are saying that they name the same number. A sentence asserting the equality of two expressions is an equation. Such a sentence may, of course, be either true or false. For example, 1 2 = 3 + 2, vis= -\0/ , and 500% = 2 2 + 1 are true statements. On the other hand, 7 = 5 + 1, 3 = -H-, and 11 = ½X 65 are false statements. To write, "is not equal to," we use the symbol ~. Thus we may write: 7 ~ 5 + 1, 3 ~ g, and 11 ~ ½X 65. When we write the inequality sign (~) between two expressions, we are saying they do not name the same number. A sentence asserting that two expressions are not equal is an inequality ._ Such a sentence may be either true or false. The inequality 7 ~ 5 + 1 is true but the inequality 7 ~ 5 + 2 is false. To determine whether such equations and inequalities are true , we depend upon our knowledge of arithmetic. For example, to determine whether 3 2 = 9 is true, recall that 3 2 (read "3 squared") means 3 X 3. Since we know that 3 X 3 is another name for 9, we know that the equation 3 2 = 9 is true. Similarly, to determine whether V 49 ~ 9 is true, we recall that V 49 (read "the square root of 49") indicates that we are to find the number which multiplied by itself gives 49. Since we know that 7 X 7 is the only product of two like numbers of arithmetic that is 49, we know that the equation V49 ~ 9 is true. V49 and 9 are unequal expressions; that is, they do not represent the same number.

°

0

EXERCISES

1. Which of the following statements are true and which false? a. 3 x 8 = 26 - 2

e. V 81

b. 4 2 = 16

f. 52 + 1

= 5+ 3 d. 1 + 1 = 40% of 5

g. 2¼ + 4¼ = 7 -

c. l/s6

=9 ~ 26

ib

h.11.4+ 2 ~ 5.7

2. Insert either the symbol = or the symbol ~ to make the following statements true.

a. 22 _? _ 4

e. 15 - 3 _? _

3 6 2

b. 15 _? _ 3 x 5

f.14 X5 _?_

3~0

c. 10 + 2 _? _ 4 + 1

d .. 8

X .7

_? _

1.; 2

g. 4 2 - 1 - ? - 17

18

CHAPTER

1

3. Read each of the following statements: If the statement is true write T on your paper. If the statement is false, change the expression on the right of the sign of equality or inequality to make the statement true. If more than one change is possible, you may take your choice. a.

tH= 2

2 -

b. 100% of c. .01 X 8

1

s ~ so

~

.08

d.

3½ X9½ = -9-N-

e. 65 + S

~

52 - 8

f. 60% of 11 = 3 X 2.2

Numerals

When we write {orange, apple, pear}, we have not put an actual orange, an actual apple, and an actual pear on this sheet of paper. We have merely written a list of words that tell other people that we are thinking of a real orange, a real apple, and a real pear. We have written the names of the kinds of fruit we are considering. Similarly, when we write {l, 2, 3, 4}, we have not put a set of actual numbers on paper. A number is an idea . It exists only in our minds. What we have written is a list of symbols, 1, 2, 3, and 4, which tell other people which numbers we are considering. These symbols are names of numbers just as the words one, two, three, and four are names of numbers. An expression which names a number is called a numeral. The symbols 1, ½, 13, 4, 7 X 6, 8, and 7 + 1 are numerals. Sometimes, when no confusion can result, we refer to the symbols ½, 27, 0, and other numerals as numbers even though we know that they are only symbols that represent numbers. Variables

In algebra we often use a symbol that names not a specific number but any member in a set of numbers. Consider the followCost Per Dozen ing example. for Pompons The Booster Club of Hall High School needs six dozen Size C paper pompons for a stunt it will present between halves of Friday's football game with McCann High Small $1.20 School. The planning committee has investigated costs Medium 1.80 and has prepared th e table at the right for the club Large 3.00 members. By writing c at the top of the list of costs the committee has really said, "You may think of the letter c as representing any one of the three numbers named by the numerals 1.20, 1.80, or 3.00."

Sets

19

In deciding which size of pompon to buy, the club members will probably reason somewhat as follows: Since we need six dozen pompons, the total cost of the pompons may be represented by 6 c. (6 c means 6 X c.) If c = 1.20, 6 c = 6 X 1.20 = 7.20; if c = 1.80, 6 c = 6 X 1.80 = 10.80; and if C = 3.00, 6 C = 6 X 3.00 = 18.00. When the club members reason, "The total cost of the pompons will be represented by 6 c," they will be using the letter c to hold a place for any one of the numerals (number names) 1.20, 1.80, or 3.00. They will then replace c by one of these numerals and proceed to find 6 times the number represented by the numeral. When we use a letter in this way we call it a variable. A variable is a symbol that holds a place for the name of any element in a given set of elements. We call the given set of elements the replacement set or domain of the variable. An element which a variable can represent is called a value of the variable. In the example above, c is a variable and {1.20, 1.80, 3.00} is the

replacement set or domain of the variable. The numbers represented by 1.20, 1.80, and 3.00 are the values that c may hav1.,. While in this example the domain of the variable is a set of numbers, there are occasions when the replacement set will not necessarily contain numbers. Would the results for the example above have been any different had the committee used the variable p (or x or y or any other letter) in the place of the letter c? Observe how Mr. Johnson, the mechanical drawing teacher, used the variable w in a problem that he gave to his freshman class. H e placed on the board the figure and notation shown below; then he said , " I want you

to draw a rectangle whose length is 2 inches more than its width. If w represents the number of inches in the width of the rectangle, w + 2 will represent the number of inches in the length of the rectangle. You may let w represent any one of the numbers listed in the column headed w . Be sure to find the length of your rectangle before you begin your drawing because otherwise you may not get your rectangle well spaced on the page."

20

CHAPTER

1

Chris Stubbins chose to let w = 3.0. He thought, "If w = 3.0, then 3.0 + 2 = 5.0. My rectangle should be 5.0 inches long." Jim Simmons chose to let w = 3.5. He reasoned, "If w = 3.5, then w 2 = 3.5 2 = S.S. My rectangle should be 5.5 inches long." Had you been in the class and had you chosen to let w = 4.0, what would have been the length of your rectangle? What would the length have been had you chosen to let w = 4.5? Suppose that you have been given the exercise: If S= {1, 2, 3, · · ·, 100}, find the square of each member of S. From arithmetic you know that 12 = 1 X 1 = 1, 22 = 2 X 2 = 4, 32 = 3 X 3 = 9, 4 2 = 4 X 4 = 16, · · ·, 100 2 = 100 X 100 = 10,000. You might restate the exercise so as to make use of a variable. For example, you might say: If n represents in turn each of the elements of set S when S = {1, 2, 3, · · ·, 100}, find n 2 • In this case you would think, "If n=l , n 2 =l2=1; if n=2, n 2 =2 2 =4; if n=3 , n 2 =3 2 = 9; • • • ; if n = 100, n 2 = 100 2 = 10,000. Have you noticed that an expression which contains a variable (for example, 6 c, w + 2, or n 2 ) does not represent a definite number until we have specified the number, or value, which the variable represents? Such expressions are merely patterns which tell us how to operate when we know the number represented by the variable. For example, 7 + a is a pattern which indicates that no matter what number is represented by a, that number is to be added to 7. You should pay particular attention to such patterns because they are very important in algebra. Sometimes we are more inter-

w

+2=

+

+

ested in the pattern than we are in the number that is obtained by using the pattern. ORAL EXERCISES

1. What is a variable? 2. What do we mean by the replacement set for a variable? 3. What word may be used in place of the words "replacement set"? 4. What do we mean by a "value" of the variable?

0

EXERCISES

1. If n is holding a place for any numeral in the set {3, S, 7, 9}, what numbers can be represented by n + 8?

l

2. If a is a variable whose replacement set is {2, 4, 6, 8, 10}, find the numbers that can be represented by :

a. a - 1

b. 3 a

c. a+ 2

d. a2

21

Sets

3. If the domain of the variable xis {0, 1, 2, 3, 4}, find the numbers that can be represented by: a. x 2 + 1

b. 9 x

4. If the replacement set ·for n is {½,

d. x+x+x

¼, ½}, what is the largest number that

may be represented by: a.

5.

½X n

b. n + 3

c. n +

'

¼ .

d. n 2

If n is a variable whose domain is the set of all counting numbers, write the set of numbers represented by:

b. ½X n

a. 2 X n

c. 1 X n

d. 0 X n

6. If n is a variable whose domain is the set of all whole numbers, W = {0,1,2,3,···},

a. For which value of n is it impossible to find~? n

b. For which values of n is it possible to find ~' if ~ is to be a whole number?

7. If y is a variable whose domain is {l , 2, 3, 4, S}, for which values of y is it impossible to find 3 - y, if 3 - y is to be a counting number?

8. If y is a variable whose replacement set is {0, 1, 2, 3, · · ·, 10}, for which values of y is it true that

a. 2 + y = 8?

b.

y is greater than 4?

c. 3 y

d.

= 4??

y is the square of a whole number?

9. If c represents the number of cents in the cost of one pencil, then 4 c represents the number of cents in the cost of 4 pencils.

a. If pencils at your school bookstore sell for 3 cents, 5 cents, and 10 cents, what will be the domain of cat your school bookstore?

b. If c = 3, 4 c = _? _ C.

If

C

= 5,

4C=

_? _

d. If c = 10, 4 c = _? _ e. Fill the blanks to make a true statement: At your bookstore four pencils of the same kind will cost _? _ cents, _? _ cents, or _? _ cents.

22

CHAPTER

1

10. During a special sale, the corner drug store is selling a well-known brand of soap at 5 cents per bar, but the advertisement states, "Limit: 6 bars per customer. " If c is a variable representing any possible number of bars of this soap that can be purchased by one customer, what is the domain of c? Write the set of possible amounts that one customer can spend for the soap that is on sale. ESSENTIALS

In this chapter we have given you some of the basic principles for working with sets of numbers. Since you will be working with such sets throughout your study of algebra, you should not leave this chapter until you are sure that you

1. Are able to define a set with both a word description and a listing of the members of the set. (Page 1.)

2. Are able to use a capital letter in naming a set .

(Page 2.)

3. Know the difference between a finite set and an infinite set and can define each clearly . (Page 4 .)

4. Know the meaning of "null set. "

(Page 4 .)

5. Understand the meaning of " universal set."

(Page 6 .)

6. Can read a Venn diagram and can make such a diagram to show the relationship between a set and its subset or subsets. (Page 9.)

7. Know what we mean when we say that a set is closed under a particular operation .

(Page 15 .)

8. Know what we mean by "expressions," "equations," and "inequalities. " (Page 16 .)

9. Understand what a variable is and can find the number represented by simple expressions containing variables.

(Page 18 .)

10. Understand the meaning of the following words and phrases, can spell them, and can use them properly. braces (Page 1.) closure (Page 15 .) complement (Page 12 .) disjoint (Page 11.) domain (Page 19.) equation (Page 17.) inequality (Page 17.) intersection (Page 11 .)

null set (Page 4.) op~ration (Page 10.) replacement set (Page 19.) subset (Page 5 .) union (Page 11 .) universal set (Page 6 .) variable (Page 19.) Venn diagram (Page 9 .)

23

Sets

11. Can express the following words and phrases by the symbols shown at their right, understand their meanings, and know how to use them. ENGLI SH PHRASE

SYMBOL

the set belongs to does not belong to the empty (null)set and so on is a subset of is not a subset of is a proper subset of intersection umon complement of X is equal to is not equal to

PAGE

{ } e

2

f.

2

0

4 4 5 5

C

g_ C

6 11 11 12

n u

X'

17 ~

17

CHAPTER REVIEW

1. By listing its members between braces, indicate each of the sets described below: a. The set N of all counting numbers less than 10

b. The set E of all even counting numbers c. The set of all three-digit counting numbers whose digits are alike

2. Write a word description of each of the sets indicated below: a. {1, 3, 5, 7, 9} b. {January, February, March, April} c. {3, 6, 9, 12, 15, 18} 3. If D is the set of whole numbers that are exactly divisible by 3, which of the following statements are true? a. 3 e D 4 . If A

b. 4 ¢ D

c. 12 e D

d. 100 e D

e. 99 e D

=

f. 1500 ¢ D

{l, 2, 3}, use the numerals 1, 2, and 3 to write five sets that equal A.

5. Indicate the following sets: a. The set of all even counting numbers between 5 and 7 b. The set of all odd counting numbers between 5 and 7

24

CHAPTER

1

c. The set of all even counting numbers d. The set of all even counting numbers less than 100

6. Given that E = {2, 4, 6, · · ·, 100}, if n represents any member of E, write the set of all numbers represented by n + 1.

7. Given that N = {1, 2, 3, · · ·, 100}, write the following subsets of N: a. The even numbers

b. The perfect square numbers c. The subset of N whose elements are represented by x + 2, where x e N

8. Given that M = {1, 2, 3, 4, 5, 6, 7, 8, 9}, which of the following statements are true?

a. If A= {l, 2, 3, 4, 5, 6, 7, 8}, A CM b. If B = {1, 2, 3, 4, 5, 6, 7, 8, 9}, B Cl. M c. If C = {9, 8, 7, 6, 5, 4, 3, 2, l}, CC M 9. If U = {1, 2, 3, 4, 5, 6, 7, 8, 9} and X = {2, 4, 6, 8}, find X'. Make a Venn diagram which shows X'.

10. Which of the following sets are finite and which infinite? a. The set of all the men in the navy

b. The set of all counting numbers c. The set of all counting numbers less than 100,000

11. Make a Venn diagram to represent the relationship of U, A, and C if U= {1, 2, 3, 4, 5, 6, 7, 8, 9}, A= {1, 3, 5, 7, 9}, and C= {l, 5, 9}.

12. If A= {0, 1, 2, 3, 4, 5} and B = {4, 5, 6, 7, 8, 9}, find: b.A nB

a.A UB

13. Letting U be the set of all whole numbers, make Venn diagrams for AU Band An B of Ex. 12 .

14. Which of the following sets are closed with respect to addition? A = {1, 2, 3}

B

=

N

{0, 1}

=

{1, 2, 3, · · ·}

15. Which of the sets of Ex. 14 are closed with respect to multiplication? 16. Insert either the symbol= or the symbol

r'c to make each of the following

statements true: a. 3 2 _? _ 6

d.

b. 3½ X 2 - ? - V17 c. .2 X .3 _? _ .6

f. 25% of 164 _? _ 40% of 102.5

f + ¼- ? -

2 X 142 e. 4 2 _? _ .1 X 160

25

Sets

17. Are the sets {1, 2, 3, 4} and {3, 4, 5, 6} equal? 18. If n is a variable whose domain is {2, 4, 6, 8}, write the set represented by each of the following:

+1

b. n

a. n 2

c. n is an odd number

19. If a is a variable whose domain N is {1, 2, 3, · · ·, 10}, for which values of a is it true that:

a. a 2 t N

b. 2 at N

c.

a+ 1 = 6

d. a is greater than 6

20. If n is a variable whose replacement set is {l, 2, 3}, write the numbers represented by:

a. 4n

d. n-1

b. n+6

CHAPTER TEST

1. Classify each of the following sets as finite or infinite: a. All counting numbers

b. All high school pupils in the United States c. All counting numbers less than 1000

d. All of the even counting numbers 2. Write all the possible subsets of {1, 2, 3}. 3. If S is the set of whole numbers that are exactly divisible by 7, which of the following statements are true? a. 21 e S

b. 40 t S

c. 1000 t S

d. 7 e S

e. 1 e S

4. Write each of the following sets: a. All two-digit counting numbers whose units digit is 5

b. All whole numbers greater than 70 but less than 71

5. Draw a Venn diagram to show that set B is a proper subset of set A which is a proper subset of universal set U. 6. If A= {1, 3, 5, 7} and B

=

{3, 5, 7, 9}, write: a. AU B

b. An B

7. If U = {l, 2, 3, · · ·, 50} and X = {l, 2, 3, · · ·, 25}, write X'. 8. If n is a variable having the domain {2, 4, 6} , find all of the numbers that can be represented by: a. n 2

b. 6n

c. n-2

d. 24 n

e. n+5

9. Is either of the following sets closed under addition? If so, which is closed? a. All counting numbers b. { 1, 2, 3}

Systematic Counting To answer the question, "How many students are in this room?" we count the students, making sure that each student is counted and none are counted twice. To do this we set up a one-to-one correspondence between the students and the counting numbers, 1, 2, 3, · · • . Our answer is the last counting number used. Other counting problems are more difficult. For example, unless we work systematically, it is difficult to find the number y z w X V of paths from A to Z along the lines of the dia70 35 1 15 5 gram at the right such that each path is eight units long. (Each of the small squares which Q T u R s compose the large square has its sides one unit 20 35 4 10 long.) Two such paths are shown-one as a p N 0 L M solid black line and the other as a dashed black 10 15 6 3 line. K H J F G Working systematically, we proceed as fol4 5 2 3 lows: Since each path must be eight units long, we can never move to the left or downward en D E A B C 1 route from A to Z . There is just one such path AB from A to B. Similarly, there is just one path to C, D, E, F, L, Q, or V. We indicate this by writing 1 near each of these letters in the diagram. There are two ways to move from A to G since we may move through B or through F. We write 2 near G. To move from A to H we must move either through G or through C. There are two paths from A to G and only one from G to H, namely GH. Therefore there are two paths from A to H, one via Band G and one via F and G. There is just one path from A to H via C. Hence the total number of paths from A to His 2 + 1 or 3. We write 3 near H. We now begin to see a systematic way to count paths. The number of fourunit paths from A to J is equal to the number of such paths through H plus the number of such paths through D. This is 3 + 1 or 4. Similarly, there are five paths from A to K. It is now clear that the numbers corresponding to M, R, and Ware 3, 4, and 5. The number of four-unit paths from A to N is now readily obtained. All such paths must pass through either Mor H. There are three such paths through M and three through H. Thus the number of four-unit paths from A to N is 3 + 3 or 6. We write 6 near N. We see that the number of paths to any intersection can be obtained by adding the number of paths leading to the intersection at the left of the given intersection and the number of paths leading to the intersection below the given intersection. Using this plan, we fill in the numbers corresponding to each lettered point in the diagram and find that the number of eight-unit paths from A to Z is 70. 26

Let us now apply what we have just discovered to answering the question, "In how many ways can we select four things from eight things, all different?" We represent the eight things by the eight units of any 8-unit path from A to Z. (I)

(2)

(3)

(4)

(5)

(6)

(7)

(8)

An eight-unit path from A to Z is determined as soon as we decide which four horizontal units are to be used. y V w X z If, for example, we select the units marked -10 l 15 5 35 with the red arrows, then there is only one choice of vertical units to complete the path Q s T R u shown by the solid black line. If we select the 4 10 -20 35 four horizontal units marked with the black p M N L 0 arrows, we have the path shown by the dashed 36 ----io - 15 black line. C F H J There is one and only one selection of four K -2 -3 4 5 horizontal units which corresponds to each path and one and only one path that corre£ A B C D sponds to each selection of four horizontal 1 l units. Thus there is a one-to-one correspondence between the paths and the selections, and we conclude that there must be seventy ways in which to choose four things from eight things, all different. Can you use the system just described to find the number of ways in which five things can be selected from ten things, all different? four things from twelve things, all different? A formula for determining the number of ways in which r things can be selected from n different things (r :::; n) is very important in mathematics. Our industrial and scientific counting problems are so complicated that many mechanical or electrical computing devices have been developed to solve them. In algebra you learn to do the kind of systematic thinking which will enable you to learn to use such high-speed computers.

27

In an experiment, Nike-Cajun rockets carrying grenades timed to explode at intervals during flight were fired to altitudes of 65 and 38 miles. From measurements made during the experiment, it is possible to determine atmospheric temperatures. Since winds affect t he time that the sounds take to reach the ground, wind direction and velocity at various altitudes can also be estimated by

the

grenade

technique.

The

grenade e x periments were the first in a series which Goddard Space Flight Center is conducting to learn more about winds and the temperature between altitudes of 25 and

75 miles. Rockets are used for many other scientific investigations.

For exam-

ple, the y are being used to determine whether we can predict the time when great clouds of dangerous cosmic ra y s will erupt from the sun. They are used to launch satellites and to serve as vehicles for space travel.

Chapter

2 The Number Line In this chapter you will Iearn to represent numbers by points on a line

In representing numbers by points on a line, we make drawings which help us to understand relationships between numbers. In this chapte r we shall represent some of the numbers that we study in arithmetic. The Numbers of Arithmetic

Let us call the set of numbers that we study in arithmetic the set of numbers of arithmetic and designate it by A. You know that in arithmetic we study such numbers as 0, 1, 29, ½, !-, and 0.5; consequently you know that 0 e A, 1 e A,½ e A, 0.5 e A, and so on. In arithmetic most of us also study the number 1r and such numbers as VS. Let us consider that these numbers belong to set A. From set A we can select many important subsets. In this chapter we shall consider the subset of whole numbers W

=

{0, 1, 2, 3, · · ·},

the subset of counting or natural numbers

C= {l , 2, 3, · · ·}, and F, the subset of rational numbers of arithmetic. In later chapters we shall consider some of the other numbers of arithmetic as well as some numbers that we do not study in arithmetic. The Whole Numbers and the Number Line

To picture the elements of set Won a line , we first draw a line and consider it as a set of points. To indicate that the line extends without end in either direction, we place a small arrowhead at each end of the portion we have drawn. 29

30

CHAPTER

2

Now we choose any point on the line and label it with the symbol 0. We refer to this point as the origin. Next we choose a second point to the right of the origin and label it with the symbol 1. Using the distance between these unit of measure

0

points as a unit of measure and beginning at the point labeled 1, we locate a series of points to the right of this point and equally spaced along the line. We think of this process as continuing without end even though we can 2

0

5

4

3

show only a few of the points. Now we label these points with successive whole numbers as shown in the figure above. There is a one-to-one correspondence between the whole numbers and the points we have located on the line. By a one-to-one correspondence between a set of numbers and a set of points we mean: 1. Each number in the set of numbers is associated with exactly one point

in the set of points, and 2. Each point in the set of points is associated with exactly one number in the set of numbers. The fact that we can establish a one-to-one correspondence between a set of numbers and a set of points is very important in our study of algebra. A line on which we associate points with numbers in a one-to-one correspondence as we did above is called a number line. The number associated with a point on the line is called the coordinate of the point and the point is called the graph of the number. Thus, the point labeled C on the number line below is the graph of the number 3, and the number 3 is the coordinate of the point labeled C. The point called the origin is the graph of the number zero, and zero is the coordinate of the point called the origin. 0

0

A

B

C

D

E

F

2

3

4

5

6

etc.

If we wish to show that a set of points continues without end, we may place etc. at the end of the line. At present we are considering only numbers whose graphs lie to the right of the origin, but later we shall consider numbers whose graphs lie to the left of the origin. Since the whole numbers are closed with respect to the operation of addition, the result of adding one to any whole number is also a whole number. We can say this in the language of sets as follows: If n t W, then (n + 1) t W.

31

The Number Line

The whole number obtained by adding one to a given whole number is called the successor of that whole number. Thus 1 is the successor of 0, 4 is the successor of 3, and 12,000,000,001 is the successor of 12,000,000,000. Clearly there is no largest whole number. If I name what I believe to be the largest whole number, you can add one to my number and get a larger whole number. Then I can add one to your number and get a still larger whole number. This process could go on forever because every whole number has a successor. The Natural Numbers and the Number Line

We have referred to {l, 2, 3, · · ·} as the set C of counting numbers. Obviously this is because the numbers 1, 2, 3, · · · are the numbers that we use in counting. Often this set of numbers is also called the set of natural numbers. The set of whole numbers {O, 1, 2, 3, · · ·} consists of zero and the set of natural numbers. It is the union of {O} and {l, 2, 3, · · ·}. As you study the number line above, you observe that when we located points corresponding to the whole numbers we also located points corresponding to the natural numbers. All that we have said about whole numbers as regards "successors" and "no largest member" applies to natural numbers.

0 0

0

R

s 2

T

u

V

3

4

5

EXERCISES etc.

1. Insert a word or phrase to make each, of the following groups of words become a true statement about the number line above . a. The point which is labeled R is the graph of __ ? __ .

b. The point which is labeled U has __ ? __ as its coordinate . c. The number 3 has the point labeled T as its __ ? _ _.

d. Another name for the point labeled O is __ ? __ . e. The coordinate of point Sis __ ? __ .

f. The point T is the __ ? __ of 3. g. The origin is the __ ? _ _ of zero . h. The number zero is the _ _ ? __ of the origin. i. Points labeled R, S, T, U, V , etc. are the graph of the set of __ ? __ numbers.

32

CHAPTER

2

j. Points labeled 0, R, S, T, U, V, etc. are the graph of the set of __ ? __ numbers. 2. What do we mean by a one-to-one correspondence between a set of numbers and a set of points on the number line? The Rational Numbers of Arithmeti c and the Number Line

Obviously there are infinitely many points on the number line between the points that we have already associated with the whole numbers. Let us consider point K which is equally distant from points C and D. It seems natural to think of this point as the graph of 3½. Let us consider the points

. • • • • • • •• • • . 0

A

0

B

C

K

D

ER S F

2

3

3½

4

5 5} 5} 6

R and S which divide the segment EF into three equal parts. It seems natural to think of R as the graph of S½ and of S as as the graph of SJ. If we divide the interval between the graphs of each pair of consecutive whole numbers into two equal parts, we obtain the graphs shown on the second line below; if we divide each of the intervals into three equal parts, we obtain the graphs shown in the third line below; and so on. Obviously this process can continue without end. 2

0 0

2

3

4

2

2

2

I

0

0

I

2

3

2

3

3

I

2

3

4

4

4

4

4

s

4

3

3

s

4

s

2 8

7

3

3

6

7

8

9

10

4

4

4

4

4

2

9

3

11

12

4

3

13

2 II

10

3

4

8

7

6

2

6

3

4

3

IS

14

4

12

T

3

4

4

16

4

Now locate on one line all of the points whose coordinates were shown on the above drawings. I

0 0

2

2

3

4

4

J.

2

2 I

4

I

2

3

3

4

4 3

3

1

4

7

4 4

3

s

4

4

s

3

8

4

6

2

2

2 6

s

II

10

9

4

4

4

6

7

8

3

3

3

2

12

4

9

3

3

8

7

2

2 13

2 IS

14

4

4 10

3

4 11

3

16

4

12

3

4

We have thus labeled a set of points which correspond to some of the elements of a set of numbers which are called the rational numbers of arith-

33

T he Number Line

metic. Let us denote this se t by F, as previously indica ted. Later we shall study some rational numbers which are not numbers of arithmetic, and we shall locate their graphs to the left of the origin on the number line. Before we define rational numbers, we should review what is meant by a fraction. Notice that the coordinate of the point on the number line corresponding to 3 has many different names, such as!, Ji . Each of these symbols is a fraction. By a f raction we mean a symbol which indicates the quotient obtained when one number, called the dividend, is divided by another number, called the divisor. In the frac tion "t the dividend is 5 and

t

the divisor is 7. In the fraction~ the numerator a represents the dividend and the denominator b represents the divisor. A rational number of arithmetic is a number which can be represented by a f raction whose numerator is a whole number and whose denominator is a n atural number. Thus if n

=

i and

a e W and b e C, then n e F . The fraction

"t

names a rational number because S e W and 7 e C. Any whole number is a rational number because any whole number can be expressed as a fraction with the denominator (divisor) 1. For example, 3 = f and 5 = f. Consequently 3 and 5 are rational numbers. The number O is special in that it can be represented not only by 5r, but also by %, 1 g0 ; in fac t by

t

Qwhere n is any natural number. The fraction V2 does not name a rational n s~ number because it cannot be represented by a fraction whose numerator is a whole number and whose denominator is a counting number. Since any whole number is a rational number, the set of whole numbers is a subset of the set of rational numbers. Since the se t of rational numbers has members (for example, ½, t and -i71 ) that are not whole numbers, the set l 7 of whole numbers is a proper subset of the se t of rational numbers; that is, W C F. The diagram at the right pictures set W as a proper subset of set F. The symbols in the diagram represent only a few members of each set. Any rational number can be rep resented in many ways, fo r example :

3= f = ½= ! = li=· .. 4½ = \3 = 266 = 399 = .. . .76= 17060 = H = H= ~86 = · · · 3.12s = ib68 = n68g = ...

34

CHAPTER

2

ORAL EXERCISES

1. What do we mean by a rational number of arithmetic?

2. When we write W = {O, 1, 2, 3, · · ·} the three dots mean that if we continue in this manner we will obtain a list of all the whole numbers. We can list the set of rational numbers in the same sense that we can list the set of whole numbers. Study the following array of numbers and observe the way the arrows are drawn. Can you describe a plan for listing all of the rational numbers of arithmetic? 0

I

I

T-T--2 2

/

2

2

T

!3 /

3

2

T 4

/

4

2

T

t/

5

5

/ / /

t-¼ t-- · 2

2

/

3/ 4 3

3

4

3

4

4

3

4

5

5

2

•

•

•

5

3

•

•

•

5

4

•

•

•

5

•

•

•

5

•

•

•

•

•

•

•

•

•

3

•

•

•

.

5

4 •

2 •

T

/

•

•

0

EXERCISES

1. Write two other names for each of the numbers indicated below: b. 1.4 d. 121 a. I c. 2½ 2. Name five rational numbers that are not whole numbers. 3. Draw a number line and on it locate the graphs of the numbers£, ¼, ¾,

¾, !, · · ·, 1i.

Be sure to label each point with the proper coordinate.

4. Below the labels£,¼,· · ·, 1i of the graphs you located in Ex. 3 write other labels which represent simpler names wherever possible.

5. On your paper draw a number line like that shown below. Place a small

..

r above the graph of each rational number, a small n above the graph of each natural number, and a small w above the graph of each whole ·

.. 0

• I

2

•

• 2

•

212

3

•

.

.

35

The Number Line

number. Allow yourself plenty of space because you will want to write more than one letter above some graphs.

6. Which of the following numbers will have the same graph on the number line: 1½, ¾, it ~t-\057} , ~~i, 1.125 , 2.250? 7. Draw a number line and on it locate the graphs of the successors of the whole numbers 0, 1, 2, 3, · · ·. Complete: The set. of the successors of the whole numbers is the set of _? _.

8. We have said that the set of natural numbers is a proper subset of the set of whole numbers. We have also said that the set of whole numbers is a proper subset of the set of rational numbers. Draw a Venn diagram showing the relationship of these sets. Use the set of rational numbers as the universal set.

9. Indicate which of the following statements are true and which false:

a. A numeral may represent a rational number even when the numeral is not a fraction. b. All whole numbers are rational numbers. c. All rational numbers are whole numbers. d. Every rational number corresponds to a point on the number line. e. Every point on the number line corresponds to a counting number. Betweenness on the Number Line

The set of rational numbers has an interesting property which is not shared by the whole numbers; namely, between any two members of the set there is a third member of the set. This is not true for the set of whole numbers because it is possible to choose two whole numbers which do not have a whole number between them. For example, there is no whole number between 5 and 6 or between 7 and 8. There is never a whole number between one whole number and its successor. Let us now consider the set of rational numbers. Let us choose two numbers such as ½and½ whose graphs are on the number line. We know that ½= ls and that ½= / 5 , so we know c M D that a number between½ and½ is a num3 4 5 15 ber between -fs and f 5 . Obviously l's (½) (~ 3 is such a number. We observe that -point M, the graph of 1\ , is between point C, the graph of ½, and point D, the graph of ½; in fact, M is the midpoint of the interval between points C and D.

t)

36

CHAPTER

2

We can use a similar procedure to find other numbers whi ch a re the coordinates of points on the number line between C and D . Since -! = 360 and ½ = ig, each of the three numbers 370 , lo, 390 is a rational number between ½ and -!- The points L , M, and N which are the graphs of these three numbers lie between C and D on the number line as shown below. This process of finding points, with rational coo rdinates, between two given ◄•--C• • -. . .~ - - ~- - ~- - ~- - ..points can continue indefinitely. There fo io lo fo ~ are infinitely many rational numbers be(½) (½) tween any two rational numbers as well as infinitely many points betwee n their graphs on the number line. This fact might lead us to believe that the graphs of the rational numbers " use up" all of the points on the right half of the number line. In other words, we might suppose that for every point on the number line there is a rational number which corresponds to it. Surprising as it may seem, this is not true. Later in this course we shall prove that there are points on the number line whose coordinates are not rational numbers.

0

EXERCISES

1. a. Find a rational number between ¼ and ½. b. On your paper draw the portion of the number line shown below. Then on it locate the graph of the number you found in pa rt a. 0

I

4

I

3

I

2

2

3

3

4

c. Find at least two other numbers between ¼ and

½.

d. Try to locate these on the number line you have already made. e. How many rational numbers are there between ¼ and ½? Could we ever find all of them? Explain.

2. If in Ex. 1 you had been asked to work with 1 ,oo6 ,ooo and

1 , 008,000

instead of¼ and½, would your answer to part e have been any different? 3. Would your answer to part e of Ex. 1 have been any different had you been asked to find numbers between .000,000,001 and .000 ,000,002?

4. Fill the blanks to make true statements: a. No matter how close together two rational numbers are, there are always __ ? __ many other rational numbers between them.

b. No matter how close together the graphs of two rational numbers are on the number line, there are always _ _ ? __ many other graphs between them.

37

T he Number Line

Order on the Number Line

If we start at any point in the number line and move to the right, we encounter points having ever increasing coordinates. Thus 5 is greater than 3 and the graph of 5 is to the right of the graph of 3. The number line below 0

2

3

4

5

6

does not show the graph of -8i°s3, but we know that if the line were extended far enough, the graph of f>1°s3 would be to the right of the graph of J/l because -8-fl is greater than -7l65 . Similarly, the graph of .101 would be to the right of the graph of .099 and the graph of 1,000,001 would be to the right of the graph of 1,000,000. We observe that the graph of the greater of two unequal numbers is to the right of the graph of the smaller. We can express the same idea by saying that the graph of the smaller of two numbers is to the left of the graph of the larger number. We shall use the symbol > for the phrase, "is greater than," and the symbol < for the phrase, "is less than." When we write 11 > 9 we have made a statement whose English translation is, "The number represented by 11 is greater than the number represented by 9." We may express the same idea by writing 9 < 11. We read this as, "Nine is less than eleven ." In true statements such as 11 > 9 and 9 < 11, the small end of the symbol > or < is directed toward the smaller number. To say that 11 > 9 and 9 < 11 have the same meaning we may write 11

> 9 +-+9
9 is equivalent to the statement 9 < 11. If a and b represent any two numbers of arithmetic, we may write a>b+-+b band b < a are both true or both false. If a= 5 and b = 11, the statements are both false; if a= 11 and b = 5, the statements are both true; if a= 7 and b = 7, the statements are both false. If A is the point on the number line whose coordinate is a and B is the point whose coordinate is b, then clearly one and only one of the following statements is true: A is to the left of B; A and B coincide; A is to the right of B. This leads us to the following agreement for a and b: If a and

38

CHAPTER

2

b are numbers of arithmetic, one and only one of the following statements is true: a < b, a= b, a > b. A

B

a

b

A

B

B

A

b

b

a

a

ab

When we say that we are making these statements for the numbers of arithmetic, we should remember that the graphs of the rational numbers of arithmetic do not account for all of the points to the right of the origin on the number line. We shall see that there are other points whose coordinates are such numbers of arithmetic as v'io and 1r which are not rational numbers.

0

EXERCISES

1. Arrange the numbers of each group below in order from smallest to largest.

a.¼,½,½,½ b. --lo,½, i2s, ¾, J

c. .701, .071, 7.01, .0071 d. 1.46, 1.64, 6.41, 6.14

2. Which of the numbers in each pair below will have its graph to the right of the graph of the other?

b. 3.1, 3.9

a. 8, 7

d.

c. 7f, 7H

½, ½

f.

e. 3 2 , 9

t, i41

3. In the figures below, the capital letters name points which are the graphs of numbers of arithmetic. In each case write the letter which corresponds to the point having the largest coordinate. p

M

B

A

a.

N

C.

0

0 E

b.

T

C

0

d.

S

R

A

0 ------------►

4. Insert the symbol , or = to make each of the following groups of symbols state a truth. a. 9 _ ?_ 1 d. 1f _? _ 1¼

b. ½_? _ ½

e. 100% of 7 _?_ -2l

c. 1¼ _?_ 1.25

f. 3.001 _?_ 3.010

g. 85 - 4 - ? - 9 2

h. 3 x 8½ i. 3

_? _ 2 X 12¾

X 0 _ ? _ 0 X 100

5. Indicate which of the following statements are true and which are false. a. 9 > 7 b. 4 < 2 c. 6 < 9

39

The Number Line

d.

f. 4 XO > 1

i94 < t+½

18 e. _02

= 90

g.

½+½+¼

to mean "is greater than." We use the symbol ,c. to mean "is not equal to," the symbol < to mean "is not less than," and the symbol :i> to mean "is not greater than." Such symbols as =, , ,c., are called comparison symbols because they are used in comparing numbers. Frequently in mathematics we use the symbol ~ or the symbol 2:'.: to mean "is equal to or greater than." Similarly, we frequently use either the symbol ~ or the symbol ::; to mean "is less than or equal to." We sometimes write 5 to mean "is less than or greater than." Using these symbols, we see that each statement in the column on the right is equivalent to the corresponding statement in the column on the left. If a < b, then either a= b or a > b. If a < b, then a 2:'.: b. If a ,c. b, then either a < b or a > b. If a ,c. b, then a 5 b. If a :i> b, then either a < b or a= b. If a :i> b, then a ::; b. All of the comparison symbols that we use in this course are given in the left column below and the meaning of the symbol is given in the middle column . We call the first six symbols simple comparison symbols and the rest compound comparison symbols, since their respective English translations are simple and compound sentences. Possible graphs are shown in the right column. Variables a and b represent any numbers of arithmetic. The point A is the graph of a and the point B is the graph of b. Comparison Symbol

1.a = b

English Translation

Graph

.

a is equal to b

B

.

A

•a

I

0

b

A and 8 are the same point

2.

a b

a is greater than b

4. a ~ b

a is not equal to b

A

B

0 a b A is to the left of 8 B

A

0 b a A is to the right of 8 A

B

or

.

1J

I ,., 0 ~

... A

C b 0 a A and 8 are different poinrs

43

T he Number Line Comparison Symbol

5. a b

a is not greater than b

.

A

B

I 0 a

b

•

B

I •

or

.

A

•a

• I 0

b

A is not to the right of 8

7. a :S b

.

a is ' less than b or a is equal to b

B

A

B

I 0 a

b

•

I ..

A

•a

or _,. I 0

.

b

A is not to the right of 8

8.a 2: b

a is greater than b or a is equal to b

...

...

B

A

I 0 b

a

•

B

or

...

.

A

•a

I

0

b

A is not to the left of 8

9. a '£ b

a is not less than b and a is not equal to b

...

I 0

B

A

•b •a •

A is to the right of 8

10. a

'l.

b

a is not greater than b and a is not equal to b

,. I 0

... B

A

•a

b

A is to the left of 8

11.a S b

a is greater than

12. a

1, b

b

a is not greater than b and a is not less than b

...

B A B A or I • I a b 0 a 0 b A and 8 are different points

...

• ••

a is less than b or

...

B

A

I 0

•a

•

.

b

A and 8 are the same point

44

CHAPTER

2

ORAL EXERCISES

In each of the following statements x is a variable which represents a number of arithmetic. Give the English translation of each of the statements.

1.

X


9

4. 5. 6.

9

3. X;?: 9

X

~ 9

X )>

9

X X 11. X = 9 12. X -;:6- 9

0

EXERCISES

1. Graph each of the following sets: a. {l , 3, 5, 7, 9}

b. {O, ½, 1, 1½, 2, 2½} c. The set of all natural numbers

d. The set of all natural numbers from 2 to 10, inclusive e. The set of natural numbers from 2 to 10 but not including 2 and 10

f. The set of all natural numbers greater than 4 g. The set of all whole numbers less than 7 h. The set of all rational numbers of arithmetic less than 5

i. The set of all rational numbers of arithmetic greater than 5 j. The set of all rational numbers of arithmetic greater than 4 but less than 7 2. If D

=

{l , 2, 3, 4, 5} and E

=

{2, 4, 6}, graph: a. D

n

b.DUE

E

3. If the universal set U = {l, 2, 3, 4, 5, 6, 7, 8, 9, 10} and X = {2, 4, 6, 8}, graph X'.

4. If E= {2, 4, 6, ···}and G= {l, 3, 5, ···}, graph EU G.

5. What can you say concerning the graph of En G for sets E and G of Ex. 4? 6. If a represents a number of arithmetic, will the graph of a for each of the statements below be to the right of, to the left of, or at the point corresponding to 3? a. a

>3

b. a < 3

c. a = 3

d. 3 < a

e. 3

>

a

f. 3 = a

7. In each of the following exercises x and y represent any two numbers of arithmetic. As you know, for these numbers exactly one of the statements x < y, x = y, or x > y is true. Which of these is the true statement for each of the following exercises?

45

The Number Line

a. b.

X

f:

y

c. x y, then x _?_ y.

y, then x _? _ y or x _? _ y.

h. If x b or a < b. c. a :l> b and a ~ b ++ a < b. a. a

e. a :l> b ++ a

< b.

10. If the variable x has the domain {l , 2, 3, 4, 5}, write the set represented by each of the following statements:

a. x < 4

>

1

e. X

~

C. X )>

3

f.

?: 3

b.

x

g. X '.S 4 h. x :S 2

d. x

b. y < 5

3

c. y ?: 6

e. y ~ 0

d. y :S 8

f. y l.

7

12. For each of the graphs below, x is a variable whose domain is the set of numbers which has been graphed. Describe the domain of x. a.

0

2

3

4

5

6

7

8

0

2

3

4

5

6

7

8

b.

c.

. • 0

EJES::"1==7

l

$

I

I

I

I

2

3

4

5

6

7

8

2

3

4

5

6

7

8

szsj

.

d.

e.

.

0 I

$

~

0

l

2

3

I

I

I

I

I

4

5

6

7

8

•

13. If x is a variable whose domain is the set of rational numbers of arithmetic, graph the set represented by each of the following statements: a.

X

b. x

3

C. X

'.S 6

d. x?: 2

e. x= 5

f. 2
1 and x < 6 h. x < 2 and x > 7

46

CHAPTER

2

ESSENTIALS

Before you leave Chapter 2 make sure that you

1. Recognize that the whole numbers, the natural numbers, and the rational numbers of arithmetic are subsets of the numbers of arithmetic.

(Page

2. Can set up a one-to-one correspondence between a. The whole numbers and certain points on the number line. b. The natural numbers and certain points on the number line.

(Page

29 .)

30.) 3 1.) c. The rational numbers of arithmetic and certain points on the number line. (Page 32 .) (Page

3. Understand what we mean by the phrase "rational numbers of arithmetic."

(Page

33 .)

4. Understand what we mean when we say that between any two rational numbers of arithmetic there is always another rational number of arithmetic. (Page 35.)

5. Recognize that of two unequal numbers the graph of the greater is always to the right of the graph of the lesser.

(Page 37 .)

6. Know what we mean by "equivalent statements." 7. Can graph a set of numbers of arithmetic.

(Page

(Page 37 .)

39.)

8. Know the proper symbol to use in expressing comparisons between numbers.

(Page 42 .)

9. Know the meaning of the following words and phrases and can correctly spell the words involved. comparison symbol (Page 42.) coordinate (Page 30.) equivalent statements (Page 37.) graph (Page 30.)

natural number (Page 29 .) origin (Page 30.) rational numbers (Page 33 .) whole numbers (Page 29 .) CHAPTER REVIEW

1. Insert a word or phrase to make each of the following groups of words a true statement about the graph below. 0

P Q CM

A

R

D

0

½ 2

t

3

4

5

a. The point P is the __ ? __ of the number 1.

b. The number f is the _ _? _ _ of the point labeled M. c. The point labeled O is called __ ? __ .

47

The Number Line

d. The arrowheads indicate that the line has no __ ? __ . e. Points that correspond to the whole numbers __ ? __ have been located. (Insert the names of the numbers.)

f. Points corresponding to the natural numbers __ ? __ have been located. (Insert the names of the numbers.)

g. The labeled points on the number line which correspond to rational numbers are __ ? __ .

2. Write three other names for each of the following numbers: a. j b. 2. 7 c. 11 3. For which of the following numbers have we defined successors: 0,

½,

1, 5000?

4. Insert a word to make a true statement: We know that the set of natural numbers is an infinite set because for each natural number there is a __ ? __ which is also a natural number.

5. Given S = {O, ½, ¾, 1, 1½, 1!, 2}, select from

S the following subsets:

a. The subset A of members of S that are rational numbers

b. The subset B of members of S that are whole numbers c. The subset C of members of S that are natural numbers

6. Find a number between

½and t,

7. How many rational numbers are there between O and 1? between 0 and ¼? between O and

1 0100 ?

8. Arrange the following numbers in order from smallest to largest: ¾, 1 1 1 1 3, s, 6, 2· 9. Which number in each of the pairs below will have its graph to the right of the graph of the other? a. 1.1, 1.01 b. !, ¾ c. 4t, 4 / 1

10. In each of the exercises below insert the symbol , or = to make a true statement: c. 0 X 5 _? _ 400 X 0

11. Insert a number of arithmetic in each blank below to make true statements: a. 2 < 4 +-+ _?_

>

b. 7 > 1 +-+

_? _

_? _


2

6. Draw a graph for each of the following sets: a. The set of all natural numbers less than 4

b. The set of all rational numbers greater than 3 and less than 7 c. The set represented by x ::; 5 when x is a variable whose domain is the set of all rational numbers of arithmetic

d. The set G n B when G = {1, 2, 3, 4} and B = {2, 4, 6, 8} e. The set EU D when E = {2, 4, 6} and D = {l, 3}

Systems of Numeration When we write the numeral 263 we are usually writing the name of an integer in the familiar decimal system of numeration in which ten is used as the base. Thus 263 = (2 X 100) + (6 X 10) + (3 X 1) The numerals 2, 6,- and 3 are called digits. The value assigned to a digit in any place is obtained by multiplying the digit by the value assigned to that place. In the decimal system, 3 in the units place has the value 3 X 1, 6 in the tens place has the value 6 X 10, and 2 in the hundreds place has the value 2 X 100 or 2 X 102. The value represented by the numeral 263 is the sum of the values accorded to the digits in their respective places. You may have had some experience with non-decimal systems, that is, systems which use bases other than ten. If so, you know that 263 could be a numeral written with base seven, base eight, base nine, etc. Let us consider (263)seven• To find the corresponding base ten number we may write (263)seven = (2 X 72)

+ (6 X 7) + (3 X

l) = (143)ten•

We read "(263)seven" as "two-six-three base seven." When 263 is a numeral i,n the base seven system we should not read it as "two hundred sixty-three" since such names are reserved for numbers in the decimal system. To express a number in the base seven system we need the seven symbols 0, 1, 2, 3, 4, 5, 6; to express it in the base five system we need the five symbols 0, 1, 2, 3, 4; and so on. We find it convenient to refer to these symbols as "digits" even though there are not ten of them except in the Lase ten system. In the base two system (the binary system) 0 and 1 are the only digits required. How many digits are required in- a base twelve system? If an integer has been expressed in a non-decimal place-value system of numeration, we can always find the corresponding decimal numeral as shown above. Thus (1432)6ve = (l X 5 3) + (4 X 5 2) + (3 X 5) + (2 X 1) = (242)ten (4t7)e1even = (4 X 112) + (t X 11) + (7 X l) = (601)ten• Note. t is the digit representing the number ten in the base eleven system. (1001 lOl)two = (1 X 2 6) + (0 X 2 5) + (0 X 2 4) + (l X 2 3) + (1 X 2 2) + (0 X 2) + (l X l) = (77)ten In order to add or multiply in a non-decimal system we must use facts other than those used for addition and multiplication in a decimal system. The addition and multiplication tables for the system using base three are shown below.

+ -- -- --0 0 1 2 2

0

l

1

1

-2 10

2

2

--

10

11 50

2

X

0

0

0

0

0

1

0

1

2

2

0

2

11

We use our tables to perform some additions and multiplications in base three: Add

(1) 11 21 102

Multiply (1)

(2) 210 111 102 1200 (2)

21 12 112 21 1022

221 102 1212 221 101012

The check below for the second multiplication shows how we check by converting our numbers to the decimal system. (22l)three = (2 X 3 2) + (2 X 3) (102)three = (1 X 3 2) (0 X 3) (25)ten X (1 Oten = (275)ten

+

+ (1

X 1) = (25)ten 1) = (1 Oten

+ (2 X

Does (101012)three = (275)ten? Yes, because (101012)three = (1 X 3 5) 3 4) (1 X 3 3) (0 X 3 2) (1 X 3) (2 X 1) = (275)ten•

+

+

+

+

+ (0 X

For Class Discussion 1. Why does it usually require more digits to express a given number using base three numerals than using base ten numerals? For example, (237)ten = (2221 0)three• 2. An integer is divisible by two if when it is expressed in the base three numeration system the sum of its digits is divisible by 2. An integer is divisible by three if when it is expressed in the base four numeration system the sum of its digits is divisible by 3. Similar statements can be made about integers expressed in base five, six, seven, etc. Make a generalization based on the above statements. 3. What is the last non-zero digit, reading from left to right, of the square of any integer, except 0, in the base three numeration system? 4. Show that the conclusion reached in Ex. 3 holds for the first ten positive integers in the base three numeration system. 5. Suppose that an integer is expressed in base ten numeration. Can you devise a method for finding its name in base three numeration?

51

Chapt"'

3 Expressions and Sentences In this chapter you will study ways in which to m ake algebraic expressions and sentences express mathematical thoughts

Algebraic Expressi ons

In English classes we learn that a simple sentence consists of two parts- a subj ect and a predi cate. The subject is a word or group of words that names a thing, person, place, or idea about which something is said. The predicate is a word or group of words involving a verb that tells something a bout the subj ect. In mathemati cs the thing named by the subject is usually a number. The predicate is usually a group of words such as " is equal to 4," "is greater than 100," or " is less than (x + y)." In mathematical sentences we often replace words with mathematical symbols. For example, whereas in English we might write,

"Six is greater than five," in mathemati cs we are more likely t o write, 6

> 5.

The subj ect of this sentence is "6" and the predi cate is " > S." In the sentence 4 + 2 > 5 the subj ect is "4 + 2." Clearly " 6" and " 4 + 2" are two of the many numerals (names) for the number six. A numerical expression is a numeral or an arrangement of numerals and symbols of operation (+, -, X, +, etc.) which names a number. If some (one or more) of the numerals in a numerical expression are replaced with variables, th e result is called an open expression. An algebraic ex pression is either a numerical expression or an open expression . For convenience, we shall often refer to algebraic expressions simply as expressions. 52

53 In the two columns below we have numerical expressions on the left and examples of open expressions on the right. These open expressions were obtained by replacing some of the numerals in the numerical expressions with variables. (Note that the dot is used to indicate multiplication, as in 3 · 7.) Numerical Expressions

Open Expressions

5

t

3.7

3x

92

n2

3+5 7

x+5 7

--

31 .7 +3

--

11(5 + 13)

11(5 + z)

6(3 + 14)

z(x + y)

a

y+3

Order of Operations

The numerals and symbols of operation in an algebraic expression must be arranged according to certain rules. You know some of these rules. For example, you know that the first arrangement below names the number whose common name is 15 and you know that the second arrangement names the number whose cc,mmon name is 16. As you probably surmise, the last two are sheer nonsense and are not acceptable as expressions.

5x3

7 245

29- 13

3

X

3

7

+

The expression 5 X 3 + 7 is not complete nonsense but it presents the problem: Which shall we do first, the multiplication or the addition? If we do the multiplication first, we have 15 7 = 22. If we do the addition first, we have 5 X 10 = 50. At this point we have not yet established a rule to apply in cases where there is a choice between multiplication and addition. But since we do not want to be continually faced with a confusing choice between two numbers in such cases, we adopt a rule. It is: When there is a choice between addition and multiplication, the multiplication will be done first. When we say that we "adopt" this rule, we mean that we agree to apply the rule from now on. Applying the rule to 5 X 3 + 7 we have 5 X 3 + 7 = 15 + 7 = 22 .

+

54

CHAPTER

3

To care for those situations in which we want to do the addition before the multiplication, we agree to enclose the indicated addition in parentheses. Thus to say, "Do the addition before the multiplication in 5 X 3 + 7," we write 5 X (3 + 7). The expression 5 X (3 + 7) means 5 X 10 or 50. Parentheses used in this way are called symbols of grouping. But the decision about addition and multiplication is only one of many similar difficulties we face. For example, we must decide which shall be done first in a division-subtraction combination or in a multiplication-division combination. To care for all of these situations, we agree to use the following rules : When more than one of the operations, addition, subtraction, multiplication, and division, appear in an expression and when there are no symbols of grouping, we shall first perform multiplication and division in the order in which they appear from left to right and then addition and subtraction in the order in which they appear. If there are indicated operations enclosed within parentheses or other symbols of grouping, we shall perform these operations first. When grouping symbols occur within grouping symbols, the innermost group of operations is to be performed first, then the remaining innermost group, etc.

Sometimes we use the vinculum, - -, to say: Do this operation first. For example, when we write 3 X 7 + 2 + 9 we mean: First find the sum of 7 and 2, then multiply the sum by 3, and finally add that product to 9. The bar over 7 + 2 is a vinculum. We always use a vinculum with the radical sign y. We place the vinculum over the numbers for which we are to find the root. For example, when we write V98 + 2 we mean: First find the number represented by 98 + 2 and then find the square root of that number. The horizontal bar which separates the numerator from the denominator 17 + 3 of a fraction may also serve as a grouping symbol. For example, - -5. (17 +3) Thus 17 + 3 = (17 + 3) = 20 = 4. Again 1s understood to mean · 5 5 5 5

~ = 33 =33= 3 8 + 3 (8 + 3) 11 . ORAL EXERCISES

1. What is a numerical expression? -an open expression? an algebraic expression? 2. In what order do we perform additions, subtractions, multiplications, and divisions when no symbols of grouping are involved? 3. What are two ways of indicating, "Do this operation first " ?

55

Expressions and Sentences

0

EXERCISES

Perform the operations indicated below to find the number named by each numerical expression . l.2x3+1

6. (7 + 8) X 2

2. 2 X (3 + 1)

7.12-(6+2)

3. 14+ 2 - 4 4. 12 - 3 X 1

8. 9+3+8+ 2

5. 8 X 4 + 2

9. !+! 10. 3 X 6 + 4 X 7

1l 7 + 5 .

2

12. 18 + 9 X 2

14. 8 + \ 9

15. 2½ + 6 X l½ 16. 18 - (4 + 2)

13. ~ 3+1

Find the number named by each algebraic expression below when a= 2, b = 3, and c = 5. 17.4a+l

20. 3 C - a

a

23. 2

18. 3 b- 4

21. 4 b - 2 c

19. 8 + 6 a

22. 4 X (a+ b)

+b

24. a+ b

25. 6 b X (c

+ 1)

26. (a+c) X (a+b)

C

Indicating Multiplication with Parentheses

There is another use for parentheses which we should consider at this time. You know that when we write two digits side by side, as in 53, we name a number. Fifty-three is another name for the number indicated by 53. When we enclose one or both of the digits in parentheses, we indicate multiplication. Thus (5)(3), 5(3), and (5)3 all indicate 5 X 3. 534 is the numeral for five hundred thirty-four; (5) (3) (4) means 5 X 3 X 4; (5) (34) means 5 X 34; (53)4 means 53 X 4, and so on. Instead of writing 3 X (2 + 5) we may write 3(2 + 5). Do you agree that 3(2 + 5) = 3(7) = 21? Evaluation

The process of finding a common name for a numerical expression is called evaluating the expression, and the common name is called the value of the expression. Example 1. Solution.

Evaluate the expression 7(5 + 3 X 2). 7(5 + 3 X 2) meam, 7 X (5 + 3 X 2). In this case the parentheses indicate that we are to find the value of 5 3 X 2 and then multiply the value by 7. Since there are no parentheses in 5 3 X 2, we do the multiplication 3 X 2 first. This gives 5 6 = 11. Now we substitute 11 for the numerical expression (5 3 X 2) and obtain 7(11) = 77. The value of 7(5+ 3 X 2) is 77.

+

+ +

+

56

CHAPTER

3

3(1l2+9) Example 2. E va1ua t e • 13 We first find the value of 11 2 + 9. Since 11 2 = 121 , w~ have 11 2 + 9 = 121 + 9 = 130. Substituting 130 for the numerical . 11?~ +9 gives . 3(130) = 390 = 30 . express10n

Solution.

13 13

Example 3. Evaluate 7(8 2 - 13).

7(8 2 - 13) = 7(64- 13) = 7(51) = 357.

Solution.

Example 4. Evaluate 3 ( 11; / 9) + 7(8 2 - 13).

From Examples 2 and 3 we have

Solution.

30 1: : 9) + 7(8 2 - 13) = 30 + 357 = 387. Example 5. Evaluate 5(7

+ 6).

5(7 + 6) = 5(13) = 65.

Solution.

Note. In this example there is an alternate solution. In spite of the fact that the parentheses indicate, " First find 7 6," we may proceed as follows: 5(7 + 6) = 5(7) + 5( 6) = 35 + 30 = 65. Alternate solutions are not always possible, but in this case we have a very special numerical expression whose value may be found in two ways. Later you will make a study of such expressions.

+

0

EXERCISES

1. Evaluate each of the following expressions. 7+3

a.10 b. 6(4 + 7) c. (4+5)2

g. (7 2 - 1)5 h. 8(2 + 2 x 3) i. / 0 (5

81

d. 24+ 3 e. ½(8

f. 2(6+5)+4(3-1)

+ 7) + 4

X 9

+ 1) + ¼(6 X

. 4(7 - 1)

J• 8- 5

2. Which of the following statements are true and which false? a.3x4+2=½(33-5)

b. 7 + 3 X 5 = 4 X 6 - 13 2

c.(7-1)(8+2)=54

3 - 3)

57

Expressions and Sentences

e. 18 + 9(6)

= 150 -

g. !(12 + 15) = 6 2

3(26)

h. 2(8 + 5) = 2(8) + 2(5)

f. 9 2 + 18 = 36

3. If a is a variable whose domain is {2, 4, 6, 8}, find the set of all numbers that can be represented by each of the following algebraic expressions.

4+a

a

a. 3 a + 1

c. - 2-

e.

b.2a-2

d.½(4+a)

f.J(a)

2+ a

g.

a

a

2+ 3

h.(a+l)(a+l)

Simple Numerical Sentences Throughout the book we have been combining numerical expressions to form simple numerical sentences. Each such sentence, like an English sentence, contains a verb form. Have you observed that in each numerical sentence we have used one of the comparison symbols, whose translation involves a verb? Examples are 52 + 12 2

1

>0

= 13 2

3 X 2 + 4 < 12

is read is read is read

52 + 12 2 is equal to 13 2 1 is greater than 0 3 X 2 + 4 is less than 12

Each sentence compares the number named on the left with the number named on the right. For example, in 3 X 2 + 4 < 12 the expression 3 X 2 + 4 names the number whose common name is 10. Thus the sentence states that the number represented by the numeral 10 is less than the number represented by the numeral 12. Henceforth, when no confusion can result, we shall not distinguish between a number and its name. For example, when we speak of "the number 10" we shall mean "the number whose name is 10." We are now ready to define a simple numerical sentence. Let us say: A simple numerical sentence is the result of writing a simple comparison symbol between two numerical expressions. For example, 2 + 3 = 5 is a simple numerical sentence because the simple comparison symbol = is written between the two numerical expressions "2 + 3" and "S." When we say that a sentence is true, we really mean that the idea expressed by the sentence is true, and when we say that a sentence is false, we really mean that the idea expressed by the sentence is false. We say that sentences are true (or false) because it is inconvenient to use the phrase, "the idea expressed by the sentence is true (or false)" each time we speak of the truth or falsity of the sentence. Since all of the numerical sentences that we consider are declarative sentences, each has the property of being either true or false.

58

CHAPTER

3

ORAL EXERCISES

1. What do we mean when we say that a sentence is true? 2. What are the meanings of the six comparison symbols that appear in simple numerical sentences? 3. What is the distinction between "number" and "numeral"?

0

EXERCISES

1. Insert one of the comparison symbols >, 2.109 3+7

c.q..:a=4½ 8

d. -1-j + 3

< -1/

e. 2(3 + t) ,c 2(3) + 2(t)

f. 7(2 + -h) < 14/2

g. 5 + 2 X 4 = 4 7 h.3x4+3=9+2x3

. 2+-l 1 1

1.

-3

>2

1+3 J•• m_.,,.1 4 -.. 2

59

Expressions and Sentences

4. From the numerical expressions in the column on the right choose one that can be paired with each numerical phrase in the column on the left to form a true numerical sentence with the symbol=. Write the sentence. a. 3(5

+ 2)

19 ½(450- 91)

b. 4 + (8) (7) - 1 c. (9

+ 6) + (2 + 1)

d. (3

X -~)

10 2 + 10+ 2

7

+1

2(18-;-

e. 56 + 2 X 4

3)- 9

10(20 - 1) 3 X 12 - 7 2 2 X 72 - 2 (5X2)(3X7) 4 2 + / 0 (50) ½(450) - 91

f. (11 + 2)2 - 7 g. ½+ (6+8)

1.9

5. Write each of the following sentences in such a way that each numerical expression is replaced by a single numeral that names the same number as the expression.

a. 2(5 + 8) > 2(5) + 8

b. 2

t

3

= }+ 1

c. 4+6(3+5)


18 - o e. ½+ ¼> !

d. 16 +

f. 2(8 + 1) + 3(8 + 2) > 5(8 + 1) 5

5

5

g. 3 +4 < 3 +4

h. ¾+ lo < ¼+ t i. 5 + 0(8 + 2) < 5 + (8 + 2) j. (4+7)(3+2)=(3+2)(4+7)

k. (18 + 3)2 + 1 -;c 18 + [3(2 + 1)] l.f+¼X2=1+2X3

m. 54 + 9 X 3

n.

(½ + ¼)2

-;c

< 27 + 3 X 3

¼+ ½

60

CHAPTER

3

Open Sentences

There are many ways in which we may change the false numerical sentence

3(9 + 2) = 27 to make it true. For example, we may replace 27 by 33 to obtain the true sentence 3(9 + 2) = 33. Let us suppose, however, that we are interested in replacing 9, and let us also suppose that the only replacements we want to consider are the members of the set S = {l, 2, 3, 4, 5, 6, 7, 8, 9, 10}. Let us draw a circle around the numeral 9 and then remove the 9 so that we have 3(0 + 2) = 27. Now let us line up the numerals representing the members of Sand place them one by one in the opening left by the removal of 9 as if we were feeding shells into a gun having the circle as the firing chamber. Every time a numeral (shell), 1, 2, 3, 4, 5, 6, 7, 8, 9, or 10, enters the firing chamber, the expression 3(0 + 2) indicates a number (we say the gun fires a certain number). Our question is: When, if ever, will our gun be "on target"; that is, when will 3(0 + 2) name the number 27? When 1 enters the chamber, 3((D + 2) = 3(3) = 9. In this case we may say that the shell with the lightest powder charge travels only to 9. When 2 enters the chamber, 3((1) + 2) = 3(4) = 12, and we may say that the shell travels only to 12. Eventually 7 reaches the chamber. Now 3((Z) + 2) = 3(9) = 27 and we may say that the shell has hit target. Of all of the numbers in S, only 7 causes 3(0 + 2) to have the value 27.

30

The fact that the circle could be filled with a numeral naming any member of set S suggests that instead of the circle we might have used a variable having S as its replacement set. If we let x be this variable, we may write

3(x+2)=27;

XES

61

Expressions and Sentences

When x = 7, we have 3(7 + 2) = 27, a true sentence; but when x = 1, x = 2, or x = any other member of S except 7, 3(x 2) = 27 is a false sentence. Since a numerical sentence contains only numerical expressions and since xis not a numerical expression, we must say that 3(x + 2) = 27 is not a numerical sentence. It becomes a numerical sentence only when we replace x by a numeral. The fact that x represents an opening to be filled with a numeral suggests that we call the sentence 3(x 2) = 27 an open sentence. A simple open sentence is an arrangement of variables or numerals and variables which becomes a simple numerical sentence when its variables are replaced by numerals. Examples of open sentences are

+

+

+

x y > 23; x E F, y E F 3x-11-;ex+S;xEA a+7b 10 is an inequality.

+

Solution Sets Any value of the variable, that is, any member of the replacement set, which changes an open sentence into a numerical sentence which is true is called a solution of the open sentence. The solution set (or truth set) of an open sentence contains all numbers which are solutions of the open sentence and contains no other numbers. Example

Solution.

1. Find the truth set (solution set) of the open sentence 3 x when x EC.

= 51

Remembering that we have decided to designate the set of natural (counting) numbers by C, we recognize that we are to find all natural numbers, x , such that 3 times each of them yields 51. We see that 17 is such a number because when we substitute 17 for x in 3 x = 51, we have 3(17) = 51, a true sentence. Moreover, 17 is the only member of the solution set because when any other member of C is substituted for x, the resulting numerical sentence is false. The truth set is {17}.

62

CHAPTER

Example 2.

3

Find the solution set for the equation

3x=41;

XEC.

Solution.

We are to find all natural numbers, x, such that 3 times each of them yields 41. We see that there is no such number. Therefore, the equation 3 x= 41, when x t C, has the null set as its solution set; that is, the equation has no solution.

Example 3.

Find the solution set for the equation

3x=41;

XEF.

Solution.

Remember that F designates the set of rational numbers of arithmetic. Here we are studying the same equation that we considered in Example 2, but this time we are to find all rational numbers of arithmetic such that 3 times each of them yields 41. V is such a number because when we replace x by the numeral ¥, we have 3(\1) = 41, a true sentence. Moreover,¥ is the only solution. The solution set is {¥}.

Example 4.

Find the truth set of x

Solution.

We are to find all the whole numbers which are less than 9. We see that when xis replaced by any member of {O, 1, 2, 3, 4, 5, 6, 7, 8}, x < 9 becomes a true numerical sentence. Consequently, the solution set is {O, 1, 2, 3, 4, 5, 6, 7, 8}.

Example 5.

Find the truth set of the inequality x

Solution.

This time we are to find all the rational numbers of arithmetic which are less than 9. It is not convenient to list all of these; however it is easy to graph the truth set, which consists of a gray line beginning at the origin and extending to a circle at the point corresponding to 9 on the number line as shown below.

s

< 9;

x

E

4

W

6

7

< 9;

x

E

F.

11

.

ORAL EXERCISES

1. What is an open sentence? an equation? an inequality? 2. May we say that an open sentence is either true or false? May we say that a numerical sentence is either true or false?

3. What do we mean by a solution set for an open sentence? What do we mean by a solution for an open sentence?

4. What is another name for solution set?

63

Expressions and Sentences

0

EXERCISES

1. Which of the following sentences are open sentences and which numerical sentences?

a. 2 + 7

< 42 b. x + 8 = 13

3X

C.

=

15

e. f.

d. ½

17

2. Which of the following sentences are true , which false, and which cannot be listed as either true or false?

a. 7 X = 12 b. 7(9)

e. 7(x

= 63

+ 3) = 28

f. a+ 9 = 15- 4

c. 4 + (8 - 6)

=

g. 2(3 + 6 X 4) = 72

18

d. 3 + 5 = 3 - 1

h. 1 =5

4

3

3. Which of the following open sentences are true when x

= 4? f

a. 2 x = 8 b. x + 7 = 11

d 25 _ 61. • 4

c. x+ x= 19

e. x- 2 = 4

x

x 4 . -7 =-7

4. In each of the following open sentences, determine whether the sentence is true if the variables have the values shown at the right.

a. 8 X = 96;

X

f. X +

= 12

X

< 9;

X

= 4

b.17+ y =27; y =lO

g.2t+4=14; t=5

c. 5 + y < 6 ; y = 3 d. r + 7 ~ 10; r = 4

h. 3(b - 1) = 9; b = 4 i.a(6 +4)=7;

a+2 e.--=3·

3

a=7 '

a=½

j. ¾(s) + 5 < 15; s = 12

5. For which of the members of the set shown at the right of each open sentence below does the sentence become a true numerical sentence? For which members does it become a false numerical sentence? a. 2

+ a= 7; a

b.

t = 18; t

9

E

E

{l , 3, 5, 7}

{2, 4, 6}

c. b + b = 10; b E {1, 2, 3, 4, 5}

d. b + b = 2 b; b E { 1, 2, 3, 4, 5} e. x(3 - x)

= O; x E {O, 1, 2, 3}

f. 3x+2x=5;

X E

{O, 1, 2, 3}

< 6; r E {l, 2, 3, 4, 5, 6, 7, 8} h. r > 6; r E {l , 2, 3, 4, 5, 6, 7, 8} g. r

i. x+2

~ 6; X E {O, 1, 2, 3, 4 , 5}

·j. 4 Y = ½; Y E Uo, ½, ¼, ¼, ½} k. x ::I> 5; x E {2, 3, 4 , 5, 6, 7} l. r - 7

14

2X

i. 5 X + 1 = 16 j. 2 X + 3 = 7

=3

g.

x+ ½= 4½

h.

x

x

e.

h. 6 x
0

x(1) =

, X

g.. 2(. 6 + .3) = 3 X

J·

½ .003

1

3 = 36

The Set Builder

We can see that the open sentence x < 9 when x EC has the truth set S = {l, 2, 3, 4, 5, 6, 7, 8}. Do you agree that S is a subset of the replacement set C? If we were required to list the members of C for which x < 9 is false, we would write: S' = {9, 10, 11, · · ·}. We observe that S' is the complement of S when we consider C as the universal set. This example illustrates a very important conclusion : A ny open ·sentence acts as a sorter f or the members in the replacement set. It sorts the members of the replacement set into two subsets, one a subset of members for which the sentence is true and one a subset of members for which the sentence is false. The subset for which the sentence is true is the truth set for the sentence, and the subset for which the sentence is false is the complement of the truth set when the replacement set is considered the universal set . For the example above we might say: T he truth set fo r x < 9 when x EC consists of the natural numbers x such that x < 9 is true. We may designate this set by writing {x such that x

< 9 and x EC}.

We read this as, "The set of all :t's such that xis less than 9 and x represents a natural number." In place of the words "such that " we often draw a vertical bar, /. Then we may write the set as {x/x 3 g. t 2 = 16

2

e. t < 6

h.

i. t > 3 j. t 20} i, {x 13 X = 4} j. {x I ½(x) = 8}

4. State which of the following sentences are true and which are false if Xt:W.

a. {x I x

+7=

10} = {3}

b. {x / x > 7} = {8, 9 , 10, · · ·} c. {x I 3 x = 16} = 0 d.

{x I~2 < 8} = {l ' 2' 3 ' · · ·' 16}

Ix+ 2 x = 3 x} = W f. {x Ix ½(6 - 5)

e. 3.5 + 8.4 = 13.9 and 16 = ½(32)

f.

½< J and !

g .. 1(.1)

=

~

¾

.01 and .2

+ .3 = .5

h. .3(1 + 4) = 1.5 and}= .1 2. Find the values of n which will make each of the following compound sentences true if n has as its domain the set of natural numbers. a. 3 + 4

b. !(6)

=

7 and 4 + n

= 4 and~= 7

= 12

c. 4 n

+ 3.2 = 3.3 6 and 8½ + 2f = 2l

=

d. n >

17 and .1

70

CHAPTER

3

3. If the replacement set for x is the set of whole numbers, find the truth set for each of the following compound sentences.

a. x > 8 and x < 6

g. x + 4 = 7 and x + 1 = 5

b. x < 8 and x > 6

h . X > 3 and X > 1 2 3

> 8 and x = 6 d. x > 5 and x > 1 e. x = 4 and x < 5 f. x + 2 = 15 and 2 x = 26 c. x

i. 4(x + 1) = 8 and x =} j. 4(5) + x = 25 and x < 7

k. x + } = J and x < 1 l. ½(x

+ 3) = 6 and x

>

3

0

EXERCISES

4. If S1 = {3, 4, 5, · · ·} and S2 = {l, 2, 3, 4} , write a compound sentence with the connective and that has the truth set S1 n S2 . Use the variable n and let the domain of n be the set of natural numbers.

5. If A= {5} and B = {5, 6, 7, · ··},write a compound sentence with the

connective and that has the solution set A n B. Use the variable x and let the replacement set of x be the set of whole numbers.

6. If S1 is the set of all rational numbers of arithmetic which are greater than 6 and S2 is the set of all rational numbers of arithmetic which are less than 14, write the compound sentence with the connective and which has the truth set S1 n S2.

7. Write a simple sentence equivalent to each of the following compound sentences. Remember that if a and bare two numbers of arithmetic, one of the statements a < b, a= b, a > b must be true.

a. x > 7 and x 1 + 22 or 7 < 8 + 5 g. 2(7 + 4) = 14 + 8 or (3 + 4)5 = 5(3 + 4) h. 475(.1) = 47 .5 or 843(0) = 843

EXERCISES

72

CHAPTER

3

2. Find the truth sets for the following sentences. C is the set of natural numbers, W the set of whole numbers, and F the set of rational numbers of arithmetic. a. x

~

4; x EC

b. a = 5 or a < 7; a E W c. x < 3 or x > 6; x E W d. m > 1 or m > .24; m E F e. m + 6 > o+ m or m + 6 = 6 + m; m E C f. 3 x::; 6;

x

E

F

X

g. x < 9(3) or 3 < 3; x EC

h. 4 x > 0 or x = 9 :_ 3 2 ; i. 2 x + 1

~ 6; x

E

x EW

C

i. 2 x + 1 > 6 or 2 x + 1 < 6; x E C 3. Remember that set A designates the set of numbers of arithmetic. Recalling that if a EA and b EA , exactly one of the statements a < b, a= b, a > b is true, write a compound sentence that expresses the same thought as each of the simple sentences below.

a. x

~

5; x EA

b. x 6; x EA

e. 3 x

d. x = 10; x EA

f. x + 1 3.5

To graph the compound sentence x reason as follows: The graph of x




when x E F, we

2

3

4

5

6

7

2

3

4

5

6

7

3.5 is

0

Since the truth set of a compound sentence having its two clauses joined by the connective "and" is the intersection of the truth sets of its clauses, the graph of the truth set of the compound sentence will be the intersection of the graphs of its clauses. Therefore the graph of "x < 7 and x > 3.5" is 2

0

3

5

4

We often write the conjunctive sentence "x compact form "a < x < b." Note that if a sentence "x < b and x > a" is the null set.

6

7

< b and x > a" in the more < b the solution set of the

0

EXERCISES

1. Graph the truth set of each of the following compound sentences. In each case x is in the set of all rational numbers of arithmetic.

f. x + 2 = 3 and x + 4 = 5 g. x + 5 = 7 or x + 6 = 7 h. 2 x ~ 7 i. x > 1 + 3 and 4 + x = 9 j. 2 x + 1 > 5 or 3 x + 2 < 17

a. x = 3 or x= 4

b. x = 3 and x = 4 c.x

d.

~

6

x _::; 6

e. x = 6 and x

>6

2. Write a compound sentence that expresses the same thought as each of the sentences below when x EA. Graph the compound sentence.

a. x ,;c 3

c. x ::I> 4

e. 3 x ,;c 9

b. x
o} b. {x I x ::I> -lo} c. {x Ix= 1 or x ,;c 1}

E

g. x + 2 < 5 h. x + 5 ::I> 8

::I> 6

F.

d. {x I 3 x

~ l}

X e. { x I 16

< 4 and x > 4

1

f. {x I x ,;c 3 or x < 4}

1}

74

CHAPTER

Q

3

EXERCISES

4. Draw the graph of the truth set of each of the following compound sentences. In each, x E F.

g. x + 3

a. x ~ 7 and x > 2 b. x < 1 or x 2".: 5 c. x

~

4 or x

h.

>6

d. x + 3

~ 8 and x

e. x + 2

~

>3

5 and x + 2 2".: 5

f. x + 2 ~ 5 or x + 2 2".: 5

i

~

¥- 3 and 7 x

• X

1.

9 and x

2 ¥- 3 or

j. 4 x

>

+ 5 2".: 8 28

7 x > 28

¥- 20 and 2 x ¥- 6 ESSENTIALS

Before you leave Chapter 3 make sure that you

1. Know what we mean by an "algebraic expression."

(Page 52 .)

2. Know the order in which operations should be performed. (Page 54.) 3. Can use parentheses to indicate multiplication. (Page 55 .)

4. Can evaluate numerical expressions.

(Page

55.)

5. Can write simple numerical sentences and can determine whether such sentences are true. (Page 57.)

6. Know the meaning of "open sentence," "equation," and "inequality." (Page

61.)

7. Know the difference between a "solution" and a "solution set." (Page 6 1.) 8. Can find the set of numbers for which a given open sentence becomes a true numerical sentence. (Page 6 1.)

9. Are familiar with the set-builder notation. (Page 64 .) 10. Can graph the truth set of a simple sentence. (Page 66 .) 11. Understand the difference in meaning of compound sentences with the connective and and those with the connective or. (Pages 68- 71.)

12. Can graph compound sentences having either the connective and or the connective or. (Page 72 .)

13. Can spell the following words and can use them correctly. compound sentence (Page 68 .) conjunctive sentence (Page 68 .) connectives (Page 68 .) disjunctive sentence (Page 70 .) equation (Page 6 1.) evaluate (Page 55.)

expression (Page 52 .) inequality (Page 61 .) numerical sentence (Page 57.) parentheses (Page 54.) solution (Page 61.) vinculum (Page 54.)

75

Expressions and Sentences

CHAPTER REVIEW

l. What is a numerical expression? 2. Explain the order in which operations should be performed when no parentheses or other symbols of grouping are used.

3. Explain how the presence of parentheses or other symbols of grouping affects the order in which operations should be performed. 4. Evaluate:

a. 7(5 + 4)

c. 8(4) + (9 - 3)

b. 8(6) + 2(5)

d 7+8 . 3

2 e. 12 + 4

f. 4(3 2 + 2)

5. What do we mean by an "open expression" ? 6. If xis a variable whose replacement set is {O, 1, 2, 3}, find the set of numbers that can be represented by each of the following phrases:

a, 2 X + 1

d. x(x+ 2)

c.3(x+2)

7. Replace each _? _ with one of the symbols > , 18 < 34

c. 4 a= 32 d 8 + 14 _, 12 . 11 .,,... 6

10. Which of the sentences in Ex. 9 are equations and which inequalities? 11. Which of the sentences of Ex. 9 can be labeled "true," which "false," and which neither "true" nor "false"?

12. Express each of the following sets in another form. a. {x / x

+7=

15 and x E W}

b. {x Ix < 6 and x E W}

c. {r / 7 r = 11 and r E W}

i !and s W}

d. {s I =

E

76

CHAPTER

3

13. For which members of the replacement set shown at the right of each open sentence below does the sentence become a numerical sentence that is true?

a. 3 +a= 11;

a E {2, 4, 6, 8, 10}

b. 4 x < 20;

E

{0, 1, 2, 3, 4, 5, 6} 1. {1 1 1 l} C. 3 Y < 3, YE 4, 3, 2, d. r > .2; r E {.01, .02, 1.0, .l} x

e. (a+ l)(a-1)

½x -

f.

= a 2 - l;

3 > 0; x

E

a

E

{l, 2, 3, 4}

{0, 1, 2, 3}

14. Find the truth set for each of the following sentences. In each case X

EF.

> 32 + 22

a. x+½=-1/

d.

X

.4+ .3=x C. 75 + 5 + X = 11

e.

x+ 1 = 1 +x

b.

8

h. 3 x+ 5 = 11 i. 6 + 2 X = 24

f. x(.4 + .5) = 1.8 g. (.2)(.2)

>X

j. X + 1 > 100

15. Make a graph of the truth set for each of the sentences below. f. X < 6; XE W a. X > 4; XE W b. 2 x= 5; x E W g. x + 1 = 4; x E C c. 3 x = 14; x E F h. x + 1 > 4; x E C

d.

x

> 6;

x

E

i. x + 2 = 2 + x; x E F j. x(3 - x) = O; x E {O, 1, 2, 3}

F

e. x < 5; x EC

16. Make a graph for each of the following sets. a. {x I x

< 5; b. {x I x < 1;

x EC} x

E

C}

< 2; d. {x I x > 5; c. {x Ix

x

E

F}

x EF}

17. Complete: A compound sentence having two clauses and having the clauses joined by "and" is true only if __ ? _ _.

18. Complete: A compound sentence having two clauses and having the clauses joined by "or" is true if _ _? __.

19. Which of the following compound sentences are true and which false? a. 3.2 +4.2

= 7.4 and 3.2 < 3.02

b. ½(4 + 5) =!or 6 > ½(8 + 6) c. d.

½> ½and .8 > .08 i + f ~ t or 10 ~ 8 + 2

e. 6+ 1


9 and 4 + 5 = 9 g. 32 + 22 = 52 and 6 > 1 h. 8.6 + 8.6 = 2(8.6) or 3 < 7

77

Expressions and Sentences

20. Graph the solution sets for each of the following compound sentences. In each case x

E

F.

a. x > 2 and x < 7

f. x = 6 or x > 8

b. x = 3 and x > 3 c.x :s; 4 d. x > 5 or x < 2

g. x

>

3. Each sentence below is true for which members of the set shown at its right? a.x(3-x)=2; XE{0,1,2,3}

b. x

:t> 3; x

{l, 3, 5, 7, 9}

E

4. Find the truth set of each of the following open sentences. In each, X

EF.

c. 2 x + 1 = 7 and x < 2 d. 5 x;?: 20

a. 2 x+ 1 = 9

b. 4x < 10

5. Make a graph of the truth set for each of the following sentences. a. x(5 - x) = 4; x

b. 4 y > 10;

:s; d. x > c. x

e. x

3; x

y E

E

E

{0, 1, 2, 3, 4}

F

C

3 and x


6; x E F

Reaching Logical Conclusions Consider the two statements: 1. Any member of a varsity squad is excused from physical education. 2. Henry is a member of the varsity football squad. Our common sense tells us that if we accept these two statements as true, then we must accept the following third statement as true: 3. Henry is excused from physical education. We say that the third statement follows logically from the other two. In drawing logical conclusions it does not matter whether the statements we accept as true are reasonable or sensible. This is because we depend entirely upon the form of the statements and not upon what we are talking about. Thus, if we accept the following statements as true: 1. All whales are mammals; 2. All mammals are warm-blooded animals; 3. All warm-blooded animals are subject to colds; then we must conclude that 4. All whales are subject to colds. Do you see that statements 1, 2, and 3 are arranged in logical order? In the diagram at the right the set of whales is represented by W, the set of mammals by M, the set of warm-blooded animals by B, the set of animals subject to colds by C, and the set of all animals by A. The diagram shows that Wis a subset of M A as required by statement 1, that Mis a subset of B as required by statement 2, and that B is a subset of C as required by statement 3. The only conclu\__), sion that uses all of our given statements is that W is a subset of C, as asserted by statement 4. Had our third statement been "No warm-blooded animals are subject to colds," our diagram would have been the one shown at the right and A our conclusion would have been "No whales are subject to colds." If you have read Alice's Adventures in Wonderland or Through the Looking-Glass, you know that their author, Lewis Carroll, delighted in giving sets of nonsense statements which lead to logical conclusions. One such set is the following: I. Babies are illogical; 2. Nobody is despised who can manage a crocodile; 3. Illogical persons are despised.

B@IW\

78

When these statements are arranged in logical order we have: 1. Babies are illogical; 2. Illogical persons are despised; 3. Nobody is despised who can manage a crocodile. From these we can draw the logical conclusion: 4. Babies cannot manage crocodiles. Other sets of statements written by this author follow. To draw a conclusion from each set of statements, first arrange the statements in logical order. A diagram such as those on the preceding page may help you. The correct conclusions are given at the bottom of the page, but do not look at them until you have written your conclusion. I. 1. Everyone who is sane can do Logic; 2. No lunatics are fit to serve on a jury; 3. None of your sons can do Logic. II. 1. No ducks waltz; 2. No officers ever decline to waltz; 3. All my poultry are ducks. III. 1. No kitten that loves fish is unteachable; 2. No kitten without a tail will play with a gorilla; 3. Kittens with whiskers always love fish; 4. No teachable kitten has green eyes; 5. No kittens have tails unless they have whiskers. IV. 1. There is no box of mine here that I dare open; 2. My writing-desk is a box made of rose-wood; 3. All my boxes are painted except what are here; 4. There is no box of mine that I dare not open, unless it is full of live scorpions; 5. All my rose-wood boxes are unpainted. Conclusions:

I. None of your sons are fit to serve on a jury. II. My poultry are not officers. III. No kitten with green eyes will play with a gorilla. IV. My writing-desk is full of live scorpions. With this brief introduction to Lewis Carroll type problems, you will find it worthwhile and interesting to construct your own problems of this type.

79

Chapter

4 Logic In this chapter you will learn to draw correct conclusions

In this chapter you will learn some of the principles of logic. Logic is a set of rules which we can use to draw valid conclusions. Statements

If we can determine that a sentence is true or false, we call it a statement. We can usually determine whether a closed sentence is a statement by reading it. For example, we know that the closed sentence 2 + 3 =Sis true and the closed sentence 3 + 4 > 9 is false. These sentences are statements. An open sentence is a statement if we can determine that it is true or false for each value in the domain of each of its variables. Let us study the open sentence 2 + x = 8. This sentence is neither true nor false as it stands. However, if we let x have the domain {S, 6, 7}, we see that when x = 5, 2 + x = 8 becomes the false statement 2 + S = 8 when x = 6, 2 x = 8 becomes the true statement 2 6 = 8 when x = 7, 2 + x = 8 becomes the false statement 2 + 7 = 8

+

+

Since we can determine that the sentence is true when x = 6 and false when x represents either of the other members of the domain, we can say that the sentence is a statement. Had the domain of x been the set of all the numbers of arithmetic, we could still call the sentence a statement because we would know that the sentence is true when x = 6 and false when x is any other number of arithmetic. Such sentences as "Write 49" lack the capacity to be either true or false. 1 They are not statements. Statements which are closed sentences are called closed statements and statements which are open sentences are called open statements. 80

81 Simple statements are statements with only one verb . Let us agree to use a small letter enclosed in parentheses to represent a simple statement. Thus we may use (a) to represent the simple statement x + 2 = 8. We read "(a)" as "statement a." The parentheses may be omitted if there is no doubt that the letter represents a statement. ORAL EXERCISES

1. What do we mean by "logic"? 2. When is a sentence a statement? 3. How can we decide whether an open sentence is a statement?

0

EXERCISES

1. Which of the following sentences are statements? a. Please feed the dog. b. Steve is at home. c. 3 +4

= 15

d. Do 1he exercises on page 15. e. 20 is an even number.

f. 3+6= 1 +4+2

2. Is the sentence 3 + x = 15 a statement if x represents a number of arithmetic? Explain.

3. Is the sentence x - 2 2'.: 0 a statement if x represents a number of arithmetic? Explain .

4. Is 8 x > 16 when x

E

{0, 1, 2, 3, 4, 5} a statement? Explain.

Equivalent Statements

In Chapter 2 we spoke of equivalent statements. We now define equivalent statements as follows: Two statements are equivalent if when either is true the other is true and when either is false the other is false. If two statements are open statements, they are said to be equivalent if they have the same truth set.

Thus the two open sentences 10 x = 20 and 5 x = 10, when x E A , are equivalent because they have the same truth (solution) set, {2}. We continue to use the double-headed arrow to indicate equivalence. Thus a++ b represents the statement "statement a is equivalent to statement b." This statement is true if when either a orb is true, the other is true and when either is false, the other is false; and this statement is false otherwise.

82

CHAPTER 4

On the basis of the definition for equivalent open statements, we draw the following conclusions: 1. Any open statement is equivalent to itself. (a)+-+ (a).

2. If (a) is equivalent to (b), then (b) is equivalent to (a) . If (a)+-+ (b) , then (b) +-+ (a). 3. Two open statements which are equivalent to the same statement are equivalent to each other. If (a)+-+ (b) and (b) +-+ (c), then (a)+-+ (c). 4. If we substitute one of two equivalent open statements for the other in any compound statement, we obtain a statement equivalent to the original compound statement. For example, if given the compound statement (a) and (c) and (a)+-+ (b) , then (a) and (c) +-+ (b) and (c) .

0

EXERCISES

1. If the domain of

x is {O, 1, 2, 3, 4} , which of the following are pairs of equivalent statements?

a. 6 x = 12 3x=6

b. x + 2 < 5

c. x

+6 =

d. 4 x > 8 2x > 4

9

x=3

x+5=7

2. Complete the second statement of each pair in such a way that it is equivalent to the first statement of the pair. Assume that x represents a number of arithmetic.

a. 5 X = 15 x=?

b.

= 16 ? = 32

2x

c.

x+ 1 = 6

d.

x=?

x+ 5 > 8 X

>?

3. Which statements in each of the following columns are equivalent to the statement at the top of the column when x EA?

a. 5 x = 20 5

X

+ 1 = 21

2x=8 tx=,¥

b. 2 x X 10 5

xt;5 2x+l>ll

C, X ~

2 X 5 2 x~2 3x ~ 6

4. Replace each question mark below to make a true statement; then state which of the conclusions above assures you that your statement is true.

a. If 5 x = 10 +-+ x = 2, then x = 2 +-+ __ ? _ _

b. If x = 3 +-+ 2 x = 6, and 2 x = 6 +-+ 3 x = 9, then x = 3 +-+ __ ? __ c. If "2 x > 10 or x = 5" is a true statement and 2 x > 10 +--+ x > 5, then " __ ? __ or x = 5" is a true statement.

83

Logic

Contradictions Contradictory statements are two statements that are so related that each is true when and only when the other is false. In this case we say that each statement is a contradiction of the other. If two statements are open statements, they are said to be contradictory if their truth sets are complementary. Example l. In the column of statements at (a) The coat is blue. the right, statement (3) is a contradiction of (1) The coat is red. statement (a) because if (a) is true, then (3) is (2) The coat is green. false, and if (a) is false, then (3) is true. Simi(3) The coat is not blue. larly, statements (4) and (a) are contradictory (4) (a) is false. statements. If we are not careful, we may assume that (a) and (1) are contradictory. This is not true because it is possible for both to be false at the same time. For example, if the coat is brown, both (a) and (1) are false. Thus, even though (a) and (1) cannot both be true, it is possible for both of them to be false. We say that such statements are contrary rather than contradictory. Statements (a) and (2) are also contrary.

(1) x > 7; x EA (2) x :I> 7; x EA

Example 2. Let us compare the truth sets of the

open statements (1) and (2) at the right by means of a graph. The graph of x > 7 consists of all points on the number line which are to the right of the point P, whose coordinate is 7 as shown below. p

0 2

0

4

3

5

6

7

9

8

10

11

12

13

14

The graph of x :I> 7 consists of the point P, the point O whose coordinate is 0, and all points between these points as shown below. p

0 0

2

3

4

5

6

7

8

9

10

ll

12

13

14

If we place these two graphs on the same line, they do not overlap, but taken together they account for the complete graph of set A. The truth set for the open sentence x > 7 is the complement of the truth set for x :I> 7. Since the graphs of x > 7 and x :I> 7 have no points in common, we know that there is no value of x for which both statements are true. Since the two graphs combine to fill the line to the right of the origin completely, we know that in A there is no value of x for which both statements are false. Consequently we know that x > 7 and x :I> 7 are contradictory.

84

CHAPTER

4

Example 3. Let us consider the two open statements x + 2 = 6 and x+ 2 ~ 6 when x has the domain {l, 2, 3, 4, 5, 6, 7, 8}. We see that the truth set for x + 2 = 6 is {4} and the truth set for x + 2 ~ 6 is {l, 2, 3, 5, 6, 7, 8}. {4} and {l, 2, 3, 5, 6, 7, 8} are complementary. Since the two truth sets have no member in common, there is no value of x for which both statements are true. Since the two truth sets together include every member of the domain, there is no value of x for which both statements are false. Therefore x + 2 = 6 and x + 2 ~ 6 are contradictory statements.

0

EXERCISES

1. Which of the following pairs of closed statements contain contradictory statements?

+

c. 4 2 1 = 17 8+2=10 d. 3 X 4 + 2 X 5 = 22 3 X (4 + 2) X 5 = 90

a. 4+5=9 8+1=9

b. 3 +2 = 7 3+2=5

2. Use a graph to show that each of the following pairs of statements is composed of contradictory statements when x E A.

a.

x

2?

< 2?

4. Suppose that x has the domain {2, 4, 6, 8, 10}. a. What is the truth set for the sentence 3 x = 6?

b. Write a statement which contradicts the statement 3 x = 6. c. What is the truth set for the statement that you wrote for part b?

5. a. If the domain of xis the set of all the numbers of arithmetic, what is the truth set of any contradiction of x + 4 = 9? b. Write a statement that has the truth set you indicated for part a. 6. If the truth set for statement (a) is {l, 2, 3, 4} and the truth set for statement (b) is {4, 5, 6}, are (a) and (b) contradictory statements? Explain.

85

Logic

7. Complete to make a true statement: If (a) and (b) are contradictory statements, the intersection of their truth sets is the __ ? __ set.

8. If x has the domain {1, 2, 3., 4, S, 6}, are the statements x < 4 and x

= 6 contradictory?

Explain.

9. In the diagram at the right, A represents the set of all the numbers of arithmetic and G represents those members of A that are greater than 10. Complete the following statements: a. The points of A which are not included in G form a subset called the __ ? __ of G. Let us name this subset G'. b. If x is a variable which represents any member of A, the truth set for the statement x > 10 is the set __ ? __ . c. The truth set for the sentence x > 10 is the set __ ? __ . 10. Which of the statements below are contradictions of x

a.x < 9

b.

xi:.

9

c. x=9

> 9? d. x > 9

More about Contradictory Statements and Equivalent Statements

A given statement has many contradictions just as it has many equivalents. Suppose, for example, that the given statement is x = 7. Then each of the following statements is a contradiction of x = 7.

x-5 ;;e 2 x+2:;;e9 X ,S 7 2 X ;;e 14 What relation exists between two statements which are contradictions of the same statement? Let us investigate such a situation by means of a table. -equivalent ~ make such a table we let (a), (b), and (c) represent three statements as shown at the right. By writing T under (a), (b), or (c), we indicate that the statement is true, and by writing F, we indicate that the statement is false. To indicate that (b) and (c) are contradictions of (a), we write T under (b) and (c) when we write Funder (a), and we write Funder (b) and (c) when we write T under (a). contra- contraNow we observe that in our finished table (b) is true if and dietary dietary only if (c) is true and (b) is false if and only if (c) is false. Therefore we conclude that two contradictions of the same statement are equivalent.

86

CHAPTER

4

The table at the right was constructed to show the situation in which (d) contradicts (c) and (c) contradicts (a). The completed table shows that (d) is true if and only if (a) is true, and false if and only if (a) is false. Therefore (a) and (d) are equivalent. We conclude that a contradiction of a contradiction of a statement is equivalent to the statement itself. This reminds us of the "double negative" that we sometimes encounter in everyday conversation. For example, if we say, "It is not true that Jim is not a member of the club," we are really saying that Jim is a member of the club. Can you think of other "double negatives"?

0

EXERCISES

1. Write a contradiction of a contradiction of each of the following statements.

a.

x+ 1 = 3

b. x ::t>

9

c. 2 + 3 = 1

d.

x represents a natural number.

2. Write a contradiction of a contradiction of a contradiction of each of the statements in Ex. 1.

3. a. Complete the table at the right so as to make (c) a contradiction of (a), and (d) a contradiction of (b).

b. Complete: Contradictions of two equivalent statements are __ ? __ .

4. a. Complete the table at the right to make (c) a contradiction of (a), and (d) a contradiction of (b).

b. State a conclusion involving (a), (b), (c), and (d) of the table you have just made.

5. a. Make a table indicating that (d) contradicts (c), (c ) contradicts (b), and (b) contradicts (a).

b. State a conclusion involving (b) and (d) of the table you have just made. c. State a conclusion involving (a) and (d) of the table you have just made.

6. a. Make a table indicating that (a)+-+ (b) +-+ (c).

b. State a conclusion involving (a) and (c) of the table you have just made. c. Is the conclusion of part b consistent with Conclusion 3 of page 82?

87

Logic

A Special Notation for "Contradiction" You have learned that contradictory statements are two statements which cannot be true at the same time nor false at the same time. We often designate a contradiction of a by '""a, a contradiction of b by '""b, etc. We read '"" a as "a contradiction of a" or "not a." Using this notation we may write a contradiction of the statement "a contradiction of a" by writing'"" ('"" a). Recall that'"" ('"" a) ++ a. What is the meaning of '"" ['"" ('"" a)]? You learned in Chapter 1 that we often represent the complement of set S by S'. Since when two open statements are contradictory their truth sets are complementary, we may give the following example of the use of mathematical notation: If the truth set of the open statement s is S, then the truth set of '"" s is S . Negation Among the contradictions of statement a there is one which has a special form and name. This contradiction of a is called the negation of a. It is written in one and only one way, namely, "a is false." Thus, while the statement x > 7 has many contradictions, for example, 2 x > 14, "x > 7 is false," x + 3 > 10, x > 7, x ~ 7, etc., only one of these contradictions is called the negation of x > 7. It is written, "x > 7 is false." While it is easy to form the negation of a statement, it is not always easy to know what the negation means. Let us consider the statement

p: All boys are swimmers. Because p involves all boys, it is a general statement. If there exists even one boy X who is not a swimmer, then the general statement pis false. This illustrates a very important fact: A single exception will destroy the truth of a general statement. A single exception is called a counterexample. In this case, the boy X, who is not a swimmer, is the counterexample. Of course, if there are several boys who are not swimmers, pis false, but it is just as false if there is only one boy who is not a swimmer. The negation of p is '""p: The statement "All boys are swimmers" is false. We could express this in another way by saying, "Some hoys are not swimmers." The latter statement is not the negation of p, but it is a contradiction of p expressing the thought of the negation. A contradiction of the statement

p: All B's are S's is the statement

'""p: Some B's are not S 's. (That is, at least one B is not an S.)

88

CHAPTER

4

Let us now consider the statement

q: Any friend of Jim is a friend of Harold and its negation

,.__, q: The statement "Any friend of Jim is a friend of Harold" is false. We see that if Jim has even one friend who is not a friend of H arold, q is false. Consequently, ,.__, q means, "Some friend of Jim is not a friend of Harold ." A contradiction of

q: Any J is an H is the statement

,.__, q: Some J is not an H. Finally, let us consider the statement r: Some of our seniors are scholarship winners and its negation ,.__, r: The statement "Some of our semors are scholarship winners" is false. We see that r is true if at least one senior is a scholarship winner. Let S represent the set of our seniors and W represent the set of scholarship winners. When we say, "Some of our seniors are scholarship winners," we mean that the intersection of sets S and W is not the empty set (Sn W '=;t- 0). Consequently, r is false only when the intersection of sets S and W is the empty set, that is, when no senior is a scholarship winner. A contradiction of

r: Some S 's are W's is the statement ,.__, r: No S 's are W's.

Q

EXERCISES

Write the negation of each of the following statements. Then express the meaning of the negation by another contradiction of the statement.

1. All birds have feathers. 2. Some of our students come to school on buses. 3. Any boy under sixteen is eligible to enter the contest.

4. Every student has prepared his homework .

5. Some people are poor drivers. 6. Sue is never tardy.

89

Logic

7. Hank is sometimes tardy. 8. All even numbers are divisible by 3.

9. Some members of {a, b, c, d} represent odd numbers. 10. Every member of the t ruth set for x - 4 = 0 is an even number. 11. No student in this room likes algebra. 12. Some st udents in this room like algebra. 13. Some squares a re rectangles. 14. The statement "Some squares are rectangles" is false. Disjunction Let us now turn our attention once more to compound sentences . First, let us consider compound sentences containing the connective or. We call these sentences disjunctive sentences and define them as follows: If a and b a re simple statements, the sentence "a or b" is the disjunction of a and b. Since a and bare statements, it is only natural t o wonder whether the sentence "a or b" is also a statement. To help us decide this matter we have a ruling that we made in Chapter 3, namely : A compound sentence with the connective or is true if either of its clauses is true and is false if both of its clauses are false. Thus a disj unctive sentence which is constructed of two statements is a statement since its truth or falsity can be determined. We give the following examples as illustrations. Example l .

Construct a truth table for the disjunctive statement "a orb" in which a and b are statements.

Solution.

To construct a truth table for the disjunctive a b a or b statement " a or b," we must consider all possible combinations of true and false for the stateT T T ments a a nd b. T T F If a is true and bis true, the sentence "a orb" T F T has both of its clauses true and by our ruling F F F in Chapter 3 must be listed as true. This information is shown in line 1 of the truth table. If a is true a nd b is false , the sentence "a or b" has one of its clauses true a nd by our ruling of Chapter 3 must be listed as true. This information is shown in lin e 2 of the truth table. If a is fa lse a nd b is true, the sentence " a or b" has one of its clauses true. Thus, by the rulin g of Chapter 3, it must be listed as true. This information is shown in lin e 3. If a and b a re both false, our ruling tells us that our compound senten ce is false. This information is show n in line 4.

90

CHAPTER

4

Example 2.

Construct a truth t a ble and a graph for the disjunction of "x < 7" and "x > 5" when x E F.

Solution.

The disjunction of "x < 7" x5 x5 -~ and "x > S" is "x < 7 or .. X > S." T T T i If both the clause "x < 7" T F T and the clause "x > S" are T F T true , certainly the disjunctive sentence has one clause true and by the ruling of Chapter 3 is a true sentence. This information is shown in line 1 of the truth table. Lines 2 and 3 of the truth table show the situation when one clause is true and the other false. The table shows no line for the case in which both x < 7 and x > S are false, because both cannot be false at the same time. The graph of "x < 7 or x > S" is shown below. Observe that the graphs overlap and account for the complete graph of set F. graph of x>5

0

2

3

4

5

7

6

8

9

10

11

12

13

graph of x 4 4 =

3x 2X

5 5

8

e. 4 x X

< > >

12 1

f. 3 x 12 x=6

Conjunction

Let us now consider compound sentences having the connective and. We call such sentences conjun ctive sentences and define them as follows: If a and b are simple statements, the sentence "a and b" is the conjunction of a and b.

92

CHAPTER 4

In Chapter 3 we ruled tha t a compound senten ce having the connective and is true if both of its clauses are true and tha t otherwise it is false . Since thi s ruling mak es a conjunctive sentence either true or fal se in all cases, we know tha t a conjunctive sentence is a sta t ement. Example l.

M ake a truth t able for the conjunctive statement " a and b."

Solution.

Considering all possible combinations of true and false for th e sta tements a and b and following th e ruling of Chapter 3, we construct th e truth table shown at th e right.

Example 2.

Solution.

0

a

b

a and b

T T

T F T

T F F F

F F

F

M ake a truth ta ble and a graph for the conjunction of "x > 5" and " x < 9" when x e A. x>S x5andx S and th e T T T clause x < 9 are true, our T F F ruling tells us th at the conT F F juncti ve sentence "x > S and x < 9" is true. This information is shown in line 1. Since our ruling states that fo r all oth er cases a conjunctive sentence is fal se, we have th e in formation displayed in lin es 2 and 3 of th e table. It is impossibl e for x > S and x < 9 to be false at the same time. Hence there is no line 4 in the table. Since a conjunctive sen tence is true only when both clauses are true, the graph of "x > S and x < 9" is th e one shown below. 2

3

4

5

6

7

8

9

10

11

12

13

We can also illustrate our ruling on the truth of conjunctive statements by means of electrical switches. This time we a rrange the switches a and b in series as shown below. This a rrangement , called " a and b," is closed (electrons will flow) if both switches a re closed a nd is open in all other cases. If we substitute " closed " for T (true) and " open" for F (false), the situation correspond s t o the truth ta ble fo r the example a bove.

.. ___-~ ~----

~ Ba ttery

-

Ammeter

-

93

Logic

We have the following two parallel statements: The arrangement of switches "a and b" is closed if both a and b are closed The conjunctive statement "a and b" is true if both a and b are true and open in all other cases.

and false in all other cases.

0

EXERCISES

1. Write the conjunction of each pair of statements below. a. It is raining. It is snowing. b. 3+6= 9

2X8 =4

c. The book is blue. 4 + 2 = 6 d. Bill is the president of our class. Mary is the vice-president of our class.

2. Write the conjunction of the two statement s in each of the following pairs of statements. Tell whether the conjunctive statement is true or false .

a. 3 + 9 = 12 4X2=8

b. 6 > 2

c. 8 = 6 + 2 9

0 if and only if x

>

4

5?

x > 0 if and only if x > S can be written as x > 0 +-+ x > S. To prove this statement true we must show that the following two statements are true: (a) x > 0-+ x > S, and (b) x > S-+ x > 0. The statement (a) is false because there are numbers (e.g. , 3) which are greater than 0, yet not greater than 5. The statement (b) is true, but since we cannot show that both (a) and (b) are true, we cannot say that x > 0 if and only if x > S.

0

EXERCISES

1. Write each of the following statements in two other forms. a.

X

> 7-+ X >

0

b. ~ = 18 only if n = 36. c. 3 x = 3 y if and only if x = y.

d. a + 8 = b + 8 +-+ a = b e. You are invited to the party if and only if you are a freshman.

f. x is divisible by 6 -+ x is divisible by 3 2. Explain the meaning of the statement: A man is Commander-in-Chief of the Armed Forces of the United States if and only if he is the President of the United States.

3. Is it true that if a number of arithmetic is divisible by 6, then it is divisible by 3? Is it true that if a number of arithmetic is divisible by 3, it is divisible by 6? Is it true or false that "xis divisible by 3" is equivalent to "xis divisible by 6" when x is a number of arithmetic?

4. Let a be a number of arithmetic. Is it true that if a > 7, then the graph of a is to the right of the graph of 7? Is it true that the graph of a is to the right of the graph of 7 only if a > 7? Is it true that the graph of a is to the right of the graph of 7 if and only if a > 7?

0

EXERCISE

5. Prove that ½m = ½n when m t A and n t A, if and only if m = n. You may assume the following statements true: a. If equal numbers of arithmetic are multiplied by 2, the products are

equal.

b. If equal numbers of arithmetic are multiplied by equal.

½,

the products are

Logic

121

Need for Assumptions and Definitions

In proofs many of the statements which we use as reasons are statements which we have previously proved. We may also use statements which have not been proved. Some of these are assumptions and some are definitions. Reasons used in first proofs are limited to assumptions and definitions. A statement such as "Today is Tuesday" is an assumption. In it we assume that we are not confused as to which day this is, that the scheme of naming the days has not been changed without our knowing it, and so on. We cannot get along without making assumptions. Without them there would be no logical structure because there would be no foundation for one. Of course, our proofs are satisfactory only to those people who accept the assumptions we make. A definition is an exact description of a set of objects (persons, ideas, things). In order to be an exact description a definition must be broad enough to include all objects in the set and yet detailed enough to exclude any that do not belong to the set. Consider the following definition: A flute is a tubular wind instrument of small diameter with holes along the side. If this definition is an exact description of the set of flutes, we must conclude that the set of flutes is exactly the same as the set of tubular wind instruments of small diameter with holes along the side . This means that two statements, each the converse of the other, are true. These are 1. If an object is a flute, then it is a tubular wind instrument of small

diameter with holes along the side. 2. If an object is a tubular wind instrument of small diameter with holes along the side, it is a flute. It follows that if a correct definition is put in if-then form, both this if-then statement and its converse are true. This characteristic of a definition is very important. It is often expressed by saying that a definition is reversible. By this we mean that the statement remains true when the positions of the if-clause and the then-clause are reversed. In order to be helpful to us, a definition should describe its subject in terms which are simpler than the subject itself. However, eventually we arrive at terms which are so simple that we cannot find simpler terms with which to describe them. At this stage we leave these terms undefined and hope that everyone has essentially the same understanding of them. Our definitions must rest upon undefined terms just as our logical system must rest upon certain assumptions .

122

CHAPTER

4

ESSENTIALS

Before you leave Chapter 4 make sure that you 1. Know when a sentence is a statement. (Page 80.)

2. Know what we mean by "equivalent statements" and can indicate the equivalence of two statements in symbols as well as words. (Page 8 l .)

3. Know what we mean by "contradictory statements" and recognize the graphical relationship of the truth set of an open statement and the truth set of its contradiction. (Page 83.)

4. Understand the relationships between equivalent statements and contradictory statements. (Page 85 .)

5. Can form the disjunction of two simple statements, can make a truth tabl~ for the disjunction , and can interpret the disjunction graphically, when a graph is possible. (Page 89.)

6. Can form the conjunction of two simple statements, can make a truth . table for the conjunction, and can interpret the conjunction graphically, when a graph is possible. (Pages 91-92 .)

7. Can express the contradiction of a disjunction and of a conjunction. (Page 95.)

8. Can form an implication involving two simple statements, can make the truth table for the implication, and can determine the hypothesis and conclusion of an implication not expressed in if-then form. (Pages 97-98 .)

9. Can write a contradiction of an implication.

(Page 101 .)

10. Understand the transitive property of implication and can use it m proving an implication.

(Page

107.)

11. Understand the meaning of a "sequence of implications. " (Page 108 .)

12. Can write a simple proof in any of the forms discussed in this chapter. (Page 109 .)

13. Understand the meaning of the "converse," the "inverse," and the "contrapositive" of an implication and know how they a re related to each other and to the implication. (Page 112.)

14. Understand proof by use of the contrapositive. 15. Understand indirect proof. (Page 116.)

(Page

114 .)

16. Can prove two simple statements equivalent. (Page 118.) 17. Understand what we mean by "assumptions," "definitions," and "undefined terms. " (Page 121.)

123

Logic

18. Can spell and use the following words correctly. assumption (Page 121 .) conclusion (Page 97 .) conditional (Page 97.) conjunction (Page 91 .) contradiction (Page 83 .) contrapositive (Page 113 .) contrary statement (Page 83 .) converse (Page 112 .) definition (Page 121 .) disjunction (Page 89.)

equivalent (Page 81.) hypothesis (Page 97.) implication (Page 98 .) inverse (Page 112 .) logic (Page 80 .) major premise (Page 104 .) minor premise (Page 104 .) negation (Page 87 .) statement (Page 80 .) syllogism (Page 104 .) CHAPTER REVIEW

1. Which of the following sentences are statements?

d. 3 + 2 = 5

a. Is Henry ready?

e.9x=18; xeA

b. Wait for me.

f. x > 7; x e A

c. San Francisco is in California.

g. "3 x

< 9 is false."

2. From the statements below select those which are equivalent to 6 x x eA.

a.

x+ 1 = 4

b. 2 x= 6

C. X

>

2

= 18;

d.12 x=36

3. Which of the following statements are true? Assume x e A.

+ 4)

a. (5 x = 10) +--+- (x = 2)

c. (x + 6 = 4) +--+- (6 = x

b. (9 > 4) +--+- (4 < 9)

d. (x+ 1 > 5) +--+- (x > 4)

4. Which of the statements below are contradictions of x < 7; x e A? a. x > 4 c. x = 7 e. x 2-: 7

b. x = 4

d. x

,c. 5

5. Write,...__,,...__, a in simpler form. 6. Write contradictions of:

a. All students study hard.

b. Some basketball players are tall. c. Every house on this street is painted white. d. Anyone who wishes to come is invited. e. No seniors are present.

f. It is false that x

< 7.

124

CHAPTER

4

7. The graph at the right shows the solution set for x = 4. Make a graph to show the solution set for the negation of x = 4 when x t: A . O 2 3 4 5

8. Make graphs to show the solution sets for x > 2 and its contradiction whenxt:A.

9. Write the disjunction, the conjunction, and two implications involving the two statements: I like cats. I like dogs.

10. Make a graph for the disjunction "x > 6 or x > 4"; x t: A. For which values of x is the disjunction true?

11. Make a graph for the conjunction "x > 6 and x > 4"; x t: A. For which values of x is the conjunction true?

12. Write the conjunction of "x < 6" and "x > 2"; x t: A. Write a contradiction of the statement you have just written. Compare the graphs of the two statements.

13. Write the disjunction of "x = 4" and "x > 4"; x t: A. Write a contradiction of the statement you have just written. Compare the graphs of the two statements.

14. Write a contradiction of each of the following statements. a. I shall play volley ball today or I shall play tennis tomorrow.

b. I like classical music and I like semiclassical music. 15. Name the hypothesis and the conclusion in each of the following implications.

a. If the water freezes sufficiently, I shall go skating.

b. I read the paper before I go to school if it arrives early enough. c. All of the freshmen are present.

16. Replace the question mark with the proper symbol to make a true statement. If (3 x = 12)--+ (x (3 X = 12) _? _ (x = 4).

= 4)

and (x

= 4)--+

(3 x

= 12),

then

17. Write the converse, the inverse, and the contrapositive of: If a number is a whole number, it is a number of arithmetic.

18. If P--+m--+n--+r, then P--+ _?_ 19. Write the contrapositive of: If my uncle does not live in the state of Washington, then he does not live in Seattle. Suppose you know that my uncle lives in Seattle and you accept the given implication. What can you conclude?

125

Logic

20. State the conclusion that can be drawn from each of the following pairs of statements.

a. John is a freshman and } ? All freshmen study mathematics ---+ - - · - (I)

b.

(2)

(x = 5 ; x 1: A) ---+ _ _ ? _ _ (1) Given

(2) If two numbers of arithmetic are equal, multiplying both by 4 produces equal products. c.

Reasons

Statements I. My dog is a dachshund 2. ___ ? __ _

I.

Given

2. If a dog is a dachshund, it is long

and low.

21. After investigating a robbery the police suspected Doubtful Dan. However, Doubtful Dan's lawyer succeeded in proving that Dan could not have committed the crime. Part of the lawyer's proof is given below. Complete the following argument as you think the lawyer might have. "Let us suppose that Doubtful Dan is guilty. If Dan is guilty, he must have been at the scene of the crime on the night of the robbery . This contradicts evidence which shows that on the night of the robbery ... " CHAPTER TEST

1. Complete: A sentence is a statement if __ ? __ . 2. Complete the second statemen t of each of the following pairs so that it is equivalent to the first.

b.

a.4x=20; x1:A X=

_?-

x

-,c 2

X

_?_ 2

3. Write a contradiction of each of the statements below: a. All flowers are beautiful.

b. Some members of our class were able to do Exercise 20. x < 10. On a second number line, graph the solution set of a contradiction of x < 10. The domain ofx is A.

4. Graph the solution set of

5. Write the disjunction of

"x

< 3" and

solution set of the disjunction.

"x

>

7" when x 1: A. Graph the

126

CHAPTER 4

6. Graph the solution set of a contradiction of the disjunction that you wrote in Ex. 5.

7. Complete the following truth table, assuming that

Ix < 3 •

x

>

7 x

)>

7 x

< 3 and x :I>

T

F

?

?

F F

T

? ?

? ?

F

7 x

1; < 1;

~

7; x e F

x eF

e. x

x eF

f. x + 5 = 8 and x

< 7;

c. x ~l ;x eF

g. x < 7 or x > 5; x e F

d. x > 1 and x < 10 ; x e F

h.

x

< 7 and x > 5 ;

x eF

x eF

N=

Logic

129

22. If A and B are simple sentences which are true and C and D are simple sentences which are false, which of the following compound sentences are true? a. A and B

e. A or C

i. B or C

b. C or D

f. A or D

j. C and D

c. A and C

g. Band C

d. A or B

h. A and D

k. Band D 1. B or D

23. Write a simple sentence equivalent to each of the following compound sentences.

a. x 5 and x 2

24. Write a compound sentence that is equivalent to each of the following simple sentences .

a. y

~

b. y 2

25. Complete the following truth table.

p q p or q p and q P-+q ,.._, p ,.._, q ,..._,q-+,...,P T T T F F T

F F

? ? ? ?

? ? ? ?

? ? ? ?

? ? ? ?

? ? ? ?

? ? ? ?'

26. Graph the solution set of the disjunction of "3 x < 15 " and "x > l ." Assume that x e A.

27. Replace each question mark with an expression to form a true statement . a. p and } -+ _? __ P-+q

b. [(p-+q) and (q -+ r)]-+ _?_

28. What conclusion can be drawn from the following pair of st atements? If a rectangle has two of its adjacent sides congruent, the rect angle is a square.

Rectangle ABCD has its adjacent sides AD and AB congruent.

29. Complete: If an implication and its converse are both true, then its clauses are __ ? __ .

30. Explain the plan for proving an implication by using its contrapositive.

130

CHAPTER

4

CUMULATIVE TEST [Chapters 1-4]

1. If C = {l, 2, 3, · · ·} a. Is it true that

2-l

E

C?

b. Write the subset of numbers in C which are rational numbers. c. Write the subset E of even numbers in C.

d. Write the subset of numbers in C which are solutions for the sentence

x+ 110 >

114.

e. Write the subset of numbers in C which is the solution set for the compound sentence x > 5 and x < 9.

f. Graph the solution set found in part e. g. Use the roster method to write {x I x :I> 4; x EC}.

2. Fill the blank to make a true sentence: 8

>4

_? _ 4 < 8.

3. Write the common name for each of the numbers indicated below. a.9(4+5)

b. 3(7) + 9(3)

c.

d .. 7(.4 + .6) +.3

42+ 32 52

4. If n is a variable whose domain is {l, 2, 3}, write the set of numbers that can be represented by 4 n + 2.

5. Write a simple sentence equivalent to the compound sentence

x

5.

6. Graph each of the following sentences. a. x

~

3; x E F

b. x = 2 and x < 1; x E F

7. Which of the following statements are true and which false? a. The set of counting numbers is a proper subset of the set of whole numbers.

b. The set of counting numbers is closed under addition. c. Th~ set of counting numbers less than 100 is a finite set.

d. The set of rational numbers of arithmetic less than 1 is a finite set. 8. Graph the solution set of the disjunction of "x + 3 = 7" and "x < 6" when x EA.

9. What are equivalent statements? 10. What are contradictory statements?

131

Logic

11. Determine the conclusion that can be drawn from the following statements. Then arrange the three statement s in one of the accepted forms of proof.

a. J ohn hopes to win the state algebra contest.

b. If a person hopes to win the state algebra contest, he must do much studying.

12. Assume that the following statements are true. If an elephant is vicious, it is ostracized by the herd . If an elephant is a rogue elephant, it is vicious . If an elephant is ostracized by the herd, it roams alone.

Which of the following conclusions can be correctly inferred from the given statements ? a. If an elephant roams alone, it is a vicious, rogue elephant.

b. All elephants are rogues. c. If an elephant is a rogue elephant, it roams alone. d. An elephant is a rogue elephant only if it is ostracized by the herd.

13. Complete the following truth table p q ,..., p P-+q ,..., p or q T T T F F T F

F

? ? ? ?

? ? ? ?

? ? ? ?

14. State whether each of the following is a true statement or a false statement.

a. If x EA, then x

b. If x > y and

E

C.

= y. 3 = 0-+-+ x = 3, then the graph of

x -t y, then x

c. If x ER and x {x x ER} is the same as the graph of {x I x = 3; x ER}.

Ix

- 3

= O;

Logic and Switching Circuits A switching network is an arrangement of switches and wires joining two terminals T1 and T 2 • If we connect these termia nals to an electric battery as shown in the diar1 : ~ 2 gram, a switching circuit is formed. A switch may be marked with a symbol to represent a statement. Each switch is closed or open depending upon whether the statement it represents ~--c..->--11 is true or false. A closed switch (True) permits the flow of current and an open switch (False) prevents this flow. In the switching circuit shown, the switches (pictured in red) are open and consequently no electric current can flow. If one or more of the switches were closed, current could flow and the bulb would light. The network itself represents a compound statement whose components are the statements represented by the individual switches in the network. This compound statement is true if and only if the switches in its representative network are set in such a way that the network is closed ; that is, the switches are set so as to permit the flow of current between the terminals. Since "closed" for a network corresponds to "true" for a compound statement, we can speak of the "truth table" of a network. Such a table can be obtained by writing, T for each closed switch and F for each open switch. A network represents a compound statement if the statement and the network have the same truth table. In Chapter 4 we observed that the network consisting of the switch a and the switch b arranged "in parallel" could be used to represent the disjunctive statement "a orb." In a simi lar fashion the network shown in the above diagram represents the disjunctive statement "a orb or c." Observe that this network will permit the flow of current except when all of the switches, a, b, and c, are open. This corresponds exactly to the fact that the disjunctive statement "a orb or c" is true except when all of the statements a, b, and c are false. Thus the network and the statement "a orb or c" have the same truth table. Since the same network would serve for "(a orb) or c" and "a or (b or c)," we see that disjunction is associative. The fact that the switches can be interchanged without affecting the truth table for the network suggests that disjunction is commutative. Can you devise a network to represent the conjunctive statement "a and b and c"? Is it evident from this network that conjunction is both associative and commutative? It is possible to couple two switches by a non-conductor in such a way that if one is closed the other is open. We in~q dicate this in our diagram by letting p represent one of two coupled switches and p the other. In the same manner, q and q are coupled.

~

~

132

When we examine this network closely, we observe that the switch representing p is really a "dummy switch" since it has no effect on its branch of the network. It was included for the purpose of showing you the meaning of the switch representing~ p. Now that we have given meaning to switches representing such statements as p and q, we can simplify our network by omitting the "dummy switch" representing p. The simplified diagram is shown at the right. Evidently this network represents the compound statement q or ( "-' q and p ). Checking either the original or the simplified diagram, we see that current will flow in all cases except when both the switch representing p is closed (that is, p is open) and the switch representing q is open. By our agreements this means that the compound sentences represented by these networks are true in all cases except when p is true and q is false. Since the statement p-+ q is true in all cases except when p is true and q is false, the networks discussed represent statements that are equivalent to the implication p - + q. Does the statement represented by the network at the right also represent the implication p - + q? To answer this question, it will help you to make a truth table for the statement represented by the network. Can you devise a network for (p and q) - + t? Networks are said to be equivalent if they have the same truth table. Sometimes it is possible to simplify a network by finding an equivalent one that has fewer switches. Note that in the preceding discussion we found three equivalent networks each representing statements equivalent top-+ q and each simpler than the previous network. Consider the networks:

~

~

~

~

Are they equivalent? If so, what two compound statements are equivalent? Which network is simp ler? Perhaps you would like to try to simplify the network:

qh

-~:~p-

T2

p--q

You might like to try the following application of switching circuits to a simple voting problem: A committee has three members. It takes a majority vote to carry a motion and the chairman has a veto (a motion passes only if he votes for it). Design a circuit for this committee so that each member votes for a motion by pressing a button and so that a light goes on only if the motion is passed. 133

Chapte,

5 Operations with the Numbers of Arithmetic In this chapter you will study the principles of adding, subtracting, multiplying, and dividing the numbers of arithmetic

In arithmetic you learned many principles of addition, subtraction, multiplication, and division. In this chapter you will review some of these principles and study their applications in algebra. The Numbers of Arithmetic

We have been thinking of the numbers of arithmetic, designated by A, as the numbers which can be placed in one-to-one correspondence with all the points on the number line at the origin and to its right. graph of set A

0

The diagram at the right shows the subsets of set A. Observe that C (the set of natural numbers) is a proper subset of W (the set of whole numbers), W is a proper subset of F (the set of rational A numbers of arithmetic), and Fis a proper subset of A. The members of set A which are outside of set F are called the irrational numbers of arithmetic. Examples of these numbers are 1r , V2, and V3. 134

135 ORAL EXERCISES

1. Why is C a proper subset of W? 2. Name a number which is in F but not in W.

3. Name a number which is in A but not in F. 4. Which of the following statements are true and which false?

a. x 1: A -+ x 1: C.

b.

X E

C -+ X

c.

X E

w -+ X E F.

d.

X E

F -+ X

E

F.

e. If x is a whole number, then x is a rational number.

f. If x is a natural number, then x is a rational number.

g. If x is not a rational number, then x is not a E

w.

natural number.

Name the sets indicated by each of the following:

5. CUW

7. WUW'

9. WU (C n F)

6. WnF

8.

en C'

IO.WU (CU F)

Closure and the Numbers of Arithmetic

We have said that the set of natural numbers is closed under addition and multiplication, but not closed under subtraction and division. Let us now consider whether the numbers of arithmetic are closed under these operations. Experience has caused us to accept the following:

I),,

Closure Property of Addition: If a and b are two numbers of arithmetic, then

a+ b is a number of arithmetic.

In other words, the numbers of arith-

metic are closed under addition.

Experience has also taught us that this sum, a+ b, is a unique number of arithmetic . When we say that the sum is a "unique" number we mean that there is only one number that is the correct sum. What number of arithmetic expresses the sum of 3 and 7? of and / 4 ? of 7.45 and 11.31? We also accept the following :

t

I),,

Closure Property of Multiplication: If a and b are two numbers of arithmetic, ab is a number of arithmetic. (ab means a X b.) In other words, the numbers of arithmetic are closed under multiplication.

Experience has also taught us that this product, ab, is a unique number of arithmetic. What number of arithmetic expresses the product of 7 and 9? of 11 and O? of .3 and 1.2?

136

CHAPTER

5

We have learned that we cannot divide by zero. We have learned, also, that with this one exception the quotient of any two numbers of arithmetic is a number of arithmetic. We assume that: If a and bare any two numbers of arithmetic and b ~ 0, then ~ is a number of a rithmetic. In other words, . b the numbers of arithmeti c are closed under division when the divisor is not zero. Later you will discover that this property is equivalent to the closure property of multiplication. We have learned , however, that we can subtract two numbers of arithmetic only when the number subtracted is not larger than the number from which it is subtracted. Thus, while we can write 11 - 5 = 6, there is no number of arithmetic to express 5 - 11. We assume that: If a and b a re two numbers of arithmetic, we can find a - b only when a ~ b. In other words, the set of numbers of arithmetic is not closed under subtraction.

0

EXERCISES

1. We have found that C, the set of natural numbers, is closed under addition. S 1 = {l , 2, 3}, a subset of C. Is S 1 closed under addition? S2

= {1, 2, 3, · · ·, 1000}, a subset of C. Is S2 closed under addition? Is any subset of C closed under addition? 2. Is S 1 of Ex. 1 closed under subtraction? under multiplication? under division?

3. Is any subset of C closed under subtraction? under multiplication? under division?

4. Is the set of whole numbers W closed under addition? under subtraction? under multiplication? under division ?

5. T 1 = {O, 1}, a subset of W. Is T 1 closed under addition? under subtraction? under multiplication? under division, provided division by zero is excluded? 6. T 2 = {O, 1, 2, · · ·, 100,000}, a subset of W. Is T 2 closed under addition? subtraction? multiplication? division? 7. Vis the set of rational numbers from Oto 1, inclusive. I s V closed under addition? subtraction? multiplication? division? 8. Is the set of all multiples of 5 closed under addition? under multiplication? (A multiple of 5 is a number which is exactly divisible by 5.)

9. Is the set of the numbers of arithmetic closed under the operation "doubling a" where a is a number of arithmetic?

/

137

Operations with the Numbers of Arithmetic

10. Is the set of the numbers of arithmetic closed under the operation "doubling a and adding 1" when a is a number of arithmetic?

11. Is the set of the numbers of arithmetic closed under the operation " taking the average," that is, finding a+ b when a and b are numbers 2 of arithmetic?

12. In telling time by a clock we count up to 12 and then start over again. Imagine that you are telling time by a clock which shows only the numbers 1 through 5. Now you will count only to 5 and then start over again. Thus 5 + 1 = 1, 5 + 2 = 2, et c. A complete addition table and a complete multiplication table for these numbers is shown below. Study the tables and then answer the questions.

a. What is a simpler name for the sum (2 + 5) + 4 in this kind of arithmetic? for1+2+3+4+5?

b. What is a simpler name for the product (4 X 3) X 2 in this kind of arithmetic? for 1 X 2 X 3 X 4 X 5? c. Is {l , 2, 3, 4, 5} closed under addition in this kind of arithmetic?

d. Is {l , 2, 3, 4, 5} closed under multiplication in this kind of arithmetic? Addition Table

Multiplication Table

+12345

X 12345

1 2 3 4

5 1

1 1 2 3 4 5

2 3 4 5 1 2

2 2 4 1 3 5

3 4 5 1 2 3

3 3 1 4 2 5

4 5 1 2 3 4

4 4 3 2 1 5

5 1 2 3 4 5

5 5 5 5 5 5

Binary Operations

Try adding 5, 11, and 13 . Can you add all three of the numbers at one time? Of course your answer is, "No," because addition is a binary operation on numbers, that is, an operation involving only two numbers at a time. To add the numbers above, you must choose two of the numbers, add them, and then add their sum to the third number. Do you agree that subtraction , multiplication , and division are also binary operations?

138

CHAPTER

5

Since addition involves only two numbers at one time, we must define what we mean by the sum of three or more numbers. We say: If a, b, and c are three numbers of arithmetic,

a+ b + c= (a+ b) + c. If a, b, c, and d are four numbers of arithmetic,

a+ b + c+ d = (a+ b + c) + d. To add more than four numbers we continue this pattern.

Observe that the first of the definitions above says: To add three numbers, add the first two numbers; then add their sum to the third number. The second definition says : To add four numbers, use the first rule to add the first three numbers; then add their sum to the fourth number. Thus a+ b + c + d =[(a + b) + c] + d. Since multiplication involves only two numbers at one time, we must define what we mean by the product of three or more numbers. We say: If a, b, and c are three numbers of arithmetic,

a X b X c = (a X b) X c. If a, b, c, and d are four numbers of arithmetic,

a X b X c X d = (a X b X c) X d. To multiply more than four numbers we continue this pattern.

ORAL EXERCISES

1. What do we mean when we say that addition is a binary operation? when we say that multiplication is.a binary operation?

2. What is the definition for the sum of three numbers? 3. What is the definition for the product of three numbers ?

4. How do we find the sum of four numbers? the product of four numbers?

0

EXERCISES

1. Replace the question marks below with an expression which will make the statement the conclusion of a syllogism. 1. 2, m, and n are three numbers

I. Given

of arithmetic. 2. 2 + m + n = __ ? _ _ +

2. If a, b, and c are numbers of arith-

__? __

metic, then a+ b + c =(a+ b)

+ c.

2. According t o the definition for the multiplication of three numbers, which numbers should be multiplied first in finding the product 3 X 4 X 2?

139

Operations with the Numbers of Arithmetic

3. Replace the "Why?" below with the implication which completes the syllogism shown. 1.

4, a, and 7 are numbers of a rithmetic

2. 4 X a X 7 = (4 X a) X 7

I. Given

2. Why?

4. Use the definitions of addition to find each of the following sums.

2+ 7+ 5 ~4+ 8 +1 a.

+ ¾+ / 0 d5+ 4+7+2 c. / 0

e . .4 +

.2 + 1.5 + 3.0 £ 9+1+½+i+3

5. Use the definitions of multiplication to find each of the following products. a. 3 X 7 X 2

c. 8(m)(m)

b. 9

d. 2

X 8X 1

X 3 X

e. 8 X

5 X 10

f.

½X £ X

½X i

X

4 X 5

t X 36

The Properties of Equality

We now need to consider some of the properties of statements containing the equal sign. By the "properties" of statements we mean the characteristics that they possess. First let us observe that we may write a = a. This is true because a variable cannot represent two different numbers at one time. If the a on the left represents 7, then the a on the right also represents 7. We see that the statement a= a is true for any value of a. Let us name and state this property as follows:

r>,

The Reflexive Property of Equality: If a is a number of arithmetic, then a=a.

Next we observe that when we write a= b we mean that a and bare different names for the same number. Likewise, b = a means that band a are different names for the same number. Since the number is the same in both cases we have :

r>,

The Symmetric Property of Equality: If a and b are numbers of arithmetic and a =

b, then b = a.

Now we observe that if a= b and b = c, then a = c because a and c are names for the same number-the number named by b. Thus we have:

r>,

The Transitive Property of Equality: If a, b, and c are numbers of arithmetic such that a = b and b = c, then a = c.

We sometimes state the transitive property by saying: If two numbers are equal to the same number, they are equal. This statement is permissible if we interpret it as having the meaning above .

140

CHAPTER

5

Another important property of equality is:

f'),,

The Substitution Property of Equality: If a and b are numbers of arithmetic such that a

= b, the truth

or falsity of any statement in which the number is

referred to by the name a is not changed if the name b is used instead.

For example, if a= b then (a+ 2 = 8) ~ (b + 2 = 8). We sometimes state the substitution property of equality as : If a number is substituted for its equal in any statement, the truth or falsity of the statement is not changed. This way of stating the property is permissible provided that we interpret it to have the meaning given above. We know that the sum of two numbers of arithmetic is a unique number of arithmetic. Therefore, if a= b and c = d, then a+ c and b d are two different names for the sum of the same two numbers. Consequently we have the following property:

+

f'),,

The Addition Property of Equality: If a, b, c, and d are numbers of arithmetic such that a = b and c = d, then a c = b d.

+

+

We sometimes state this property by saying: If equal numbers are added to equal numbers , the sums are equal. This statement is permissible if the interpretation above is intended. As an example of this property observe that if 4 = 3 1 and 8 = 6 2, then 4 8 = (3 1) (6 2). We know that the product of two numbers of arithmetic is a unique number of arithmetic. Therefore, if a= b and c = d, then ac and bd are two different names for the product of the same two numbers. Consequently we have:

+

f'),,

+

+

+ + +

The Multiplication Property of Equality: If a, b, c, and d are numbers of arithmetic such that a = b and c = d, then ac = bd.

We sometimes state this property by saying: If equal numbers are multiplied by equal numbers, the products are equal. This is permissible if the above interpretation is intended. There are two other properties which we find temporarily useful. They are:

t>,f'),,

The Subtraction Property of Equality: If a, b, c, and d are numbers of arithmetic such that a= b, c = d, a ~ c, and b ~ d, then a - c = b - d. The Division Property of Equality:

arithmetic such that

If a, b, c, and d are numbers of

a= b, c = d, c -,e 0, and d -,e 0, then~ = ~ -

141

Operations with the Numbers of Arithmetic

0

EXERCISES

1. Assuming that a = 3 and b =2, state the property or properties of equality which explain why each of the following conclusions is true; a EA, b EA.

a. a+ b = 3 + 2

d. ~=~ b 2

b. ab = 3 x 2

e. (a= 3) +-+ (3 = a)

f. (3 > 2) ++ (a > b)

c. a - b = 3 - 2

2. Assuming that m = n and n = 6, state two properties of equality either of which explains why the conclusion m = 6 is true.

3. What conclusion can you draw from each of the following pairs of statements?

a. p: a and 3 are numbers of arithmetic such that a= 3. P---+ q: If a and bare two numbers of arithmetic such that a= b, then (a = b) +-+ (b =a). b. p: m, n, and 4 are numbers of arithmetic such that m > 4 and m = n. p---+ q: If a number is substituted for its equal in any statement, the truth or falsity of the statement is not changed.

c.

p:

2 is a number of arithmetic.

p---+ q: If a is a number of arithmetic, then a = a.

d. p: r, s, 3, and 10 are numbers of arithmetic such that r = 3 ands = 10. P---+ q: If a, b, c, and d are numbers of arithmetic such that a= b and

c=

d, then

a+ c =

b

+ d.

4. Replace each "Why?" below with the statement of the property which completes the syllogism.

a. 5 is a number of arithmetic. .·. 5 = 5 b. a, y, 9, and 4 are numbers of arithmetic such that and a = 4. .·.4+ y < 9

Given Why? Given

a+ y < 9 Why?

c. a, c, 7, and 6 are numbers of arithmetic such that a= 7, c = 6, and c < a . .·. a - c =7-6

Given

Why?

d. a, b, and 6 are numbers of arith-

Given

metic such that a= b and b = 6. .·. a= 6

Why?

142

CHAPTER

0

5

EXERCISES

5. Replace each "Why?" below with an implication to form a proof. Reasons Statements I. x, y, 10, and 4 are numbers of I. Given arithmetic such that x = 10 and y =4 . 2. Why? 2. xy = 10(4) 3. Arithmetic fact: If 10 and 4 are 3. 10(4) = 40 multiplied, the product is 40. 4. Why? 4 . . ·. xy = 40

6. Replace each "Why?" below with a statement of a property of equality to form a proof. Reasons

Statements I. a, b, 2, and 3 are numbers of arith-

metic such that a= 2 and b = 3. 2. a+ b = 2 + 3 3. 2 4.

+3= 5

I. Given

2. Why? 3. Arithmetic fact: If 2 and 3 are

added, their sum is 5. 4. Why?

a+ b = 5

Properties of Addition and Multiplication

We have been considering properties of equality. Now we turn our attention to some properties of addition and multiplication. You know that 3 + 7 = 7 + 3, 4 + 2 = 2 + 4, and so on. In other words you know that the order in which two numbers of arithmetic are added does not affect the sum. We call this property the commutative property of addit ion and state it as follows:

I),,

Commutative Property of Addition: If a and b represent any two numbers of arithmetic, then a b = b a.

+

+

The word "commutative" is suggested by the fact that either member of the equation a + b = b + a may be obtained from the other by interchanging (i .e., by commuting) the variables a and b. You have known ever since you learned the multiplication tables that 3 X 4 = 4 X 3, 6 X 7 = 7 X 6, and so on. We state this property as follows:

I),,

Commutative Property of Multiplication: numbers of arithmetic, then ab = ba.

If a and b represent any two

143

Operations with the Numbers of Arithmetic

Subtraction and division do not possess the commutative property. For example, 5 - 2 -;e. 2 - 5 and¾ -;e. l Another important property shared by addition and multiplication is illustrated by the following examples: (5 + 7) + 3 = 12 + 3 = 15 5 + (7 + 3) = 5 + 10 = 15

(7 X 5) X 3 = 35 X 3 = 105 7 X (5 X 3) = 7 X 15 = 105

.·. (5 + 7) + 3 = 5 + (7 + 3)

.·. (7 X 5) X 3 = 7 X (5 X 3)

Since similar statements can be made about the sums and products of any numbers of arithmetic, we have the

I),,

Associative Property of Addition: If a, b, and c represent any three numbers of arithmetic, then (a+ b) + c =a+ (b + c).

I),,

Associative Property of Multiplication: If a, b, and c represent any three numbers of arithmetic, then (ab)c

= a(bc).

Some operations are not associative. For example, division is not associative. This is illustrated by the following example: (36 + 12) + 3 = 3 + 3 = 1 36 + (12 + 3) = 36 + 4 = 9 .·. (36 + 12) + 3 -;e. 36 + (12 + 3) The commutative and associative properties sometimes help us to simplify computations. For example, finding 4 X 9 X 25 is easiest when we first find 4 X 25 and then multiply that product by 9. To see how the commutative and associative properties enable us to perform the operations in this order, we reason as follows: Reasons Statements 1. Definition of multiplication of 1. 4 X 9 X 25 = (4 X 9) X 25 three numbers 2. Commutative property of multi2. (4 X 9) X 25 = (9 X 4) X 25 plication 3. Associative property of multipli3. (9 X 4) X 25 = 9 X (4 X 25) cation 4. Arithmetic fact 4. 9 X (4 X 25) = 9 X 100 5. Arithmetic fact 5. 9 X 100 = 900 6. Transitive property of equality 6. 4 X 9 X 25 = 900 The proof above may easily be written in the form below.

(1) (2) (3) (4) (5) 4 X 9 X 25 = (4 X 9) X 25 = (9 X 4) X 25 = 9 X (4 X 25) = 9 X 100 = 900 The numbers above the equal signs refer to the corresponding reasons in the proof above.

144

CHAPTER

0

5

EXERCISES

1. Give reasons for (1) and (2) in the proof below. (1)

(2)

a+ b + c =(a+ b) + c =a+ (b + c) 2. Assuming that each first statement below is true, replace each "Why?" with either the name of a property or the statement of a property to make the conclusion (the second statement) true.

a. 9 + a is the sum of two numbers of arithmetic .·. 9+a=a+9 h. 7(3) is the product of two numbers of arithmetic .·. 7(3) = 3(7)

Given Why? Given Why?

c. (5 + 2) + 8 is the sum of three Given numbers of arithmetic .·. (5 + 2) + 8 = 5 +(2 + 8) Why?

d. (a · 3) · 4 is the product of three Given numbers of arithmetic .·.(a·3)·4=a·(3·4)

Why?

3. Each of the following computations is easier if we associate the numbers in a different way. On your paper indicate the easier way; then perform the computation.

a. (9 + 2) + 98 b.(18+25)+75

c. (37

X

2) X 5

d.95+(5+14)+6

e. 20(5

X

62)

f. 9 + (11 + 17) g. ½(20 X 28) h. 9999 + (1 + 457) ' 8 X (18 X 23 41) 1.

j. 4

X

(25 X 758)

4. Use the definitions of addition and multiplication , the commutative properties, or the associative properties, to do each of the following exercises in the easiest way you can find. On your paper indicate what you have done. a. 18 + 75

+2

h. ½X 5 X

4

c. 570 + 142 + 30

d. 15 + [(5 + 3) + 7] e. 901 + 84 + 7 + 3 + 16 + 99

f. 7¾+ 2½+ 2¾ g. 6 X (5 X ½) X 16 h. .5 x 1.97 x 2 i. (2 X 4) X (8 X 25) j. 8.57 + 74.9 + 2.43 + 25 .1

145

Operations with the Numbers of Arithmetic

333 ." One student rewrote X 982 before she performed the multiplication. What prop-

5. On a certain test, one exercise said , "Find the exercise as X

~~~

erty of multiplication tells us that she will get the correct product for the question, provided she makes no mistake in multiplication? Example.

Remembering that a· a can be written as a2 , use the properties of multiplication and the multiplication facts to write ¼· a · 8 · a · b in simplest form .

Solution.

By the commutative and associative properties we may write

¼·a · 8 ·a • b =

(¼ •8)(a • a)b =

2 a2 b

6. Use the associative and commutative properties of multiplication to write each of the following products in simplest form. a. (3 x)(2 x)

c. ½(3 a)(2 a)

e. (4 x)(2 y)

b. 4 · m · n · 8 · m

d. m(2 m)(6 n)

f. 8 · r · r · ¼

Evaluating Open Expressions

An algebraic expression containing variables can be evaluated when we know the values of the variables; that is, when we know the numbers which the variables represent. To do this we first use the substitution property of equality to replace each variable by its indicated value. This substitution produces a numerical expression. Then we perform the indicated operations to find a simpler name for the number represented by the numerical express10n. The process of substituting specified values for the variables of an open expression and finding a simple name for the number represented by the expression is called evaluating the expression. The following examples and exercises provide practice in the evaluation of algebraic expressions. Example 1.

Evaluate (3 y) 2 when y = 4.

Solution.

Substitu ting 4 for yin (3 y) 2 we have (3 y) 2 = (3 X 4) 2 = (3 X 4)(3 X 4)

= (12) (12) = 144.

Example 2. Find the number represented by the expression 6 xy 2 when x = 2 and y = 3. Solution.

Substituting 2 for x and 3 for yin 6 xy 2 we have 6 xy 2 = 6(2)(3)2 = 6(2)(9)

= (12)9 = 108.

146

CHAPTER

0

5

EXERCISES

1. If a = 2, b = 7, and c = 5, evaluate each of the following expressions. a. a +b

d .ab

g. 4c-a

b. b - c

e. abc

h. 15

j.3a+2b

k. 4(b + c)

C

f. 7 a

i. 9 - b

l. b+6c

2. What number is represented by ab if a = 7 and b = 2? if a = .3 and b = .4? if a=½ and b = J?

3. Evaluate rst when r = 3, s = 2, and t = 7; when r = 8, s = 9, and t = 0; when r = ½, s = 8, and t = j. 4. a3 means a X a X a. What number is represented by a3 when a = 2? when a= .2? when a = ½?

5. If m = 7, n = 6, and k = ½, find the number represented by each of the following expressions . a. m - n

d. mnk

g. 4m + 5 n

j. k(n + 9)

b. mn

4n e.3

h. 9- 6 k

k. kn+9

c. nk

f. 9k + 3k

l. k(n + 9) + m

i. 4(m + n)

6. If a = 2, b = 3, and c = 5, find the value of each of the following express10ns.

a. a 2

d. (ab) 2

g. (ac) 2

b. b2

e. (a +b)2

h. (a-1) 2

c. a 2 b2

f. b3

j. 4 b2 k. (4 b) 2 l. (4 ab) 2

i. ac2

7. If r = 4, s = 3, and t = .1, find the number represented by each of the following expressions. a. 3(r + s + 5)

e. t2(r + s)

b.

8 ts

f. rs t

c.

G)1

j. t3r2

8 r+ 2 s g. 19

k. sr 2

h. t 3

l. (sr) 2

d. t(r + s)

. rs

1. -2

8. The expression rt represents the number of miles that a plane travels int hours at the rate of r miles per hour. How many miles are represented by rt when r = 400 m.p.h . and t = 3 hours?

147

Operations with the Numbers of Arithmetic

L>

9. If the variable a represents the number of inches in the length of one side of a triangle, the variable b represents the number of inches in the length of a second side, and the variable c represents the number of inches in c the length of the third side, write an expression ~ that represents the perimeter of the triangle. (Recall that the perimeter of a triangle is the sum of the lengths of its sides.) Use the expression that you have written to find the perimeters of the triangles having the sides with the lengths below:

a. a= 4 in. b = 8 in. C = 6 in.

b. a= 4½ in. b = 6 in. C = 3½ in.

c. a= 47.0 in. b = 51.5 in. C = 45.2 in.

10. If a represents a number, write an expression that represents: a. The number increased by 2

b. The number decreased by 1 c. Four times the number

d. The quotient when the number is divided by 3 e. 5 more than twice the number

11. Evaluate the expressions that you wrote for Ex. 10 assuming that a= 12. 12. If x represents the number of years in Sue's present age, write an expression that represents:

a. Sue's age 3 years in the future

b. Sue's age 3 years ago c. Twice Sue's age

d. The age of Sue's brother if he is 4 years older than Sue e. The age of Sue's grandfather if his age is 4 years more than four times Sue's age

13. Find the age represented by each of the expressions you wrote for Ex. 12, assuming that Sue is now 14 years old.

14. Albert Einstein discovered that the amount of energy (in ergs) stored in any material is represented by the expression mc 2 where m represents the number of grams of the material and c represents the speed of light in centimeters per second . Find the number of ergs of energy stored in 2 grams of uranium. Light travels with a speed of 30,000,000,000 centimeters per second.

148

CHAPTER

5

Inverse Operations

From our understanding of addition we are able to define subtraction for the numbers of arithmetic in terms of addition. For example, we know that 10 - 7 = 3 means 10 = 3 7. In general, if a and b are two numbers of arithmetic such that a ~ b, we define a - b to be a number of arithmetic that we may represent by c such that c + b = a. Thus we have the following definition of subtraction for the numbers of arithmetic.

+

If a,

b, and c represent numbers of arithmetic and a b = a.

means c

+

~

b, then a - b = c (1)

Since the first of these equations says that a - b equals c, we may substitute a - b for c in the second equation. This gives us (a - b) + b = a. This statement says: If from a we subtract b and then add b to the difference, the result is a. In other words, adding b undoes the effect of subtracting b. Thus (5 - 2) + 2 = 5. How might you undo the effect of subtracting 25 from 70? of subtracting ¾from 5? The second of the equations in (1) says that c + b = a. If we substitute c + b for a in the first of the equations in (1), we have (c + b) - b = c. This statement says that subtracting b undoes the effect of adding b. Thus (5 + 2) - 2 = 5. How might you undo the effect of adding-½ to 16? of adding .002 to 6? Two operations such that each undoes what the other does are said to be inverse operations. Addition and subtraction are inverse operations. In general, we say: If x and

d are two numbers of arithmetic, then

(x+d)-d=x

and

(x-d)+d=x; x

~

d.

When we have discussed the negative numbers we shall remove the restriction x ~ d. For the present we must keep it because we cannot find x - d when x < d. From our understanding of multiplication we are able to define division for the numbers of arithmetic in terms of multiplication . For example, we know that 12-+- 3 = 4 means 4 X 3 = 12. In general, if a and bare two numbers of arithmetic such that b ,t- 0, we define ~ to be a number of arithmetic that we may represent by c such that c · b = a. Thus we have the following definition of division for the numbers of arithmetic. If a, b, and c represent numbers of arithmetic and b

c ·b

= a.

,t- 0, then ~

= c means (2)

149

Operations with the Numbers of Arithmetic

Since the first of these statements says that~ equals c, we may substitute

~ for c in the second statement. This gives us

This statement tells us that if we first divide a by b and then multiply the quotient by b, the result is a. In other words, multiplying by b undoes the effect of dividing by b. Thus (li)4 = 12. The second of the statements in (2) says that c(b) = a. If we substitute c(b) for a in the first statement of (2), we have c(b) b

-= c

This statement says that if we first multiply c by b and then divide the product by b, the result is c. In other words, dividing by b undoes the effect 3X5

of multiplying by b. For example, - 5 -

= 3.

Multiplication and division are inverse operations. In general, we say: If x and d are two numbers of arithmetic such that d ~ 0, then

and

Subtraction and division are binary processes, but sometimes we use such expressions as 7 - 3 - 2 and 12 + 4 + 3. If we write (7 - 3) - 2, we obtain 2; if we write 7 - (3 - 2), we obtain 6. These two results remind us that subtraction is not associative. Verify for yourself the fact that division is not associative either. Let us define subtraction and division when three or more numbers are involved. If a, b, c, and d are numbers of arithmetic such that a (a - b -

cl

2:

2:

b, (a - bl

2:

c and

d, then a - b- c a - b - c- d

= (a -

bl - c

= (a -

b - cl - d

To subtract when more than four numbers are involved, we continue this pattern. If a, b, c and d are numbers of arithmetic such that b ~ 0, c ~ 0, and d ~ 0, then

a + b + c = (a + bl + c a + b + c + d = (a + b + cl + d To divide when more than four numbers are involved, we continue this pattern.

150

CHAPTER

5

Solving Equations

The process of finding the solution set for an equation is known as solving the equation. From the beginning of this course you have been solving equations, but we have said little about how inverse operations and the properties of equality are used in the process. Let us now consider what we really do when we solve an equation. Example l.

Solve the equation~= 3.2 when x e A.

Solution.

PART

1. The expression ~ in the equation indicates that the num-

ber represented by xis divided by 7. If we multiply~ by 7, the multiplication will undo the division and give us x alone in the left member of the equation. (The left member of an equation is the entire expression to the left of the equal sign and the right member is the entire expression to the right of the equal sign.) If we multiply the left member by 7, we must also multiply the right member by 7 to comply with the requirements of the multiplication property of equality. We arrange our work as follows: 1.

2.

X

7= 3.2

1. Given

1(f) = 7(3.2) a. 7(~) = X

2. Multiplication property of equality a. If d and x represent numbers of arithmetic,

b. 7(3.2) 3. x= 22.4

= 22.4

d(~) =

x

b. Multiplication fact 3. Substitution property of equality have substituted x for 7(3.2) in Statement 2. )

(We

1(~) and 22.4 for

By the principles of logic which we studied in Chapter 4 we know that statements 1- 3 prove: If

X

7=

3.2, then x = 22.4. We must

now show: If x= 22.4, then~= 3.2. We do this by showing that when 22.4 is substituted for x in ~ = 3.2, the statement is true. PART

2.

1. 227.41= 3 .2

1. Substitution property of equality

2. 3.2 ;;;_ 3.2 2. Definition of division The solution set is {22.4}.

151

Operations with the N umbers of A rithmetic

The question mark over the equal sign in statement 1 means tha t we are asking whether the statement is true when x represents 22.4. The check mark over the equal sign in statement 2 indica tes that we have performed the indicated operation and have found that the statement ~ = 3.2 is true when x = 22.4. In Part 1 of such ·a solution we find the numbers which may possibly be solutions of the original equation. In Part 2 we determine whether these possible solutions are actual solutions. In solving equations you should show both Part 1 and Part 2 of your work. Example 2.

Find the solution set for x - 8½ = 10½ when x e A.

Solution.

PART

1.

1. X - 8½ = 10½ 2. (x- 8½) + 8½= 3. x = 19

PART

10½+ 8½

+

2.

821 =? 1021 10½ i 10½

1. 19 2.

1. Given 2. Addition property of equality 3. If d and x represent numbers of arithmetic, (x - d) d = x ; addition fact; substitution property of equality 1. Substitution property of equality 2. Definition of subtraction

The solution set is {19}. We have now shown in Part 1 that if x - 8½= 10½ has a solution, the solution is a member of the solution set of x = 19. We have also shown in Part 2 that 19, the only member of the solution set of x = 19, is a solution of x - 8½ = 10½. We did this by substituting 19 for x in x- 8½= 10½. Example 3.

Find the value of x for which 3 x

Solution.

PART

e C.

1.

1. 3 x = 86

2.

= 86 when x

\x= ~

6

3. x = 28J

1. Given

2. Division property of equality 3. If d EA and x EA, dd,; = x; definition of division for numbers of arithmetic; substitution property of equality

These statements tell us that if 3 x = 86 has a solution, the solution is a member of the solution set for x = 28!- But 28J, the only member of the solution set for x = 28J, is not a natural number. Therefore the solution set for 3 x = 86 when x E C is 0.

152

CHAPTER

0

5

EXERCISES

1. Solve each of the following equations, indicating the properties and operations used. Each variable represents a number of arithmetic. a.x+9=21

f.r-6=12

b. 3 X= 36

g. s - .2 = 18.2

c. 4 y = 11

h. x + 1.4 = 15.6

d. ~ = 27

i. .4 y = 20

3

e.

m

j. 1. = 65

2 = 7.5

.3

2. Solve each of the following equations: a. 4 y = ½; y E A

f. 8 y = 18; y E W

b. _!_ 11 = 3 .3 ·,

g. y + 16.5 = 16.5; y EA

zEA

h. y + 16.5 = 16.5; y EC

c. y- 9 = 14!; y EA

d. x - 7 = 0; x E C

i. .6 t = 2.4; t EA

e.t+75=150; tEW

j. y +17=19 .3 ; y EA

3. Solve each of the following equations. Each variable is a number of arithmetic. a. .5 x= 8

b.

i. 180 =

½x= 8

• 4

J• 1 =

X

c. -= 8 2

d. y+ 1.9 = 47 .3 e. 117 =

X -

83

f. m+84= 84

X

+ 5

2X

k. 3~0 = 1

I. 6 m = 1.2

o. f=r-f

P· .09

x=

81

r q. 41 = 6

r.

t

7=

0

g. y - 2¼ = 8¾

m. 14.4 = z - 6.0

s. z - 4f = 2¼

h. 17 + z = 21

n. 8f = y + 2J

t. x +47½= 84

= bh where b represents the number of units of length in the base of the rectangle, h the number of units of length in the height, and A the number of square units in the area. Find the length of the base of a rectangle whose height is 8 inches and whose a rea is 96 square inches .

Example 4 . The area of a rectangle is found by the formula A

153

Operations with the Numbers of Arithmetic

Solution.

PART

1.

1.A=bh; A=96; h=8 2. 96 = b(8) 3 96 = b(8) . 8 8 4. 12= b

1. Given 2. Substitution property of equality

3. Division property of equality 4. Definition of division; if d e A and

xd

x eA , -d= x PART

2. ?

1. 96 == 12(8) i 96

1. Substitution property of equality

2. 96

2. Multiplication fact

The solution set is {12}. Therefore the base is 12 inches long.

4. Find b, the length of the base of each of the rectangles whose measurements are given below.

a. A

=

12 sq. ft.; h = 4 ft.

b. A= 108 sq. 5.

c. A = 62.32 sq. in.; h = 8.2 in.

rd.; h = 9 rd.

Use the formula A below.

= bh

d. A= 12 sq. yd.; h = 1-g- yd.

to find h when b and A have the values shown

c. A= 68.58 sq. in.; b = 12.7 in

a. A = 175 sq. rd. ; b = 10 rd.

b. A= 220

sq. yd.; b = 15 yd.

d. A = 9 sq.

ft.; b = 2.5 ft.

Q

EXERCISES

Example 5.

Solve the equation x - c = 10 for x when x and c are numbers of arithmetic such that x ~ c.

Solution.

PART

1.

1.x-c=lO 2. (x-c)+c= 10+c 3. X= 10+ C PART

+ d = x.

2.

1. (10+ c)-

2.

1. Given 2. Addition property of equality 3. If d e A and x e A, then (x - d)

?

C

== 10

1. Substitution property of equality

10i 10 2. If d e A and x e A, (x + d) - d = x. The solution set is {x I x= 10+ c; x, c e A}.

6. Find the expression for x for which the equation x + a = 7 is true if a and x are numbers of arithmetic and a :s; 7. b 7. Solve y- 2 = 10; y

~

b 2;

b, ye A.

8. Solve the equation~= 12 for x when x and a are natural numbers. a

154

CHAPTER

5

9. In order to find an expression for x for which the equation ax= 12 is true when a and x are numbers of arithmetic, what restriction must be placed on a? Solve the equation for x.

10. To solve the equation of Ex. 9 for x when x and a are whole numbers , what restriction must be placed on a? Equations Whose Solutions Require More than One Operation

Solutions of each of the equations of the preceding section required only one inverse operation. The solutions of each of the equations of this section require more than one . To decide which of several operations to perform first , we choose the one which will make our work the simplest . Example 1.

Find t such that 35t - 7 = 23; t e A.

Solution.

PART

1. Since 7 has beeri subtracted from the expression 351, our first

step is to add 7 to each member of the equation. This gives: 1. 351- 7 = 23 2.

(35' -

1. Given

7) + 7 = 23 + 7

2. Addition property of equality

3. 35t=30

3.If d,xeA, then (x -d)+d=x ; addition fact Since 3 tin Statement 3 is divided by 5, we multiply both members 3I of 30 by 5. This gives:

s=

4.

s(351) = 5(30)

4. Multiplication property of equality

5. 3 t = 150

5. If~, x EA, then d(~) = x; multiplication fact Since t, in 3 t, is multiplied by 3, we divide both members of 3 I= 150 by 3. This gives:

6

lJ=

. 3

150

7. t=

so

PART

2.

6. Division property of equality

3

dx x; d 1v1s1on ' . . f act 7.If d,xEA,d=

1 3(50) - 7 J 23

.

2.

5

23

i

23

The solution set is {SO} .

1. Substitution property of equality

2. Arithmetic facts

155

Operations with the Numbers of Arithmetic

Example 2.

Solve the equation 5 x

Solution.

PART

+ 23 = 58;

x e A.

1.

1. 5 x+ 23 = 58 2. (5 x+ 23) - 23 = 58- 23 3. 5 X = 35

1. Given 2. Subtraction property of equality 3. If d, x e A, then (x+ d) - d = x; subtraction fact

4. Division property of equality

dx x ; d'1v1s1on .. 5. If d, x e A, t hend= fact

5. x= 7 PART

2. ?

1. 5(7) + 23 == 58 2. 58 i 58

1. Substitution property of equality

2. Arithmetic facts

The solution set is {7}.

0

EXERCISES

1. Solve each of the following equations. The variables represent numbers of arithmetic. a. 4

X -

h. 2 t+ 7 = 7

1 = 19

b. 3 y+ 2 = 17 c.

2x

S =

. 1.

10

1.1

X

l. ½X - 12 = 12 m. 4 y+4= 4½

+ 3.4 = 8.9

5r

i

i

q.--s=s 4

4

2

7m

S + 6 = 41

p. 9 + .6 t = 11

k. ~ = o

y+ .3 = 1.5

f. 7 m = 28

g,

15-~ 3

j. 125 m - 6 = 994

d. 18 = 5 x+ 3 e•. 4

o.

n. 7 + 2 y = 25

r.

st= 2 t - 8

S,

115

X -

.86

=

2t t. 5-18 = 7

2. Determine the solution set for each of the following equations. a. 3 y - 1 = 20; y e C

g.2.1 x +3=9.3; xeA

b. 4 y + 7 = 15;

h. 20½ = 4 x + ½; x e A

c.

2r

3 = 15;

y eC

re C

i. 39t + 6 = 87; t e A

d. ~ + 3 = 17; x e C

j . .4t-2= 14.4; teA

e. 6 t - 1 = 16; t e C

k. ;

f. 7 a + 1 = 28; a e W

l. 94 = 19 y- 1; ye C

x-

½= 10;

x eW

1.44

156

CHAPTER

5

3. Find the value of x for each of the equations below when a has the value shown at the right and x E A.

a. 4 x + 1 = a; a = 1

c. 4 x + 1 = a; a = 3

b. 4 x + 1 = a; a= 2

d. 4 x + 1 = a; a= 20

0

EXERCISES

4. Use the roster method to list the members of {x I 3 x + 9 = 15; x EA}.

t- =

5. Use the roster method to list the members of { x I 7 6. Solve the following equations.

XE

2

12; x EA}•

A.

x+ 5 -1 = 3

a. x+4 3 -12 -

c.

b. 3 X + 1 = 14 2

d. 3 x + 7 + 1 = 6 + 2

3

4

7. The formula F = ¾C + 32 represents the relationship between Fahrenheit and Celsius (centigrade) temperatures. In the formula, F represents the number of degrees in the reading of a Fahrenheit thermometer and C the corresponding number of degrees in the reading of a Celsius thermometer. Use the formula to find:

a. F when C = 15°

= d. C when F = c. C when F

b. F when C = 20°

104° 176°

8. The formula V = ½1rr2 h enables you to calculate V, the volume of a cone, when you know r, the measure of the radius of the base of the cone, and h, the measure of the height of the cone. Use 3.14 as the value of 1r in the formula to find the following:

a. V when r

b.

= 3 and h = 7

c. V when r = 4 and h = 9

d. h when V = 40.035 and r = 1.5

V when r = 5 and h = 12

9. Find the value of x which makes each of the following equations true . All of the variables represent numbers of arithmetic. ~4x=a

b. 3 x + b = 15; b :::;; 15 c.

ax

4

=12;a~0

d. 3 x + b = c; b :::;; c

e.

4x 5 =p

f,

¼x - C = 0

g.3bx=12;b~0

h. 6 x- a= 7 a

157

Operations with the Numbers of Arithmetic

Using Equations to Solve Problems

Equations help us to solve many problems involving numbers. Below we show some of the steps used in solving the problem: If Jim had $3 more than twice as much money as he has, he could buy a bicycle costing $45. How much money does Jim have? To solve the problem we reason as shown in the column on the left and write what is shown in the column on the right. We write:

We think:

The problem asks, "How many dollars does Jim have?" We choose the letter n (or any other letter) to hold a place for the answer until we can determine what it is. We can represent the number of dollars Jim needs by 2 n + 3 and also by 45. Since 2 n + 3 and 45 represent the same number, we can write the equation shown at the right. We now solve this equation.

1. Let n represent the number of dollars that Jim has. PART

1. Why? 1. 2 n + 3 = 45 2. (2 'n + 3) - 3 = 45 - 3 2. Why? 3. 2 n = 45 - 3 3. Why? 4. Why? 4. 2 n=42 5. Why? 5. n= 21 PART

2. ?

1. 2(21)+3....:...45 2. 45 ✓ 45

1. Why? 2. Why?

The solution set is {21}. Jim has $21. Notice that in solving the problem above we took the following steps: 1. We read the problem carefully to determine the information given and the question asked or the request made. 2. We chose a letter to hold a place for the answer until we could determine the answer. 3. We formed an equation by finding two expressions that represented the same number. 4. We solved the equation. 5. We stated the answer to the problem. Many mistakes in solving word problems are caused by forming the wrong equation. To make sure that this has not happened it is usually best to check the answer with the information given in the problem. To check our work above we might write: Is $3.00 more than twice $21.00 equal to $45.00? 2(21) = 42. 3 more than 42 is 45. Our answer is correct.

158

CHAPTER

5

Example.

When 5 is added to 7% of a certain number, the result is 82. Find the number.

Solution.

PART 1. Let x represent the number

1. .07 x+ S = 82 2. (.07x+5)-5=82-5 3. .07 x= 77 .07 X 77 4. = .01 5. x = 1100

m

PART

1. Why? 2. Why? 3. Why?

4. Why? 5. Why?

2. ?

1. .07(1100) + 5 == 82 2. 82 i 82

1. Why? 2. Why? The solution set is {1100}. The number is 1100.

Check.

7% of 1100 = 77. Since 77 + 5 = 82, our answer is correct.

Many of the exercises in the list below are so easy that you will be tempted to use short cuts in the solutions. While short cuts are good in many situations, you should not use them at this time. This is because you are learning a procedure for solving word problems. Unless you learn to use the procedure in easy problems, you will not be able to use it in difficult problems.

Q

EXERCISES

Use the plan of the examples above to solve each of the following problems.

1. When a certain number is multiplied by 7, the product is 161. What is the number?

2. This year Mr. Adams is three times as old as his son Randy . If Mr. Adams is 42 years old, how old is Randy?

Let x represent son's age ? X x represents father's age ? Xx= 42

3. This year the number of fatalities in automobile accidents is only twothirds of the number of such fatalities last year. If there have been 68 fatalities this year, how many fatalities were there last year?

4. The sum of a certain number and 9 is 75. What is the number?

5.

The sum of twice a certain number and 5 is 17. What is the number?

6. In any triangle the sum of the degree measures of the three angles is 180°. If the measure of one angle of a triangle is 51 ° and the measure of a second angle is 63°, what is the degree measure of the third angle?

51+63+x-?

159

Operations with the Numbers of Arithmetic

7. If the measure of one angle of a triangle is 43 ° and the measure of a second angle is 63°, what is the measure of the third angle?

8. If the sum of the measures of two angles is 90°, each angle is said to be the complement of the other. Find the complement of an angle whose measure is 51 °.

9. Find the measure of the complement of a 72° angle. 10. The measure of an angle is 54°. If this measure is 6° less than two-thirds of the degree measure of a second angle, what is the degree measure of the second angle?

11. Four per cent of a certain number is 72. What is the number?

12. Newspaper carrier Bill Hodges commented, "If I can get three more customers, I shall have twice as many as I had a year ago." If

Bill now has 45 customers, how many did he have last year?

Let x represent the number of customers Bill had last year Then ? X x = 45 + ?

13. The school librarian said, "We have ten more than three times as many books as we had five years ago." If the library now has 1510 books, how many did it have five years ago? () EXERCISES

·14. When John Kirk took his first job his grandfather said, "On my first job my salary was $26.00 per year less than 12% of yours." If John's grandfather's first salary was $550 per year, what is John's?

15. Don has a job which pays him $60 per week plus 5% of the value of the goods he sells during the week. If he was paid $77 .55 last week, how many dollars worth of goods did he sell during the week?

16. Jane has 95¢ and Pam has 55¢. How much must Jane give Pam so that she and Pam will have equal amounts?

17. Jane's algebra test grades to date are 80 and 100. What must she make on the next test to have an average of 90?

18. In one year 157,452 more automobiles visited Yellowstone National Park than visited Grand Canyon National Park. The sum of the number of cars visiting each park was 766,210. Find the number of cars that visited each park.

19. The perimeter of triangle ABC is 98 inches. If side AB is twice as long as side BC, and side AC is 2 inches longer than side BC, find the length of each side.

160

CHAPTER

5

Equivalent Equations and Equation Solving

You now know that solving an equation is a two-part process. In Part 1 you find numbers which may possibly be solutions for the equation. In Part 2 you determine whether or not the numbers found in Part 1 satisfy the original equation. The second part of the solution of an equation may be accomplished by another method. In Chapter 4 we showed that if a and bare open statements and a-+ b and b-+ a, then a and b have the same truth set; that is, a+-+ b. Let us now apply this idea to equation solving. First let us consider the following simplified version of the solution of the equation 3 x = 12. Example l.

Solve3x=12; xeA.

Solution.

PART

1. 1. 3x= 12 2. x= 4

PART

1. Given 2. Division property of equality

2.

1. Given

1.

x= 4 2. 3 x = 12

2. Multiplication property of equality

The solution set is {4}. Part 1 of this solution tells us that 3 x = 12-+ x = 4, and Part 2 tells us that x = 4-+ 3 x = 12. Therefore we know that 3 x = 12-+-+ x = 4. Now we see that we might have combined Parts 1 and 2 of the solution above into a single part as follows: Example 2.

Solve 3 x = 12; x e A.

Solution.

3 x = 12 +-+ x = 4. The solution set is {4}. (1) Division property of equality; multiplication property of equality.

(1)

Notice that we wrote two reasons for (1). One of these is the reason needed when we read 3 x = 12-+ x = 4. This is the reason from Part 1 of the proof above. The other reason is the reason needed when we read 3 x = 12 ~ x = 4, (i.e., x = 4-+ 3 x = 12). This is the reason from Part 2 of the original proof. Notice that the two reasons refer to inverse processes. Now let us consider an equation whose solution involves an extra step. Let us consider its somewhat abbreviated solution which follows : Example 3.

Solve 4 x

+1=

21; x e A.

161

Operations with the Numbers of A rithmetic Solution.

PART

1.

l.4 x +1=21 2. 4 x = 20

1. Given 2. Subtraction property of equality 3. Division property of equality

x= S

3. PART

2.

x= S 4x= 20

1.

2.

1. Given 2. Multiplication property of equality 3. Addition property of equality

3. 4 x+ 1 = 21 The solution set is {S}.

Statements 1 and 2 of Part 1 of this solution tell us that 4 x + 1 = 21 4 x = 20. Statements 2 and 3 of Part 1 tell us that 4 x = 20 ~ x = S. We may write these implications together as

~

4x+ 1 = 21 ~ 4 x= 20~x= 5.

In a similar manner Part 2 of the solution enables us to write: x= 5~4x= 20~ 4x + 1 = 21. Thus we have

4 X -f- 1 := 21

~

4 X = 20 ~

X

= 5.

Now, by the transitive property of implication,

(4 X + 1 = 21 ~ 4 X = 20 ~

X

= 5) ~ (4 X + 1 = 21 ~

X

= 5).

Had we realized in the beginning that 4 x + 1 = 21, 4 x = 20, and x = 5 were equivalent, we could have combined P,arts 1 and 2 of the solution above into a single part as shown below. Example 4.

Solve 4 x + 1 = 21; x E' A.

Solution.

4x+ 1= 21 ~ 4 x = 20~x= S. The solution set is {S}. (1) Subtraction property of equality; addition property of equality (2) Division property of equality; multiplication property of equality

(1)

(2)

0

EXERCISES

Solve each of the following equations by making use of the idea of equivalent statements.

1. 4 x = 36; x E' C 2. x + 1 = 15 ; x

E'

3.2 y =ll; C

y E' A

4.4 x +7=23;

X E' A

162

CHAPTER

5. 9 y - 4 = 14;

5

8. 16 = 7 + 3 s; s e A

y eA

6. 3 z + 2 z + 1 = 11 ; z e A

9. 4 y + 2 + 3 y = 6 + 3; y e A

7.21= x+8; xeA

10. ½x+4=10; xeA

Identity Elements

From experience you know that

2+0=2 0+3=3 7¾+0= 7¾ and so on. In other words you know that when a is any number of arithmetic a+ 0 = a and O+ a= a. Moreover, you know that O is the only number which has this property. The property may be stated as follows:

J:>,-

Addition Property of Zero : There is a unique number O such that a+ 0 and O + a = a when a E A.

=a

Since adding Oto a number leaves the number unchanged, we often call 0 the identity element for addition or the additive identity. Similarly, you know that 2(1) = 2 3(1) = 3

4f(l) = 4f and so on. Thus you know that when a is any number of arithmetic a(l) = a and (l)a = a. Moreover, you know that 1 is the only number which has this property. The property may be stated as follows:

J:>,-

Multiplication Property of One: There is a unique number 1 such that a(l) = a and (l)a = a when a e A.

Since multiplying a number by 1 gives a product which is the number itself, we often call 1 the identity element for multiplication or the multiplicative (mul'ti-pl1-ka'trv) identity. (

The Multiplicative Inverse

Let us agree to the following definition: If the product of two numbers is 1, then either is called a multiplicative inverse of the other. Thus, if a· b = 1, a is a multiplicative inverse of b and b is a multiplicative inverse of a. Since every definition is reversible, the definition above may be stated as: If two numbers are multiplicative inverses of each other, their product is 1. Thus, if a and b are multiplicative inverses of each other, a · b = 1.

163

Operations with the Numbers of Arithmetic

Now the question arises: Does every number of arithmetic have a multiplicative inverse? We know that zero has none, because we know that when x represents any number of arithmetic, 0 · x = x · 0 = 0. Thus 0 · x ~ 1 and x · 0 ~ 1. Experience has taught us that all other numbers of arithmetic do have multiplicative inverses. For example, ¾has the multiplicative inverse ½, since ¾· ½= 1. What is the multiplicative inverse for f? We make the assumption :

I>

1

If a is a number of arithmetic and a ~ 0, there is a number of arithmetic -

1 such that a • -

a

= 1.

1

a

This statement tells us that - is the multiplicative inverse of

1 a and a is the multiplicative inverse of -. a

a

Not only does every non-zero number of arithmetic have a multiplicative inverse, it has only one multiplicative inverse. We can prove that this is true. Recalling that the more important implications that can be proved are called theorems, we now state Theorem la. Note. We shall label our theorems as la, 2a, etc., the "a" serving to indicate that the theorems are proved for the numbers of arithmetic. ►

Theorem 1a: If a EA and a

~ 0, then a has only one multiplicative inverse.

Proof of Theorem la: If a EA and a ~ 0, then a has only one multiplicative mverse. This theorem will be proved if we can show that if a has two multiplicative inverses, they are equal.

1.aE A;a~0

1. Given

1 2. Let us assume that - and x are

a multiplicative inverses of a. 3. a • x = 1 1 1 4. - (ax) = - · 1 a a 5.

G·a)

6. 1 · X

x

= ; ·1

= -1 · 1 a

1 7. x=-

a

3. Definition of multiplicative in-

verse 4. Multiplication property of equality 5. Associative property of multiplication 6. Definition of multiplicative mverse 7. Multiplication property of 1

164

CHAPTER

5

When a theorem leads easily to the pr00f of another statement, the statement is often called a corollary. Theorem l a has the following corollary. Corollary 1a: If a EA and a

-,e 0 , then the multiplicative inverse of the mul-

tiplicative inverse of a is a.

There are many names for the multiplicative inverse of a number. For example, ¼, / 2 , and .25 are names for the multiplicative inverse of 4. The first of these is called the reciprocal of 4. The multiplicative inverse of a

l

is called the reciprocal of a. Since the multiplicative a inverse of any number a can be expressed in the convenient form l, we shall a use this symbolism when referring in general to the multiplicative inverse. The graph below shows how numbers and their multiplicative inverses are related. In the figure some numbers and their multiplicative inverses have been joined by double-headed arrows. written in the form

From your study of the diagram do you see that a number less than ¼ will have a multiplicative inverse greater than 4? Do you a_gree that a number less than io will have a multiplicative inverse greater than 10? What is the multiplicative inverse of 1? of O? ORAL EXERCISES

1. What is the identity element for addition? Why do you think we give it the name "identity element"? 2. What is the identity element for multiplication? Why do you think that we give it the name "identity element"? 3. When are two numbers multiplicative inverses of each other? 4. What is the special name for the multiplicative inverse of a number a that . wntten . . h e f orm -1 ?. 1s mt a

0

EXERCISES

1. Replace each of the question marks below with a numeral to form a true statement. a. 9 +? = 9

b. 9 · ? = 9

¼ d. V3(?) =V3 c. ¼(?) =

e. 4.987

+ ? = 4.987

f. 37½· ? = 37½

Operations with the Numbers of Arithmetic

165

2. Write the reciprocal of each of the following numbers if there is a reciprocal. All variables represent numbers of arithmetic.

a.

s

d. .6

g.

o

j. rs; r

h. 4.7

e. 1

1,3x+y;y~O

1. 5 8

t

0, s ~ 0

k.x+4

. 1

c.

~

y

3. List the reciprocals of the members of {l , 2, 3, 4, 5} in order of size from smallest to largest.

4. Write three names for the multiplicative inverse of each of the following numbers.

b. ¾

a. 7

d.

c. 2½

1

2 3

5. Write the multiplicative inverse of each of the following numbers if such an inverse exists, in each case expressing it in simplest form. The domain of the variables is the set of numbers of arithmetic.

a. 4

d. .4

g. 2½

b.f

e. 0

1 h. a

c.

¾

j. x + 7

k 3 x+ 2 .

8

f. 1

i, 3 x;

5

1. _7_; x and y X

~

0

x + y are not both 0

6. Give the value of x which makes each of the following equations true. a. ¼:i; = 1 c. .1 x = 1 e. 1rx = 1

b. ix= 1

d. 9 x= 1

f.

1

5 X

=

1

7

7. In the formula rt= d, d represents the number of miles traveled in t hours at r miles per hour. Find the time required to travel 1 mile at each of the rates shown below.

a. 6 miles per hour

c. 60 miles per hour

e. 300 miles per hour

b. 20 miles per hour

d. 120 miles per hour

f. 1000 miles per hour

8. Use the information of Ex. 7 to answer the following questions. a. As the rate increases, what change takes place in the time required to travel 1 mile? Nate. t

= l.

r b. Does your answer for part a check with information shown in the diagram on page 164? c. Does your answer to part a check with your experience?

166

9. If

CHAPTER

5

a and b are two numbers of arithmetic (a -;e- 0, b -;e- O), write:

a. The reciprocal of a; of b

b. The reciprocal of the sum of a and b; of the product of a and b c. The sum of the reciprocals of a and b

d. The product of the reciprocals of a and b

0

EXERCISES

10. Under what condition will

a - 1 have no multiplicative inverse if a is a natural number? Under what condition will x + y not have a multiplicative inverse if x and y are numbers of arithmetic?

11. Give the value of x which will make each of the following equations true. All variables are numbers of arithmetic.

a. (rs)x = 1; r -;e- 0, s -;e- 0

b.

(4 ab)x = 1; a -;e- 0, b -;e- 0

1 e. a+ b · x = 1 ; a and b not both 0

c. (1rr 2 )x = 1; r -;e- 0

12. What is the multiplicative inverse of 3? What is the multiplicative inverse of the multiplicative inverse of 3? What is the multiplicative inverse of t? What is the multiplicative inverse of the multiplicative inverse of -t? The Distributive Property

Suppose that you are painting the walls of a long corridor which is 9 feet high. Suppose that during one hour you paint 14 feet of the corridor wall and the next hour you paint 17 feet of the corridor wall. What is the total area painted? 17 14 One way to find the area painted is to forget that some was painted during one hour and some during the next. Under this plan the total area painted is a rectangle (14 + 17) feet long and 9 feet high. Thus (l) A= (14 + 17)9 = (31)9 = 279.

,1----------------

Another way to find the total area painted is to find the area painted each hour and then to add the two areas. Under this plan the total area is found as follows: (2) A = (14)9 + (17)9 = 126 + 153 = 279.

167

Operations with the Numbers of Arithmetic

From statements (1) and (2) we obtain the statement (14 + 17)9 = (14)9 + (17)9.

(3)

The property illustrated by statement (3) is very important. Let us reexamine the procedure above, this time using variables. Consider the rectangle at the right. If we let c c represent the number of units in its height and a+ b the number of units in its length, the area A of the rectangle is expressed by a+ b

I

----------' (4)

A= (a+ b)c

If we draw a line to divide the large rectangle into two smaller rectangles, one with the length a units and one with the length b units, we have the area of one small rectangle ex- c pressed as ac square units and the area of the other .._ expressed as be square units. Consequently the total area A in square units is expressed by a b

I I

___ ____ (5)

A= ac+ be

From statements (4) and (5) it seems reasonable to conclude that (a+ b)c We summarize these results by stating the following property.

= ac + be.

r>

Distributive property of multiplication over addition: If a, b and c are numbers of arithmetic, then

(a+ b)c = cc+ be.

Observe how the multiplication indicated by (a + b)c is distributed over the addition in ac + be. Note that by using the symmetric property of equality and the commutative properties of addition and multiplication , we can write the statement (a + b)c = ac + be in other forms . Some of these are ac +be= (a+ b)c c(a + b) = ac + be 'c(a + b) = ca + cb ca+ cb = c(a b)

+

We now have two ways to find such products as 5(3 the procedure: 5(3 + 4) = 5(7) = 35,

+ 4) .

We may follow

or we may follow the procedure: 5(3 + 4)

= 5(3) + 5(4) = 15 + 20 = 35.

Sometimes one of these forms is to be preferred over the other. Examples 1 and 2 below show this.

168

CHAPTER

Example 1.

Find 15(½ + ½).

Solution l.

15(½+ ¼) = 15( 135 +

Solution 2.

15(½ + ¼) = 15(½) +

5

= 15(/5 ) = 8 15 (¼) = 3 + 5 = 8 1\ )

Do you agree that Solution 2 is better in this case because it eliminates the necessity for changing½ and½ to a common denominator? Example 2.

Find 8(3 .76 + 6.24).

Solution l.

8(3.76 + 6.24) = 8(10.00) = 80.00

Solution 2.

8(3. 76 + 6.24) = 8(3. 76) + 8(6.24) = 30.08 + 49.92 = 80.00

Do you agree that in this case Solution 1 is preferable? Why?

0

EXERCISES

1. Find the common name for each of the following phrases in two ways. a. 4(7 + 2)

f. 13(11) + 13(19)

b. (9 + 11)14

g. 7(15.6) + 7(14.4)

c. 10(8. 7 + 9 .3)

h. 8 x 11 +

(½ + t)140 e. }(¼ + t)

d.

9

x 11

i. ½(150) + ½(SO) j. 14 X .7 + .7 X 16

2. Which of the following are true sentences? a. 2(6) + 3(6) = 5(6)

d. 8(5) + 6(8) = 14(5 + 8)

b. 18(14) = 9(12) + 9(2)

e .. 71(.8) + .2(.71) = 1(.71)

c. 12(½ + ¾)

= 12(½) +

12(¾)

Example 3.

Express 26 + 39 as the indicated product of two numbers.

Solution.

26 + 39 = 13(2) + 13(3) = 13(2 + 3) = 13(5)

3. Write each of the following sums as a product of two numbers. a. 12 + 16 c. 30 + 20 e. 16 + 48 g. 330 + 121 b.6+15

d.25+35

f.17+17

h.121+33

4. Answer each of the questions below. a. 9(7 - 4) = _? _ c. ½(6 - 2) 9(7) - 9(4) = _? _ 9(7 - 4) J 9(7) - 9(4) b. 8(10) - 8(6) = _? _ 8(10 - 6) = _? _ 8(10) - 8(6) J 8(10- 6)

= _? _ ½(6) - ½(2) = _? _ ½(6 - 2) J ½(6) - ½(2)

d .. 7(1.2) - .7(.2) = _? _ .7(1.2 - .2) = _? _ .7(1.2) - .7(.2) J .7(1.2 - .2)

Operations with the Numbers of Arithmetic

169

e. Do you agree that when a, b, and care numbers of arithmetic such that b ~ c, it appears to be true that a(b - c) = ab - ac? This statement can be proved. Perhaps you would like to try writing such a proof. Hint . a[(b - c) + c] = ab. Why?

5. Write reasons for each of the statements of the following proof. Prove that a(b + c + d) = ab+ ac + ad. Proof. a(b + c + d) = a[(b + c) + d] Why? =a(b+c)+ad =ab+ ac + ad

(The brackets [ ] serve the same purpose as parentheses.) Why? Why?

6. Do you think a multiplication can be distributed over an addition containing more than three addends?

7. Use the distributive property to find a numeral to replace each"?". a. 8(4 + 7 + 2)

= 8(?) + 8(?) + 8(?) b. 7(5 + 1 + 3 + 11) = 7(?) + 7(?) + 7(?) + 7(?) c. 9(4) + 9(6) + 9(10) = 9(? + ? + ?) Equivalent Expressions

We have said that a numerical expression is a numeral or an arrangement of numerals and symbols of operation that names a number. Two numerical expressions that name the same number are said to be equal. Thus, since 2 + 2 and 3 + 1 each name the number 4, we may write: 2 + 2 = 3 + 1. Moreover, if two numerical expressions are known to be equal, we know that they name the same number. Ilan algebraic expression contains one or more variables, it is an open expression. Let us agree tl, the following definition: Two open expressions are equivalent if and only if they become equal numerical expressions when each variable is replaced by any number in its replacement set. Thus, if both a and b have the set of all numbers of arithmetic as their replacement sets, we may write a+ b = b + a because, regardless of the number chosen for a and for b, a+ band b + a represent the same number. Moreover, since it is known that a+ b = b + a, it is known that regardless of the values chosen for a and for b, the two open expressions a+ band b a represent the same number. The two expressions 2 a+ 1 and a+ 3 are not equivalent because they do not always represent the same number. For example, although it is true that 2 a+ 1 =a+ 3 when a= 2, 2 a+ 1 ~ a+ 3 when a= 3.

+

170

CHAPTER

0

s

EXERCISES

1. Which of the following expressions are equivalent when a, b e A? e. ~; ~

a. a+ a; 2 a

(a ,t- 0, b ,t- 0)

b. 2 a + b; b + 2 a c.2(a + b) ; 2a+2b

d. ab;

3

g. a(b + 3); ab+ 3 h. ab; a+ 2

2. Copy the expressions that you find in the left column below. Beside each expression write the equivalent expression, or expressions, that appear in the two columns on the right.

a. 3 r

(1) 9 + r

(7) s + r

b. r + s c. 9 r

(2) 8(r+ 1g2 s)

(8) s + s + s

(3) (t + 3)r

(9) r + r +

d. r(t + 3)

(4) r(9)

(10) t(r + s)

e. rt + st

(5) r(t + s)

(11) 5 r + 4 r

f. 8 r + 12 s

(6) 4(3 s + 2 r)

(12) ts + tr

r

3. In each of the following equations x and y represent numbers of arithmetic. State whether the two members of the equations are equivalent expressions. If they are not equivalent, state the value of x or y for which the expressions do represent the same number.

a. x+ 2 = 2 +x

g. x + x= 2 x

b. xy = yx

h. (7 + y)7 = 49 + 7 y

c. 3(x+4) = 3 x+ 12

i.4 (3x) =12 x

d. 12 y = 10 + 2 y

j. 4(3 x) = 12

e. 9 X

=

k. y + .9 = .9 + y

18

f. (x + 4) + y =

X

+ (4 + y)

l. 9(x + 2) = 9 X + 18

Using the Distributive Property to Write Equivalent Expres sions

Often we want to change an algebraic expression into an equivalent algebraic expression having a different form . There are many ways to do this. In this section we shall consider changes that may be mad e by use of the distributive property.

171

Operations with the Numbers of Arithmetic Example 1.

Write the indicated product a(b + 5) as an indicated sum.

Solution.

a(b

+ 5) =ab+ a(5) = ab +

Distributive property Commutative property of multiplication.

5a

Note. It was not necessary to change a(5) to 5 a, yet we usually prefer having the numerals appear before the variables. We left ab unchanged because we find it convenient to have variables appear in alphabetical order. Example 2.

Write 11 k + 11 w as an indicated product.

Example 3.

+ 11 w = 11 (k +w). Distributive property Write 11 t + 31 tin simpler form

Solution.

11 t + 31 t = (11 + 31)t

Solution.

11 k

= 42 t

Distributive property Addition fact

Example 4.

Write the expression 5 x + 7 y + 3 x + 8 y in simpler form.

Solution.

5 x +7 y + 3x + Sy = [(5 x + 7 y) 3 x]

+

+8 y

=[5 x + (7y+3 x) ]+Sy =[5x + (3x + 7y)]+Sy =[(5 x + 3x) +7 y] +S y = (5 x +.3 x) (7 y + 8 y)

+

= (5 + 3)x + (7 + 8)y = 8x+ 15 y

Definition of the sum of two or more numbers Associative property of addition Commutative property of addition Associative property of addition Associative property of addition Distributive property Addition fact

You have seen in this example that a combination of the definition of addition of two or more numbers and the associative and commutative properties of addition enables us to justify each step from 5 x + 7 y + 3 x + 8 y t o the equivalent form (5 x + 3 x) + (7 y + 8 y) . In fact, the definition and rules mentioned permit us to add the terms of an indicated sum in whatever order we find most convenient. You may shorten the solution shown here by going from 5 x + 7 y + 3 x + 8 y to (5 x + 3 x) + (7 y + 8 y) in one step. Be sure, however , that you are able to justify your work as we have done if you are called upon to do so. Example 5. Solution.

Simplify x + x + x. x + x + x = (l)x + (l) x + (l )x = (1 1 l )x =3 x

+ +

Multiplication property of one Distributive property Addition fact

172

CHAPTER

Example 6. Solution.

5

Write 5 a(x + 3 w) as an indicated sum. The closure property of multiplication allows us to consider the indicated product 5 a as a single number. Consequently, 5 a(x+ 3 w)

= 5 a(x) + 5 a(3 w) = 5 a(x) + (5 · 3)(a • w)

Distributive property Associative and commutative properties of multiplication Multiplication fact

=5ax+15aw Example 7.

Write the indicated sum 5 xy + 5 xz as an indicated product.

Solution.

5 xy + 5 xz = (5 x)y + (5 x)z

Definition of multiplication of three numbers Distributive property

= 5 x(y + z)

Note . We might have so proceeded that we obtained 5(xy+ xz) or we might have so proceeded that we obtained x(5 y + 5 z), but the procedure we have used is the preferred one.

+ 2?

Example 8.

Write the indicated sum 24 x

Solution.

24 x + 21 y = (3 · 8) x + (3 · 7) y = 3(8 x) + 3(7 y)

as an indicated product. Multiplication fact Associative property of multiplication Distributive property

= 3(8 x+ 7 y)

0

EXERCISES

1. Write the indicated products as indicated sums. a. 4(a + 7)

f. 1(7 + 5 + t)

b. r(s + t)

g. (25 m + 100 n)-!

c. (9 + a)5

h. 18(½ a+½ b + / 8 c)

d.6(½ x+½y)

+ .01 y + .001 z) j. 0(15 a+ 19 b + c)

e. ¼(20 a + 16 b)

i. 100(.1

X

2. Write each indicated sum as an indicated product. a. 4 x+ 8 y

f. 6 m+ 6

b. ½a+½ b

g. 6 a+ 15 b

c. 9 m+4m

h. 15 m + 10 n + 20

+ rt

i. 2 a+ 14 b + 16

d.

r

e. ab+ a

j. ½r + ½s

173

Operations with the Numbers of Arithmetic

3. Write each of the following expressions in a simpler form. a,

9t+ 2t

g. 3 X + 7 y + 4 X + 2 y

b. .6 a + .4 a

h.

c. ½a+¾ a+¼ a d . m+m+m

i. (9 y + 12 y)½ . (16 x +n 1 J• Y + 81 x )2 4 k. ½x + ½x + ¼x 1. .25 t + .375 t + .125 t

e. 2 X

+X +5 X

f.4 y + y + y

½m + J m

4. Combine as many terms as you can in each of the following expressions.

f. 4 a + 2.5 b + 3 e + 3.5 b g. 14 X + 7 y + 5 X + 9 Z h. ½r + ; r + 7 s + 4 s i, 7 X + 7 y + 9 + 4 X + 4 y + 4 j. 3.2 + 4 a+ 16 + 7 a+ 8 b

a. 6 a+ 3 b + 9 a+ 4 b

b. ½a + ½b + J a + ½b

+ 3 y + 2.8 X + 2 y d. m + n + 3 m + 6 n C• . 6 X

e. r + s + 4 s + 3 r

5. Write each of the following indicated products as an indicated sum. a.9a (2 + 3b) b. 4 x(6 y + 2)

d.4 x(x +2) e. ½ab (6 + 3 d)

c. 5 rs(t + 3)

f. .1 mn(lO + 100 t)

g.27 (½a+Jb) h. .625(8 y + 24 z)

i. ab(a + ab + b)

6. Write each of the following sums as an indicated product. a. 5 ab+ 5 ae

c. 9 mn + 9 m

e. 4 abed+ 4 abe g. 15 xy + 20 xz

b. 4 rs+ 4 rt

d. 3 rs + 6 rt

f. 4 ab+ 12 ae

h. ½ab+½ be

+ 5)(x + 3).

Example.

Use the distributive property to find (x

Solution.

x and 5 are numbers of arithmetic. By the closure property of addition for the numbers of arithmetic, x 5 is a number of arithmetic. This means that we may treat (x 5) as we would 4 or 7 or any other number of arithmetic. Thus we have

+ +

(x + 5)(x+ 3) = (x + 5)x+ (x + 5)3 = x2 + 5 X + 3 X + 15

= x2 +

(5 + 3) x + 15

= x2 + 8 x +

15

Distributive property Distributive property and multiplication facts Distributive property Addition fact

7. Use the distributive property to find the following products.

a. (x+2)(x+3) b. (x +4)(x +6) c. (y +l )(y +2)

d. (a+6)(e+3)

g. (2+x)(x +3)

e. (x + y)(x + y)

h. (a+b)(e+d) i. (x+l)(y+l)

f. (a +b )(a +4)

174

CHAPTER

5

Using the Distributive Property in Equation Solving

The distributive property is useful in solving equations as shown by the following examples. Example 1.

Solve 8 x + 4 x = 24.

Solution.

PART 1. 8 x+ 4 x= 24 (8 +4)x=24 12 x= 24 x= 2

Given Distributive property Addition fact Division property of equality

2. ? 8(2) + 4(2) == 24 ? 16+ 8 == 24 24i 24 The solution set is {2}. PART

Example 2.

Solve 3 y + 4 + 7 y = 24.

Solution.

PART

Substitution property of equality Multiplication facts Addition fact

1.

3 y + 4+ 7 y= 24 3 y+ 7 y+4= 24 (3 + 7)y+ 4= 10 y +4= 10 y= y=

24 24 20 2

Given Associative and commutative properties of addition Distributive property Addition fact Subtraction property of equality Division property of equality

2.

PART

?

3(2) + 4+ 7(2) == 24 ? 6+4+ 14= 24 24 :;I,, 24

Substitution property of equality Multiplication facts Addition facts

The solution set is {2}. Example 3.

Find the value of y for which 9 y

Solution.

1. 9 y= 9y+2y= (9 + 2)y= 11 y=

= 55 -

PART

55- 2 y (55-2y)+2y 55 55 !!..2'._ 55

Given Why? Why? Why? Why?

11 - 11

y=5

Why?

2 y is a true sentence.

175

Operations with the Numbers of Arithmetic PART

2. ?

9(5) = 55 - 2(5) ? 45 = 55- 10 45 i:45

Why? Why? Why? The solution set is {5} . The equation is true when y = 5.

0

EXERCISES

1. Solve each of the following equations. The variables represent numbers of arithmeti c.

+ 3 X = 15 b. 39 = 8 r + 5 r

g. x+x+x= 45 h. 96 = 3 r + 2 r + r

c. 18 y - 2 y = 64 ·

i. ½y + ¾y + ¼y = 15 j. .1 X + .2 X + X = 3.9 k. .6 x + .9 x + .3 x = 5.4

a. 2

X

d. ½m+¼m= 24

+ .8 m = .88 f. 5 X + 4 X + 2 X = 99

e. .3 m

l. 3 = ½b+ ½b+ fb

2. Find the value of the variable for which each of the following equations are true. The variables are numbers of arithmetic.

a. 4 X + 5 X

-

1 = 17

e. 15

= 8y-

+5

3y

b. 6 x + 7 x + 9 = 35

f. 3 X + 7 + 2 X = 42

c. 10 y - 3 y - 4 = 24

g. 10 = 4 + a + 2 a h. 7 x + 1 + x + x = 20

d.

f x + ½x + 7 = 12 +

2

3. Solve each of the following equations. The variables are numbers of arithmetic.

a. 8 X = 24+ 2 X

e. 9 y = 7 y + 48

b. 4 y = 35 - 3 y

f.

c. 11 r

= 2.6 -

2r

d. 6 a+ 2 a= 12 + 5 a

½y = 8 + ¼y

g. .3 r + .4 r + r

= r + .14

h. 9 y + 2 y = 20 + 3 y + 4

4. Find the truth set for each of the following equations. The variables represent numbers of arithmetic. a. 3 (x + 5)

= 60

b. ½(6 r + 8) =19 c. ½(x + 6) = 8 d. 9 x + 2 (x + 1) = 24 e. 4(" + 7) = 2(x + 15)

f. 4(a +

1) + 3(a + 2)

= 94

g. 6 y + 8(y + 3) = 4 y + 74 h. (r + 8) + (r + 6) = r + 25 i. 7 s = 0(s + 14) j . .4 b + .2(b + 3) = .5(b + 2)

17 6

CHAPTER

5

Example 4.

The perimeter of a rectangle is 70 inches. If the length of the rectangle is 5 inches greater than the width of the rectangle, what are the length and width? (Perimeter is the sum of the lengths of the sides.)

Solution.

PART

1.

Let w represent the number of inches in the width of the rectangle. Then w 5 represents the number of inches in the length of the rectangle.

+

= 70 4w+ 10= 70 4w=60 w= 15

w + (w + 5) + w + (w + 5)

PART

w

2. ?

15+ 20 + 15+ 20 == 10 70 ;i,_ 70

w+S

The solution set is {15}. The width of the rectangle is 15 inches and the length is 20 inches.

5. The perimeter of a rectangle is 62 inches. If the length of the rectangle is 3 inches greater than the width, what is the width?

6. The length of a rectangle is twice the width of the rectangle. What are the length and width if the perimeter of the rectangle is 48 rods?

7. The sum of a certain number and twice the number is 81. What is the number?

8. The total cost of a raincoat and umbrella is $20.00. The raincoat costs three times as much as the umbrella. Find the cost of each.

9. The sum of a certain number and its successor is 109. What are the numbers?

10. The successor of a certain number when added to twice the number produces 61. What is the number?

11. If three times a certain number equals the number increased by 8, what is the number?

12. A man distributed $515 among his three sons so that the eldest received twice as much as the youngest and the middle son received $15 more than the youngest. How much did each receive?

0

0

177

Operations with the Numbers of Arithmetic

13. a. How many cents are there in 1 quarter? in 2 quarters? in x quarters?

b. How many cents are there in 1 dime? in 2 dimes? in

x

+ 3 dimes?

c. How many cents are there in x quarters and (x + 3) dimes?

d. Write an equation to state that the number of cents in x quarters and (x + 3) dimes equals the number of cents in $2.05.

Example 5.

In his pocket Chuck has a collection of dimes and quarters. If the value of the coins is $2.05 and there are three more dimes than quarters, how many dimes and how many quarters has Chuck?

Solution.

PART

1.

Let x represent the number of quarters that Chuck has and x + 3 represent the number of dimes that he has. Each quarter is worth 25¢ and each dime is worth 10¢. Then 25 x + lO(x + 3) = 205 25 X 10 X 30 = 205 35 x + 30= 205 35 x= 175 x =5

+

PART

+

2. ?

25 (5) + 10(5 + 3) ='= 205 ? 125 80 ='= 205 205 :f:. 205

+

The solution set is {5}. Chuck has 5 quarters and 8 dimes . To make sure that we made no mistake in forming the equation we check our answer against the information in the problem, as follows: Check.

The value of 5 quarters is $1.25 The value of 8 dimes is .80 The total is $2.05

14. In her purse Mary has nickels and dimes whose total value is $1.10. If she has 2 more dimes than nickels, how many dimes and how many nickels has she?

15. Tickets for the school play cost $.75 for adults and $.50 for pupils . If Mike sold five more tickets for adults than for pupils and collected $16.25, how many of each kind did he sell?

178

CHAPTER

5

16. Tom has twice as many nickels as quarters and 3 fewer dimes than quarters. If the value of his coins is $1.95, how many of each kind of coin has he ?

17. A man invest s some money at 5% and twice as much money at 4%. If the yearly income from the two investments is $455, how much is invested a t each rate?

18. A man invested a certain amount of money a t 5% and $1000 more than thi s amount at 6% . H ow mu ch did he invest a t each rate if his yearly income from the t wo investments is $335? Multiplication Property of Zero

In arithmeti c we learn that 5 X O= 0, 483.7 X O= 0, 0 X ¾= 0, et c. We assume that if a is any number of arithmetic, a· 0 = 0. We call this property the multi plication property of zero. We no longer need t o be satisfied with merely assuming this property of zero. Our previous assumptions, our definitions and undefined terms, and the st atements we have already proved true, show us that this property can be proved. ~ Theorem 2a: If a is a nu mber of arith metic, a • 0

=

0.

Proof of Theorem 2a: If a is a number of arithmetic, a · 0 = 0. 1. 1 and a are numbers of arithmetic

2. 1 + 0 = 1 + 0) = a· 1

3. a (l

+

4. a · 1 a · 0 = a · 1 5. a+ a· 0 = a 6 . . ·.a· 0

=0

1. Definiti on of the numbers of arith-

metic and given 2. Addition property of zero 3. Multiplication property of equality 4. Distributive property 5. Multiplication property of 1 6. Since O is the only number which has the property that its sum with any number is the number itself , we must recognize a · 0 (step 5) as another name for 0.

179

Operations with the Numbers of Arithmetic

Theorem 2a may also be stated as follows: If a and bare two numbers of arithmetic such that a = 0 or b = 0, then ab = 0. In arithmetic you also learned that the converse of Theorem 2a is true. In other words you learned that the following statement is true:

►

Theorem 3a: If a and b are numbers of arithmetic such that ab= 0, then

a= 0 orb= 0.

In the case of Theorem 3a, as in the case of Theorem 2a, we do not need to rely upon experience alone to conclude that the statement is correct. The properties that we have already established and the definitions that we have agreed upon can be used t o prove the theorem. The proof follows.

Proof of Theorem 3a : If ab = 0, then a= 0 orb= 0. (a EA, b EA) There are two possibilities for a. E ither a = 0 or a ~ 0. If a = 0 the theorem is true. If a follow s:

~ 0, then a has a multiplicative inverse ! and we reason as a

0~! · 0~\ ·(ab) ~)(.!. · a) b ~ 1· b ~ b a

a

a

Thus if a ~ 0, then b = 0. Therefore at least one of the numbers a orb is zero. This means that the disjunctive statement "a= 0 or b = 0" is true and our proof is complete. Reasons:

(1) Multiplication property of zero.

(2) Substitution property of equality (We are given that ab= 0.) (3) Associative property of multiplication (4) Definition of multiplicative inverse

(5) Multiplication property of one

Theorem 3a has a useful contrapositive. Recall that we form the contrapositive of the statement "If p, then q" by contradicting and exchanging the hypothesis and conclusion to obtain "If,,._, q, then ,,._, p." The contradiction of "ab= 0" is "ab ~ 0." The contradiction of the disjunctive statement "a= 0 orb= 0" is "a ~ 0 and b ~ 0." Thus we have

►

Contrapositive of Theorem 3a: If a

~

0 and b

~

0, then ab

~

0.

In other words, if neither of two numbers is zero, then their product is not zero.

180

CHAPTER

0

5

EXERCISES

1. Write a simpler name for each of the following products. The variables represent numbers of arithmetic. a. 0 X 794

e. (3.5 - 3.5) (7 .5 - 7 .5)

b.17½(8-8)

f.3(m-m)

c.

i50 (10-5)

g. i(6 a - 5 a) h. (r - r) (r - s)

d. 0(1,000,000)

2. In each of the following exercises a and b represent numbers of arithmetic. a. If ab= 0 and a b. If ab= 0 and b

~

0, what can you say about b?

~

0, what can you say about a?

c. If ab= 0, can both a and b represent zero?

d. If ab

~

0, can a represent O? can b represent O?

3. Find the value of each of the following expressions if a= 4, b = .5, and c = 0.

+

e. a(b + c) f.c(a+b)

a. a2 c2 b.16c 2

c.

b14 + 14 c

g.10(a+b+c)

h• a~+Eb

d. 18 abc

4. In which of the following equations must x = 0 to make the equation true? a. ½ x = 0

c. ax = 0

e. 0 x = 0

b. 37984.5 x = 0

d. 9(x - 7) = 0

f.

:Jf55 x = 0

5. In each of the following equations the variables represent numbers of arithmetic with the restrictions noted at the right of each exercise. If the equation is always true, write T; if always false, write F; and if true only under certain conditions, state the conditions.

= 4, as 8 f. m (8 - a) = 0; m = 0, a < 8 g. (9 - b)(4 - c) = 0; b < 9, c < 4 h. (5 - x)(6 - y) = 0; x < 5, y S 6

a. ab= 0; a~ 0, b ~ 0

b. rs = 0; r = 0, s

e. m(8 - a)= 0; m

~ 0

c. (7 - 7)s = 0; s ~ 0

d. (½- ¼)r = 0; r

~

0

6. Find 9874 X ¾X 847 XO X 636¾. 7. If (a - b)(c - d) = 0 when a~ band c statements.

(a-b)=0or(c-d)=0 a= b or c = d

Why? Why?

~

d, give reasons for the following

181

Operations with the Numbers of Arithmetic

Expressing Division by the Multiplicative Inverse

You learned in arithmetic that multiplying by½ produces the same result as dividing by 2, multiplying by ½produces the same result as dividing by ¾, et c. In other words, multiplying by the multiplicative inverse of a number produces the same effect as dividing by the number. Many of you will be interested in the proof that this is true.

l)i,-

Theorem 4a: If a and b are numbers of arithmetic and b ,6- 0, then

f= ~ •

a.

Proof of Theorem 4a: If a and bare numbers of arithmetic and b ,6- 0, then a 1 b=b. a. I. If

2.

3.

4. 1.

5.

d (~)

i (b ·~) = l· G·b) ~ = !· a

then

= x.

a

2. Multiplication property of equal-

a

3. Associative property of multipli-

ity cation 4. Definition of multiplicative inverse 5. Multiplication property of one

1

b = b. a

a

d, x EA and d ,6- 0,

1

b = b. a

The division property of equality was stated as follows: If a, b, c, and d are numbers of arithmetic such that a= b, c = d, c ,6- 0, and d ,6- 0, then a = d. b N ow we see, h owever, t h at ~a = d b.1s eqmva . 1ent to ~·a 1 1 · b an d =d

~

thus may be considered as a case of the multiplication property of equality.

0

EXERCISES

1. Express each of the following indicated quotients as an indicated product and each indicated product as an indicated quotient . a. 14 + 5

b.

1

½

b2

c.C

d. x+9 x+3 e. ½(7) 1

f. - · 8 a

1

g.-·b

a+ c

h. -

1

r+s

(r+8)

182

CHAPTER

5

2. Which of the following statements are true?

a.

1

x+ y

1

d. a + 5 = _1_ (a + 5)

2 X + 2 y = -2-

a + 2 a+ 2 1 1 r+s e. 3 r + 4 s = -7-

b. 3x14b =i (3x+4b) c.

4 rst

4

f.

=rst

7ry2

+ 7r = y2 + 1 7r

3. Write in two ways: a. one fourth of c

c. one fifth of (3 x + y)

b. one third of (x + 4)

d. one half of 1rr 2

4. Solve each of the following equations.

a.

x

+5 3 -_

d.

12

½(x -

5)

=0

x- 8 e.--=0

b. ¼(x + 2) = 10 7x - 1 c.---= 8

g. 7 y + 5 y = 0

h. m+m+m= 18

4

f. 4 x + 8 x= 96

6

Fractions

Recall that a fraction of arithmetic is the indicat ed quotient of two num. h met1c, . prov1"d e d t h e d"1v1sor . . not zero. Th us ""j' l l51.½' VU 1r b ers of a~1t 1s 7 ✓- ' 3 ~ 3 are fract10ns. 3 In arithmetic we learn that X ½= / 1 , ½X f 1 = is, etc. In the sent ence "t X ½= ,fT," is the reciprocal of 7, ½is the reciprocal of 3, and / 1 is the reciprocal of the product of 7 and 3. Thus we may say : The product of the reciprocals of 7 and 3 is the reciprocal of the product of 7 and 3. Stating thi s idea so that it applies to any two non-zero numbers of arithmetic we have:

+

+

►

Theorem Sa: If a and b are numbers of arithmetic such that a ',t- 0 and

b ',t- 0,

1 1 1 = -. a b ab

then - • -

Proof of Theorem Sa: If a and bare numbers of arithmetic such that a ',t- 0 1 1 1 and b ',t- 0 then - · - = -.

'

a b

ab

. 1· . . 1 I. a h as t h emu l tip 1cat1ve inverse -;

a

b has the multiplicative inverse

i

1. Why?

183

Operations with the Numbers of Arithmetic

1 1 2. a · - = 1 · b · - = 1 a ' b 3.

2. Why?

(a· ~)(b ·i) = 1 · 1 ab G ·i) = 1

3. Why?

4. : .

4. Why?

5. Since a and b represent non-zero numbers, ab ,6- 0.

5. Why?

6. ab has the multiplicative inverse 1b. 1 a 7. ab· - = 1 ab 1 1 1 8. From steps 4 and 7 we conclude - · - = -· a b ab

6. Why? 7. Why? 8. Why?

Multiplication of Fractions

In arithmetic we learn that the product of two given fractions is obtained by forming a new fraction whose numerator is the product of the numerators of the given fractions and whose denominator is the product of their de. F' or example, -2 X -1 = nommators. 3 5 write a rule for multiplying any two Theorem 6a.

I),,,-

2x1 - = -2 • 3 X 5 15

u smg ·

'bl es, we can vana

fractions of arithmetic. Let us call this

Theorem 6a: If a, b, c, and dare numbers of arithmetic such that b ,6- 0 and

d -;t- 0, then

a

c

ac

b X d= bd.

Proof of Theorem 6a: If a, b, c, and d are numbers of arithmetic such that

b -;t- 0 and d -;t- 0 then !!'. X !:. = ac_ ' b d bd

!!'.

b

Reasons:

(1) Why? (2) Why? (3) Why? (4) Why?

X

!:.

d

~ (a •!)(c •!)d C;} ac (!b .!)d ~ ac (1-) ~) bdac b bd

184

CHAPTER

Example 1.

Find¾ X}.

Solution l.

3 S 3XS 3XS 4X9=4X9=3X3X4

3

S

S

s

S

3 X 3 X 4 = 1.X 12 = 12

Sometimes much of the work carried out in Solution 1 is done mentally, with the results written down as shown in the following solution.

Solution 2.

1 3 S ~ S lXS S -x---x-----4 9 4 If 4 X 3 12 3

Example 2.

Find 7 X Sf.

Solution.

7 43 7 X 43 301 7 X Sf= 1 X 8 = 1 X 8 = 8

Example 3.

. c2 5 d2 Fmd - X - · d C

Solution l.

~XS d2 = c2 . 5 d 2 = (c · d)(S cd) = cd. 5 cd = l. S cd = S cd d c d •c (c • d) · 1 cd 1

d

C

Solution 2.

c2

5 d2

_e1

5#

1

1

c •5d 1

d X -c-= ,,rX 7= -

-= S cd

Example 4.

Find x + 6 • x + 2. 4 3

Solution.

x+6 x+2 (x+6)(x+2) -4- .-3-= 4.3

=

(x+ 6)x+ (x+ 6)2 12

x2 +6x+2x+12 12

'

x2 + 8 x+ 12

.

12

Example 5.

Perform the indicated multiplication ( 1 + ~)

Solution.

(1

+ ~) = ~ + ~ = a · ! + 3 · ! = (a+ 3)! a a a a a a 4 a= (a+ 3)! · 4 a= (a+ 3) ·!·a·_!_ ( 1 + ~) a Sb a Sb a Sb = (a+ 3)(;

•a)• 5\

= (a+ 3) • 1 • 5\

= (a+ 3) . _!_ = 4(a + 3) Sb

Sb

!:·

185

Operations with the Numbers of A ritltmetir,

0

EXERCISES

Write a single fraction which expresses each of the following indicated products.

1.Jxt 2. ½X 170 3. ¾X 5

2

5. 2} X f

15. s2

6. 3! X 5} 16.

a b

7• 4· 3

a2 a

9• b. b lO, x +7 __ 4_ X

+5

+~)\a 26. (; +3) ~

a 5

25. (1

a3

27.

19. _1. rsz r2 s 7

28 · - 4 - · 6 a - 30

2a

21 x + 3_x + 2 5

24

4 29 _3(x + y) . 8 Sx + Sy

12 , x +S. _6_

'

x;

a- 5

20. a3 . b3

x +5

G+ s)

18 _ 4_. 2 a2b '6 a3b 8

11 • ·x 2 18 6

+

24. ( 1 + ~) b

5 · ;·a

b

x2

2

d c+ d a+ b

G)

' 5

3

23 . a + b . c

17 2 a2 . ~

8 ~.2... ' b 4a

9

.

14 • .!!!._ 7 · 14

4, ! X !

4

22 r + s . r + s

l3 4 cd . 10 ' 5 cd

30 , 4 X

3

+20 .

3Z

6 z3 3X

Q 31 (a +b)6. '

4

12 3(b+ a)

32 2 X + 2 . 3 + 3 X ' 5 15

bm

1- r

36 _ (x + y) 2 . 4 abc 12 ab

x +5

34 , am + an . 2 m - 2 n

+ bn

EXERCISES

(1 + ~ )(s )(s) s s+ r

37 . 1 + 7 X

33 , ab + b . ab - b a +l a -1

bm - bn

35.

+15

x

+y

1+ ~ 5 x + .!. 7

• __

2x + 8 4x + 8 3x + 9 15. 6 X 24 . 7 X 14

38 • 5 X

+

+

+

186

CHAPTER

5

Simplifying Fractions

It is now easy to prove the following theorem.

►

Theorem 7a: If a, b, and c are numbers of arithmetic such that b

c --' 0 then ~

.,- '

b

~

0 and

= ac • be

Proof of Theorem 7a: If a, b, and c are numbers of arithmetic such that a ac b ~ 0 and c ~ 0, then b = be· 1.

If, in Theorem 6a, we let c = d we have

1. Substitution property

of equality

a c ac -·-=b c be

2.i(c·~)=::

2. Theorem 4a

3 ~. 1 = ac

3. Definition

. b

4.

of multiplicative inverse. 4. Multiplication property of 1

be

i = ~:

We call this theorem the fundamental principle of fractions. You will find this principle very useful in changing fractions from one form to another. Example l.

Express j as a fraction with the denominator 12.

Solution.

Note. We chose to multiply by 4 rather than some other number

because 3 X 4 gives the 12 that we want in the denominator. We may also use Theorem 7a to simplify fractions. By simplifying a fr action we mean writing the fraction in a form which contains no indicated operations that can be performed and no factors (except 1) common to both the nu5 is simpler than 5 X 3 because 5 X 3 can be merator and the denominator. 14 7X2 1

~4 is

simpler than !~x + Y~ x+ y because x + y is a factor common to both the numerator and the denomi.nator. written as 15 and 7 X 2 can be written as 14.

187

Operations with the Numbers of Arithmetic

i ~.

Example 2.

Simplify

Solution.

15 5X3 5 -=--= 12 4X3 4-

Example 3.

s·1mpnl y-53x -;

Solution.

3x -- 3 X • 1

Example 4.

Find the common name for the complex fraction

1 xy

15 xy

X

-:;c. 0, y -:;c. 0. 1

3 X · Sy - Sy ;!_

¾· 7

Solution.

In Examples 3 and 4 we say that we have expressed the fraction in simplest form. Example 5.

Express in simplest form: S(x - y) ·

Solution.

S(x - y) _ j__ 7x(x-y)-7 x

Example 6.

Simplify

7x(x-y)' x

t+

-g-.

Solution 1.

Solution 2.

1+!

Example 7.

Simplify

7 ;

a -:;c. 0, b -:;c. 0.

2+b Solution.

ab(1+~) ab+b 2+1= ab(2 +~) =

Example 8.

Simplify xy + Y . x + 1 -:;c. 0. x + 1'

Solution.

~ - y(x + 1) - y _ ' x + 1 - l (x + 1) - 1 - )

1+¾

-:;c. 0

,x

>

y.

188

CHAPTER

0

5

EXERCISES

In the following exercises the variables represent numbers of arithmetic such that no denominator is zero.

1. Express in simplest form: a. ~~

7 m2 2m

d. 7 xyz

b.

1~

xy

c.

s;;

4a

xy e, 12 X

h 7 a3

f.-

· 15 a

S(x + 1) g. 7(x+ 1)

2. Simplify: 11 r 2s 2

4xy

c.-12 xyz

a.~ a3b2

d.

b. a2b2

35 x 2y 2 s xy z

12 a2b e. 72 ab

5 X + 10 y g. 4 x + 8 y

f. 4x+4y

h. ab+ b2

12

X

+ 12 y

ac+ be

Q

EXERCISES

Ih the following exercises the variables represent numbers of arithmetic such that no denominator is zero.

3. Find the common name for the following. 2

_Q_

a. s3

4 c. 15

3

TT

3

b, l 8

3

d. f8

7 19 e.-+a a

14 g,17 2+4

£. 1-+ll

h

5y

5y

2+1

3 s . !+½

4. Simplify: a2 b

x+l

b

d.ax+2 a2

a+b 7 b.a -b 7

12 abc 7 xy e. 14 abc -14 xy

7rr 2 + r 14 c. 2 7rr + r 7

5 x+S y f. 3 x + 3 y x+y 3 x+3 y

a.C

r+s r-s g. -5-

1

h.-b aa+b r-s . r+s 1. - r-s r+s

. 1-¼ 3+½

1.--

189

Operations with the N umbers of Arithmetic

5. Simplify: m+l a.-1 m+n

m+n m-n c.-m-n m +n

f

rs

· (r + s) 2

_,___1_ •

r+s

3 x2 - 3 X d. 2 x(x - 1) Addition of Fractions

We may use Theorem 4a and the distributive property to add fractions. The following examples show how this is done. No denominator is zero. Example 1.

Express the sum of 1+ !.! as a single fraction. a a

Solution.

3 11 (1) 1 1 =14 -+-=3 - +11 (1) - = (3 +11)-=14 •a a a a a aa

Example 2.

a + b = ? 3 x +2 3x +2 3x+2

Solution.

a b 1 1 3 x + 2 + 3 x + 2 =a· 3 x + 2 + b · 3 x +2 1 a+ b = (a+b)3 x +2= 3 x +2

Sometimes in adding fractions we use the fundamental principle of frac. · property. tlons, ba = ac be' as we11 as t he d'1stn"b ut1ve Example 3.

Add -fo- + /4 .

Solution.

The least common denominator is 70. 3 S 3·7 S · S 21 25 1 1 10 + 14 - 10 · 7 + 14 · S - 70 + 70 - 21 . 70 + 25 . 70 1 1 46 23 = (21+ 25)- = 46 · -= - = 70 70 70 35

Example 4.

Simplify 5 - ~b

Solution.

a 5 a Sb a 1 1 S-b=l-b=b-b=Sb·b-a·,; = (S b- a)!= Sb- a b

b

190

CHAPTER

Example 5.

. d -4 + -3 Fm a3 a

5

+ -· S a2

Solution.

0

EXERCISES

In the following exercises the variables represent numbers of arithmetic such that no denominator is zero.

1. Find the single fraction which expresses each of the following indicated sums or differences.

4

7

x

X

d.-9-+

a.-+-

4

Sx+3

Sx+3

h ~+6a

+ _1_1_

e. _1_2_ 3r+s

• 8

4

. 7b

2b

• 3

S

3r+s

f_ _ 7 __ x+3y

2

I ---

x+3y

2. Simplify:

6 7 a.2+ 2 a a

s a c.-- m m

b. .!+~ ab ab

d . 4--b

1

2 e. 6+-

C

4

1

.

a

g. 1 + b2

I,

1

71"-2 r

2

h. a+-

f. 2 +s C

C

3. Write a single fraction which expresses each of the following sums.

a.½+½ a

b

b.4+8

c.

2 4 -+a b

e.

d 4a+6b • S

10

1 4 -+a2 a

g.

1 4 1 -+-+m n p

i.

__!_

4a

+ ___£

2a

r. ~+Z X

y

Equations Involving Fractions

Not all equations involving fractions are called fractional equations. The term fractional equation is reserved for an equation in which the variable appears in the denominator. Thus

3 11 lS . not. - + 5-=

X

1= 7 X

is a fractional equation while

191

Operations with the Numbers of Arithmetic

The following examples show how we solve equations containing fractions. Example l.

Solve the fractional equation 5 =

Solution.

PART

Z; X

x EA.

1.

The form of this equation implies that x zero is excluded. 5=2

7 S(x) = - (x )

Multiplication property of equality

X

S x= 7

If x,

¼(5 x) = ! (7)

a E A,

(~)a= x.

Multiplication property of equality Theorem 4 a

7

5 - 5

x=f PART

0 because division by

Given

X

SX

~

If x, d E A,

dx

d = X.

2.

sr z s 1.1....:1

Substitution property of equality Theorem 7a

.M. S 1. 7

Multiplication fact; if x,

7

5

- f ·S

d E A,

sis

(~)d = x.

Definition of division

The solution set is {t}. Example 2.

Find the solution set for the open sentence~+~= 8; x EA, X ~

Solution 1.

X

0.

PART

1.

PART

X

2. ?

1+~=8 X

X

~) x= 8 X (i+ X X 3 5 --x+-•x=8 x X

X

3+5=8x 8= 8x 1=

X

t+t==8

sis

The solution set is {1}.

192 Solution 2.

CHAPTER PART

1.

PART

2.

~+-2..ls 1 1 sis

~+~=8 X

5

X

3(;) + 5(;) = s

The solution set is {1}.

1 (3+ 5)-= 8 X

(8).!= 8 X

~=8 X

8=8x l=x

x+4

Example 3.

Solve3 -+7=x+l; xeA.

Solution.

PART

1.

PART

x+ 4+7=x+l 3

2.

11 + 4 +11=11+1 3

?

3(xt 4 + 1) =

3(x+ 1)

3(xt 4) + 3(7) = 3(x) + 3(1) x+4+21=3x+3 x+ 25= 3 x+ 3 X + 25 - 3 = 3 X + 3 - 3 x+ 22= 3 X X 22 - X = 3 X - X 22= 2 X

¥+7==11+1 5+ 1 J 12 12 i 12 The solution set is {11}.

+

ll=x Example 4.

Solve the equation .006 t = .03; t EA.

Solution 1.

PART

1.

.006 t may be written as nfoo t and .03 may be written as Therefore .006 t = .03 may be written as

nfoot= Th Then lOOO(nfoo t) = lOOO(rto) 6 t= 30 t= 5

rto"·

193

Operations with the Numbers of Arithmetic

2. ? .006(5) == .03 ✓ .03 = .03

PART

The solution set is {5}. 1. .006 t= .03 1000(.006 t) = 1000(.03) 6 t= 30 t= 5

Solution 2.

PART

2. ? .006(5) == .03 ✓ .03 = .03 The solution set is {5}.

PART

0

EXERCISES

1. Find the truth set for each of the following sentences. The variables represent numbers of arithmetic. 4

e.. 4 r

b. 3 x = 18 5 C• . 4 X

=

X

d. .006 x = 3.6

a.~= 12

f.

16

21

=

g. 15 =

12.8

1

5

h. 2½ r = 30 . 1

=3X

l,

7

5m

7= 21

2. Solve the following equations. The variables represent numbers of arithmetic. a a 4r 1 a.-+-= 10 e. 2/=25- 7/ g. 5 - 2 = 3 2 3 b

b

f. 5 = ~+ !

b. 5 = 8 - 15

3

h. 3 p _p = 16

6

2

6

3. Solve, assuming that the variables represent numbers of arithmetic. a. x + 2 = 14 3

c. P+ 6 + 3 = 24

b. p + 1 + 1 = 5

d.

4

e. 10 = 1 + 3 x

2

11

=

19 p 5

+2+3

f.

x

+3

4

+4 5 + 1 = 13

4. Find the solution sets for each of the following equations when the variables represent numbers of arithmetic other than zero. 3 a. -

x

+ -5x = 2

b)y + ~y - 3 = o

4

6

18

c. - + - = .5 r r

e. y

d. 14 = 28

f.

X

=9

!y + !y = ~y

180

g. 3 = -

2y

h.. 4 = 1306c

194

CHAPTER

5

5. The sum of one half of a certain number and one fourth of the same number is 48. What is the number?

6. The difference between one third of a number and one fifth of the same number is 2. What is the number?

7. The width of a rectangle is two thirds of the length. If the perimeter is 60 feet, what are the length and width?

8. If 6 is added to seven ninths of a number, the sum is 20. What is the number?

9. If a number is added to the numerator of

and subtracted from the denominator, the resulting fraction is equal to 1. Find the number. 173

10. One half of a certain number is 7 less than the successor of the number. What is the number?

11. One number is one third of a second number and one fifth of a third number. If the sum of the three numbers is 108, what are the numbers?

12. What number increased by 12% of itself equals 138.88? 13. What number decreased by 23% of itself equals 616? Example.

If Jim can mow the yard at home in 2 hours and his brother Bill can do the job in 3 hours, how long will it take the boys to do the job if they work together? Assume that the work of neither boy interferes with the work of the other.

Solution.

PART

1.

If Jim can mow the whole yard in 2 hours, he can mow ½of the yard in 1 hour. If Bill can mow the whole yard in 3 hours, he can mow ½of the yard in 1 hour. Let n represent the number of hours needed by the boys to complete the work if they work together. Then n( ½) represents the part of the yard Jim can mow in n hours and n(½) represents the part of the yard Bill can mow inn hours. Since both boys working together mow the whole yard, we may let 1 represent one whole job completed.

Thus

n(½)

+ n(½) = 1 ~+~=1 2 3

6(~

+i) = 6(1)

3n+2n=6 5n=6 n= !=

1¼

195

Operations with the Numbers of Arithmetic PART

2. ?

!(½) + !(½) ==? 1

!+¾==1 1i 1

The solution set is {1! }. The boys can mow the yard in 1! hours when they work together.

14. One window washer needs 5 hours to wash the third floor windows of a downtown office building, but a second window washer needs only 4 hours for the same job. How long will it take the two men to do the job if they work together?

15. Jean can type 100 pages of a manuscript in 15 hours while Mary can type the same number of pages in 12 hours. If the girls work together, how long will it take them to complete the job? Assume that the work can be so divided that the two can work together without slowing either of the girls.

16. If 10 hours are required to fill a pool with the water flowing from a pipe, but 12 hours are required to fill it with the water flowing from a hose, how much time will it take to fill the pool if both the pipe and the hose are used? Age six years Age now from now

17. What number divided by 15 will give a quotient of 12 and a remainder of 7?

Ann

Jane

Ann

Jane

6

~

fx+?

x+?

! of Jane's. Six years from now Ann's age will be ¾of Jane's. How old are Jane and Ann now?

18. Ann's age is

fx+? = ¾(x+?)

Q

EXERCISES

19. Solve, assuming that the variables represent numbers of arithmetic. 8a. - 7 - +x+ .1 x+.1

=5

d. ~ = 2

b. ...!i_ + ~ = 2

e.

c. 9 = ...1Q_ x+5

f. - 8 -

x+3

x+3

1 +! x g.-1-=10

x+4

~+x+2

y+3

3- = 3

x+2

+1=

__!!_ y+3

X

196

CHAPTER

20. Assuming that a, b, e, and d are numbers of arithmetic such that b and d ~ 0,

~

5

0

a. Show that if ~ = ~, then ad= be.

b. Show that if ad= be, then~=

i

c. Replace the question mark with the proper symbol to make the following statement true: Since parts a and b of this exercise are true, it follows that ~-~ b - d - ?. - ad-b - e. ESSENTIALS

Before you leave Chapter S make sure that you

1. Know what we mean by "a number of arithmetic." (Page

134 .)

2. Know what we mean by "closure" of the numbers of arithmetic under addition and multiplication. (Page 135 .)

3. Know what we mean by a "binary operation." (Page 137.) 4. Understand and can use properly the following properties of equality: a. Reflexive (Page 139 .)

b. Symmetric (Page

139.)

c. Transitive (Page 139.)

d. Substitution (Page

140.)

e. Addition (Page 140.) f. Multiplication (Page 140.) g. Subtraction (Page 140.) h. Division (Page 140 .)

5. Understand the commutative properties of addition and multiplication and the associative properties of addition and multiplication and can use them. (Pages 142-143 .) 6. Can apply the substitution property in evaluating expressions. (Page 145 .)

7. Know-what we mean by "inverse operations." (Page

148 .)

8. Can make use of the principle of inverse operations in equation solving. (Page 150 .)

9. Recognize 0 as the identity element for addition and 1 as the identity element for multiplication. (Page 162 .)

10. 11. 12. 13.

Know what we mean by "multiplicative inverse." (Page 162 .) Understand the distributive property and can use it. (Page 167.) Understand the multiplication property of 0. (Page 178 .) Can express division by the multiplicative inverse. (Page 181.)

197

Operations with the Numbers of Arithmetic

14. 15. 16. 17. 18. 19.

Can multiply fractions. {Page 183 .) Can simplify fractions. {Page 186.) Can add fractions. {Page 189.) Can solve equations involving fractions. {Page 191.) Are learning to solve problems by use of algebraic equations. {Page 194.) Can spell and use the following words correctly: associative {Page 143 .) binary {Page 137.) commutative {Page 142 .) distributive {Page 167.) equality {Page 139.) equivalent equations {Page 160.) identity {Page 162 .)

inverse operations {Page 148 .) multiplicative inverse {Page 162 .) reciprocal {Page 164 .) reflexive (Page 139.) substitution (Page 140.) symmetric (Page 139.) transitive (Page 139.) CHAPTER REVIEW

1. What do we mean when we say that the numbers of arithmetic are closed under addition? 2. What do we mean when we say that addition is a binary process?

3. Is multiplication a binary process? 4. a. If you are asked to find 3 X 8 X S by the definition of the multiplication of three numbers, which of the following products do you find first: 3 X 8, 3 X 5, or 8 XS?

b. Provide reasons for the following steps: (1) 3 X 8 X S = (3 X 8) X S; (2) (3 X 8) X 5 = 5 X (8 X 3); (3) 5 X (8 X 3) = (5 X 8) X 3.

5. Give the property which tells us that each of the following statements is true.

a.3+1=1+3

b. 5 + 1 = 6 ++ 6 = 5 + 1 c. 4=4 d. If 2 + a = 6 and 6 = 3 X 2, then 2 + a = 3 X 2. e. If 3 + 8 = 11 and b = 8, then 3 + b = 11. f. If 9 + 1 = 10, then 9 + 1 - 1 = 10 - 1. g. If 6 x = 12, then x = 2. h. If r = 3, then 3 r = 9. i. If x = 4 and y = 3, then xy = 12.

198

CHAPTER

5

6. What number is represented by each of the following expressions if a= 2, b = 3, and c = 5?

a. a+b

d 15

g. c- a

• b

h. 2 b + 4 a

b. be+ 1

i. (b + c) 2

c. 4 a+ 2 b

7. Do all numbers of arithmetic have multiplicative inverses? 8. Not all equations which contain fractions are called fractional. Describe those which are.

9. State the multiplication property of 1. 10. State the multiplication property of 0. 11. State the addition property of 0.

12. State the theorem which explains how to multiply fractions. 13. Complete: Multiplying a number by _? _ gives the same result as dividing it by 7.

14. a. Write an expression which represents the perimeter

□

of the rectangle at the right. b. Use the expression you a have written to find the perimeter if a= 2¼ and b = 3½.

15. How do you know

that (8

+ 5) -

5 = 8? that G-)5

= 7?

b

16. Use the distributive property to state each of the following indicated sums as an indicated product.

a. 8 + 12

c. rs

+ rt

b. 4 a + 7 a

d. 3 r + 9 s + 6 t

e. 9 a+ 18 b

f. 4 x 2 y + 2 xy 2

½a2 +½a h. .01 t + .001 t2 g.

17. Use the distributive property to state each of the following products as a sum.

a. 4(a + b)

+ 2 x + 1) f. 5 ab(a + 2 b + c)

e. 3 x(x 2

b. 9 x(x + y) c. ½(12

+ 16 m)

g.

(a + 2) (a + 3)

h. (m+ l)(m+ 7)

d.5 (x +2 y + z)

18. Simplify: a.

r+r+r

b. t + s + t c. a + b + 3 a + 4 b

d. 6 a+ 9 b + 2 b

e, 4 X + 7 y + 2 X + y £. m+m+3m g.

5 7 -+a a

199

Operations with the Numbers of Arithmetic

19. Solve the following equations. a. 5 X

-

e. 7 X + 4 = 6 X + 9

1 = 14

f. 8 y - 1 = 2 y + 17

b. ~ = 0 7 C.

½X -

g. 1 + 3 y = 9-y

8 = 17

h. 3 x + 2 x + 6 = x + 14

i.. 7y+5+ .2y=23

d. 14 = 2-.! 3

j. ½x + ½x +6= 18

20. Solve: a. 6(x + 3) = 36

d. 10y=2(y+8)

b. 4(r + 5) = 2(r+ 18)

e. 6 y = 3 (y + 1)

c. 6 y + 2(y + 3) = 30

f. 2 X + (x + 7)

= 14

21. Express in simplest form: 4a

12 abc c -• 3a

6 x2

d 18(x + y)

a.~

b. 2 x

15x+12y 5x+ 4y

e. - - - ~

f. 2 m + 3 m 4m+6m

• 6(x+y)

22. Write a single fraction which expresses each of the following indicated products in simplest form.

a.

a 3

3. b

c.

6 2 b.-. r r

3a

3 a 5 ab e.--·-10 ab 6 a2 b

5

b . 3b

d. r+ s . _5_ 5

f. ( 1 +

r+ s

23. Simplify each of the following fractions. a 2a+c r+ s 1

2 a. T

b.~

3

-

5 c.--

C

r-s

b

5

d.6a+ 2 C 3

~)a 2(m + n) m- n f. m+ n m-n

1

a+ c e.-a-c

24. Simplify:

c.

r s -+s t

d. 1-+ ~ xy

y

4 6 e.-+a2 a

g. 3 X

f. __!._ + __!._

h x + y +5 x +5 y

3a

2a

7

.

3

+y-

2 3

X

6

+y

200

CHAPTER

5

25. Solve each of the following equations. 4

7

X

X

a.-+-= 22

6 4 c. -+-= 5

e.

b. 2 = 14

d. x- 3 = 12

f. 9 = 2 + 4 x+ 5

a

a

2

X

x+ 3 +4= 8 2

2

26. A painter can paint a house alone in 5 days and his helper can do it alone in 7 days. In how many days can the house be painted if the two men work together?

27. John is 4 years older than Henry. In two more years John will be f as old as Henry. How old is Henry now?

28. A collection of coins contains nickels, dimes, and quarters. If there are 2 more nickels than dimes and twice as many quarters as dimes, how many of each kind of coin are there if the value of all the coins is $2.05?

CHAPTER TEST

1. Replace each "Why?" with a property that justifies the statement at the left. 1. 2 X - 7½ = 12½ 2. (2 X - 7½) + 7½ = 12½ + 7½ 3. 2 X = 20 4. x= 10

1. Given 2. Why? 3. Why? 4. Why?

2. Supply the missing reasons in the following solution. Solution.

PART

1.

1.

3 X + 3= 12 4

2. (3t+3)-3= 12-3

1. Given

i Why?

3.

3 x= 9 4

3. Why?

4.

4(34x) = 4(9)

4. Why?

S. 6.

7.

3x=36 3 x 36 -3 3 x= 12

S. Why? 6. Why?

7. Why?

201

Operations with the Numbers of A rithmetic PART

(!

2.

1. 3 2)

2. 3.

34 6

+ 3 J. 12 + 3 J 12

1. Why?

2. Why? 3. Why? 4. Why?

9+31=.12

4.

12i12

The solution set is {12}. 3. Evaluate each of the following phrases when a= 6, b = 7, and c = a. 3 a+ 2 b

b. c(a + 8)

½-

c. abc

4. Write each of the following in simplest form. a. 7 x + 2 y + 4 x 3 4

b.

-+x X

d. 12 m + 12 n 9

a+l

m+n 7

f.

a

1

+2a a

e. 2 ab+ 4 ab

5. Find the truth set for the sentence~+~= 4. X

X

6. Write in simplest form: 12 · a· a· ¼,

7. a. Express the indicated sum 10 a+ 5 bas an indicated product. b. What is the name of the property that you used in writing your answer to part a?

8. Solve: 4 a+ 1 = 2 a+ 15. 9. What properties do we use when we write (3 + 5) 10. If aK = a, what is the value of K if a ~ O? 11. If aK = 0, what is the value of Kif a ~ O?

+ 7 as 3 + (7 + 5)?

Boolean Algebra An algebra is a set of elements and a set of operations defined for these elements. Let us consider an algebra in which the elements are 0 and I. We define three operations by the following tables: "Addition"

Ai A2 A3 A4

"Multiplication"

"Complementation"

M1 0· 0 = 0 M2 0 · 1 = 0 M3 1 · 0 = 0 M4 1 · 1 = l

0+0=0 0+ 1= 1 1+ 0 = 1 1+ l = 1

C1 C2

0' = 1 1' = 0

This algebra js called a Boolean algebra after George Boole (1815-1864), whose object was to provide an algebra for the study of logic by methods like those of the elementary algebra you are studying. From the above definition of "addition," we see that in Boolean algebra a + l = 1. To prove this we consider the only values a can represent, namely 0 and I. If a = 0, we have 0 + 1 = 1 (by A2); and ifa = 1, we have 1 + 1 = 1 (by A4). Also we note that a + 0 = a, because if a = 0, we have 0 + 0 = 0 (by A 1); and if a = I, we have 1 0 = 1 (by A 3). Can you use our definition of "multiplication" to prove that a • 0 = 0, that a • l = a, and that a • a = a? Can you use our definition of "complementation" to prove that a • a' = 0 and that a+ a'= l? Additional study of our three operations reveals that {0, l} is closed under each of them. Moreover, just as in ordinary algebra, "addition" and "multiplication" are both commutative and associative. To determine whether the distributive property, a · (b c) = ab + ac, is true, we make c b c a • (b c) ab ac ab+ac a table in which we consider 0 0 0 0 0 all possible combinations 1 0 0 0 0 of the two values that the 1 0 0 0 0 variables a, b, and c can 0 0 0 0 0 represent. Since a • (b c) .J 1 I 1 0 and ab ac are equal for 1 1 1 0 1 each set of values of a, b, 0 0 0 l O and c, we see that "multil 1 1 1 1 plication" is distributive over "addition" as in ordinary algebra. In Boolean algebra, however, we have a second distributive property-"addition" is distributive over "multiplication"; that is, a+ (b • c) = (a b) • (a c). Prove to yourself that this is true by making- a table to include columns headed b • c, a+ (b • c), a+ b, a + c, and (a + b) • (a + c) . When the table is complete, you will see that the columns headed a + (b · C) and (a + b) • (a + c) are the same.

+

+

+

+

+

+

+

202

+

If we let T (true) replace 1 and F (false) replace O in the Boolean algebra we are studying, the above table becomes a truth table and we can use the networks studied on page 132 as a model for our Boolean algebra. Recalling that a network with two switches in parallel can represent the compound statement a orb, compare the truth table for the expression a + b with the truth table for a or b.

-0a

b a ;t- b

0 0 1 1 0 1 1

a orb F T T T

0 1 1

1

We see that the table for the network is the same as the table for the expression if we indicate false (open) with O and true (closed) with 1. Therefore the expression a + b can be represented by a network consisting of two switches in parallel. The statement 1 + 1 = 1 merely says that our network consisting of two switches in parallel is closed when both switches are closed. Thus we have a sensible ~~ interpretation for A4. open switch Another comparison reveals that the expression ab can be represented by a networkT ...1 _ ___ _.___ , / ._____.:2 consisting of switches in series. The statea •' open switch ment a • 0 = 0 says that a network consisting of any switch a in series with an open switch ~ is always open. The statement a • I = a says a ~ - ---- --closed switch that a network consisting of any switch a in series with a closed switch is open or closed T1 according as a is open or closed. a closed switch Let us return to the second distributive property of our Boolean algebra. The following networks correspond respectively to a + (b • c) and to (a + b) • (a + c). In "Logic and Switching Circuits" we pointed out that these networks are equivalent.

-l2

-------=--.,,.____ _ __._ . .

By using variables whose replacements are the elements O and 1 and by using the properties and operations that we have defined for Boolean algebra, we can simplify switching networks. Can you sketch the network represented by the Boolean algebraic expression xw + yzt +1(x + y)(x + z) and then by means of Boolean algebra find a simpler network? 203

Chapte,

6 The Real Numbers In this chapter you meet some numbers which are not numbers of arithmetic

The numbers that you are now ready to study will give you greater mathematical power and enable you to do some problems that you could not do using the numbers of arithmetic alone. The Real Numbers If we confine ourselves to the numbers of arithmetic, it is impossible to solve the simple equation 5 x = 0. This is because there is no number of

+

arithmetic which when added to 5 gives the sum 0. Yet it would be helpful to have such a number. We meet the situation by extending our set of numbers to include the number ·we need. We use the symbol - 5 to represent this number. Now we have 5 + (- 5) = 0. If the sum of two numbers is zero, each is called the additive inverse of the other. Consequently, the number represented by - 5 is the additive inverse of the number represented by 5. We read - 5 as "the additive inverse of 5." Similarly, it would be helpful to have a number which produces 0 when added to 6, a number which produces 0 when added to½, a number which produces 0 when added to 3.5, and so on for all the numbers of arithmetic. We let - 6, - ½, - 3.5, and so on represent these numbers. We read these as the additive inverse of 6, the additive inverse of ½, etc. We do not need to include a new number which when added to 0 produces 0, because the set of numbers of arithmetic already contains such a number. It is 0 itself, since 0 + 0 = 0. That is, zero is its own additive inverse. 204

205 The set of numbers composed of the numbers of arithmetic and their additive inverses is called the set of real numbers and is designated by R. We call the additive inverses of the non-zero numbers of arithmetic the set of negative numbers and designate it by N. Thus R = A UN. When we do this we call the non-zero numbers of arithmetic the positive numbers. Thus R may be considered as the union of the positive numbers, zero, and the negative numbers. We sometimes use a plus sign when we want to indicate that a number is a positive number. For example, we may write+ 4, +½,and+ VS. More often, however, we write 4, ½, and VS.

0

EXERCISES

1. Rectangle R at the left below encloses the symbols for a few of the real numbers. On your paper draw a rectangle and a circle similar to those at the right below. Place the symbols shown in the left rectangle so that all of the symbols which represent numbers of arithmetic are within the circle labeled A and all of the symbols which do not represent numbers of arithmetic are outside the circle but within the rectangle. /

S'"'

~

-1

\

4

✓"'

\ '..?~/

.,.

,r

R

'➔

0

'/9'

-\

\

",

14

-4

0

2. Use the diagram you have made for Ex. 1 to do the following exercises. a. Name the numbers represented by symbols within A whose additive inverses are represented by symbols outside of A.

b. Name the numbers represented by symbols within A whose additive inverses are represented by symbols within A. c. What is the name of the set of numbers represented by symbols outside of A?

d. Name all of the positive numbers represented by symbols within A. e. Name the numbers represented by symbols within A which are not positive numbers.

f. Is it true or false that the additive inverses of all positive numbers are negative numbers?

g. Is it true or false that the additive inverses of all numbers of arithmetic are negative numbers?

206

CHAPTER

6

3. Write the additive inverse of each of the following :

a. 7

c. 0

e.

b. }

d . .74

f.V2

1r

g. r; re A h. (2+7)

i. (a + b); a, b e A

4. Fill each blank below with a symbol which will make the statement true .

a. 3 + _?_ =

o

b. 4 + _?_ = 0 c. ½+ _?_ = 0

d. _?_ + .9 = O e. a+ _?_ = O; a e A f. 0 = 1r + _?_

g. J + _?_ = o h. 0 = ¾+ _?_ i. V8 + _?_ = 0

Integers and Rational Numbers

Since the set of whole numbers, W = {0, 1, 2, 3, ···}, is a subset of the set of the numbers of arithmetic, it is a subset of R. As we have seen, R also contains the additive inverse of each whole number. Thus R contains {0, - 1, - 2, - 3, · · ·}. The union of the set of whole numbers and the set of additive inverses of the whole numbers is {· · · - 4, - 3, - 2, - 1, 0, 1, 2, 3, 4, · · ·} . We speak of this set as the set of integers and designate it by I. We call {· · · - 4, - 3, - 2, - l } the set of negative integers and {l, 2, 3, 4, · · ·} the set of positive integers. Observe that while zero is an integer , it is neither a positive integer nor a negative integer. The set of rational numbers of arithmetic (set F) is also a subset of the numbers of arithmetic and consequently a subset of the real numbers. Remember that a rational number of arithmetic is a number which can be expressed as a fraction whose numerator is a whole number and whose denominator is a natural number. For example, -!, 2, i, and 0 represent rational numbers of arithmetic. Set R also contains the additive inverses of the rational numbers of arithmeti c. The additive inverse of! is represented by - -!, that of i by - i, and so on. The union of the set of rational numbers of arithmetic and the set of the additive inverses of these numbers is called the set of rational numbers and is designated by Q.

0

EXERCISES

1. Set B , indicated below, has as its .elements a few of the real numbers. B

= {-

14, - 3, - {,

-V3, - ½, 0, ¾,

1, ½,

1r ,

10}

Write the following subsets of B: a. The element s that are not numbers of arithmetic; b. The elements that are negative numbers; c. The elements that are whole numbers; d. The elements that are rational numbers of arithmetic; e. The elements that are rational numbers; f. The elements that are positive numbers; g. The elements that are integers

207

The Real Numbers

which are numbers of arithmetic; h. The elements that are negative integers; i. The elements that are non-negative integers; j. The elements that are positive integers. The Real Numbers and the Number Line

The real numbers can be graphed on a line. To construct a real number line we begin with the number line for the numbers of arithmetic and extend it to the left. Let us suppose that O is the origin, the point whose coordinate is 0, A the point whose coordinate is ¾, B the point whose coordinate is 1, etc. on the part of the line representing the numbers of arithmetic. On the el'.(tended line we now locate a point A 1 as far to the left of Oas A is to the right of 0 . We assign this point the coordinate - f We locate point B1 as far to the left of O as B is to the right of O and assign this point the coordinate - 1. We read A 1 and B 1 as "A sub one" and "B sub one," respectively. A B C

i

-3

-2

-¾

-y-1

3

0

5

4

3

D

E

F

2

2

5

3

Proceeding in this way we can find a point on the left half of the line corresponding to each point on the right half of the line. In each case the coordinate of the point on the left is the add itive inverse of the corresponding point on the right. Zero is its own additive inverse. Thus our line provides graphs for both the numbers of arithmetic and the additive inverses of the numbers of arithmetic. The graphs below show the relationship between the set of real numbers and some of its subsets.

0

0

0

Set N

Set A

Set R=N U A

etc.

0

2

3

4

Set W

5

6

7

etc.

etc.

-4 -3 -2 -I

0 Set /

0

0

Set F

Set Q

2

3

4

208

CHAPTER

6

Recall that set A contains some numbers such as V2 which are irrational. Consequently R contains some numbers which are irrational. Sets F and Q do not contain these numbers. Since the set of rational numbers is dense (see page 40), the graphs of sets F and Q appear to be solid lines even though there are gaps in each. To distinguish the graphs of F and Q from those of A and R, we have used gray lines to indicate that the points are dense but that some points of the line are not included, and heavy black lines to indicate that the points are dense and that all points of the line are included. Order on the Real Number Line

Let us suppose that a and b are two numbers of arithmetic and that A and Bare their graphs. We have agreed that

a a a

> b means

A is to the right of B and B coincide b means A is to left of B

= b means A
• I

•

I

I

I

-5 -4 -3 -2 -1

•

0

I

♦

I

I

-5 -4 -3 -2 -1

I

I.,

I.,

0 1

Any positive number is greater than any negative number. Why? The examples below make use of these ideas. Example

1. Graph the truth set for x 2:: - 5; x ER.

Solution.

-6 -5 -4 -3 -2 -1 Example

0 1 2 3 4

2. Graph the truth set for x

< 3.5;

5

6

x ER.

Solution.

-6 -5 -4 -3 -2 -1 Example

0

1 2

3

4

3. Graph the truth set for x 2:: - 5 and x

5 6

< 3.5;

Solution.

-6 -5 -4 -3 -2 -1

0

1 2

4.

1

Since the graph of - 5 is to the left of the graph of - 3, then - 5 • ·

-

3 4

5

6

x ER.

< - 3.

209

T he Real Numbers

Q

EXERCISES

1. Indicate which number of the following pairs of numbers is the greater. a.7 , -7 c.- 2, 0 e.4, -14 g.-7 , - 4 i.-5 ,½ b. o, 2 d. - 9, - 15 · f. - 5, o h. 1r , - 1r 2. Arrange the numbers of each of the sets of numbers below in order from greatest t o least.

d. ¾, - ¾, o e. - !, - / o, -

a. 9, - 2, 0

b. 1, - 2, - 3 c. ½, - 7, - 5

g. (1 + ½), 1 + (½)2, (1 + ½)2 h. - 1.21 , 1 + (.1)2, (1 + .1) 2

½

1~, - ¾

i. - ~ ~, -

3. Insert one of the symbols ,or= to form true statements.

a. 9 _? _ - 2

d. - 6 _? _ -

b. 3 _ ? _ - 3 c. - 4 _? _ 1

e. -

-V

g.

J -? - ¾

l's

h. -

_? _

9 30

t -? - 2\

f. - 1000 _? _ .0001

4. Graph the truth set fo r each of the following statements; x e R .

- 2

d. x

~ - 1

g. x > - 3 and x

~ - 5

e. x

< 4.5

h. x > - 3 or x < 3

b. x

f. x 3 we mean that a positive number must be added to 3 to give 7. This number is 4. When we say that 5 > - 3 we mean that a positive number must be added to - 3 to give 5. This number is 8. What do we mean when we say that 8 > 2? In general, for real numbers a and b we define a > b as follows: a > b means that there exists a positive number p such that a That is, a > b ~ a = b p, p > 0.

+

= b + p.

When b = 0 this statement becomes a > 0 ~a= 0 + p, p being a positive number. In other words, saying "a > 0" is equivalent to saying "a is a positive number." When a= 0, the statement becomes 0 > b ~ 0 = b + p, p being a positive number. In this case bis the additive inverse of p. Why? Since p is a positive number and since the additive inverse of a positive number is a negative number, we know that bis a negative number. Consequently, saying "0 > b," or " b < 0" is equivalent to saying " bis a negative number. " By the addition property of equality a= b+ p~a+ (-p) = b. We have previously stated that a> b ++ b

< a.

Consequently, it follows that if a, b, and p represent real numbers and p > 0, then b b ~a= b p-+-+ a+ (- p) = b. Thus we know that if any of the statements b < a, a> b, a= b + p, or a + (- p) = bis . true, the others are true. We can now prove the following theorem.

+

►

Theorem 17: For real numbers a, b, and c, if a

> b, then

Proof of Theorem 17: For real numbers a, b, and c, if a

>

a+

c

> b + c.

b, then a+ c

b+c. l. 2. 3. 4.

a

>b

+

a= b p; p > 0 a+ c = (b + p) + c; p > 0 a+ c = (b + c) + p ; p > 0

5. a+ c

> b+c

1.

2. 3. 4.

5.

Given Definition of a > b Additton property of equality Commutative and associative properties of addition Definition of a > b

We speak of this theorem as the addition property of inequality.

>

257

The Real Numbers

The converse of Theorem 17 is true. Its proof is left as an exercise. Converse of Theorem 17: For real numbers a, b, and c, if a+ c > b + c, then a> b.

Theorem 17 and its converse may be combined into the single statement: For real numbers a, b, and c, a > b ++a+ c > b c. We may interpret Theorem 17 on the number line when c > 0 as follows:

+

b

b+c

a+c

a

+

The theorem says, "If a > b, then a+ c > b c." The graph indicates that if the graph of a is to the right of the graph of b, the graph of a+ c is to the right of the graph of b + c. If the graph of a is to the right of the graph of band the graph of bis to the right of the graph of c, then the graph of a is to the right of the graph of c. b

C

a

This suggests the following theorem. ►

Theorem 18: For real numbers a, b, and c, if a> band b > c, then a> c.

Proof of Theorem 18: For real numbers a, b, and c, if a

> b and b > c,

then a> c. 1.

I. Definition of a 2. Definition of a

a=b+Pi; Pi> O

+ +

2. b = C P2; P2 > 0 3. a= (c h) Pi 4. a= c + (P2 +Pi); (P2

5. a>

+

+ Pi)

3. Substituting c + P2 for bin (1)

>

0

Solution.

4. Associative property of addition

and the sum of two positive numbers is a positive number. 5. Definition of a > b

C

Example.

>b >b

Solve the inequality x PART

14 when x ER.

1.

x+ 7 > -

14 (x+ 7) + (- 7) x+ [7 7)]

+ (-

X

+7 > -

> - 21

>>-

14+ (- 7) 14+ (- 7)

Given Theorem 17 Associative property of addition Definition of addition of real numbers

258

CHAPTER 6

Thus we have shown that if x + 7 > - 14 has a solution set, then each member of that set is a real number greater than - 21. Now we show that each real number greater than - 21 is a member of the solution set of x 7 > - 14. We do this in Part 2.

+

PART

2.

Given Theorem 17 Definition of addition of two real numbers

>-21 x+7>-21+7 x+ 7 > -14

X

The solution set is {x I x

> - 21} . ORAL EXERCISES

1. State our definition for a > b. 2. What is the shortest way you can think of to write, "a is a positive number"? to write, "a is a negative number"? 3. What single statement expresses the combined thoughts of the two statements below?

+

If a, b, and c are real numbers such that a > b, then a+ c > b c. If a, b, and c are real numbers such that a+ c > b c, then a > b.

+

Q

EXERCISES

1. Use one of the symbols >, c + 7 b>C

Given Why?

b.

2 < 9 2+c < 9+c

Given Why?

3. If r + s < r + t, what relationship exists between s and t? Why? 4. Find the solution set for each of the following inequalities.

a. x + 4 b. x - 2

>9 1 e. x +¾ < 0 f.9 2 + r h. S(x + 1) < 4(x - 1) i. x 2 + 9 x > x 2 + 8 x + 7 j. 3(x+7)

< 2x-5

k. x(2 x + 7) - (2 x 2 + 6 x + 4) < 2

1. 4(x 2 - 2 x + 1) - 3(x 2 - 3 x + 1) > x 2

259

The Real Numbers

5. For what values of a is each of the following sentences true? a. 2 + a > 5 c. 6 - a < 4 - a e. 4 > 1 + a b. 3 + a > 2 + a d. a + ½> a - ½ f. 4 < 1 + a 6. Arrange the following numbers in ascending order of magnitude (from least to greatest).

a. 9, -100, -45

b. - 5, - 14, 14

c. - 2, - 20, - 6 d. - 57, 16, - 250

e. - 1, 5, - 10

f. - 4, o, - 3

Multiplication Properties of Inequality

Let us consider what happens when we multiply two unequal numbers by the same non-zero number. In the column on the left below we compare two numbers, in the middle column we indicate the number to be used as a mult iplier, and in the column on the right we compare the resulting products.

a> b

7>5 4 > -7 -1

> -5

7>3 4>- 7 -1

> -5

ac _? _ be

C

35 > 25 8 > -1 4 -4 > - 20 - 35 < - 15 -8 < 14 4 < 20

5 2 4 -5 -2 - 4

In the first three cases the multiplier is a positive number. In these cases the comparison symbol between ac and be is the same as the comparison symbol between a and b. In the last three cases the multiplier is a negative number and the comparison symbol between ac and be is"< " while that between a and bis "> ". This suggests Theorems 19 and 20.

►

Theorem 19: For real numbers a, b, and c, if a

> band c >

Proof of Theorem 19: For real numbers a, b, and c, if a ac > be. 1. a > b 2. a= b 3. ac = be

+ p; p > 0 + pc; pc > 0

4. ac

> be

0, then ac

>

be.

> band c > 0, then

1. Given 2. Definition of a > b 3. Multiplication property of equal-

ity; given c > 0; the product of two positive numbers is a positive number. 4 . Definition of a > b

260

CHAPTER 6

~ Theorem 20:

For real numbers a, b, and c, if a

> b and c < 0, then ac < be.

The proof of Theorem 20 is left as an exercise. The following converse of Theorem 19 is true. Converse of Theorem 19: For real numbers a, b, and c, if ac c

> 0, then a > b.

> be

and

The proof of the converse is left as an exercise. Can you state and prove a converse of Theorem 20? When we were studying the numbers of arithmetic, we learned that for a and b, any two numbers of arithmetic, a < b, a= b, or a > b. We now state the comparison property for real numbers .

I,)-

Comparison property: For any real numbers a and b, exactly one of the following statements is true: a < b, a = b, or a > b.

We shall refer t o the comparison property and Theorems 17-20 as the order properties for real numbers. These properties enable us t o determine the order of the graphs of the real numbers on the number line and to arrange real numbers in order of magnitude. There are, of course, many other order properties but these are sufficient for our work. Moreover, all of the other order properties can be proved from these . Example Solution.

l.

Find the truth set for 2 x PART

+ 5 < 11 -

1.

2x + 5 < 11- x 2 x +5+ x < 11- x + x 3x+5 < 11 3 X + 5 + (- 5) < 11 + (- 5) 3X

½(3 x) X

PART

b, then - 2 a _? _ - 2 b. b. If 8 < c, then 3(8) _?_ 3 c. c. If r > s, then r + 5 _? _ s + 5. d. If m < n, then m + (- 2) _? _ n + (- 2). e. If y > 8, then - 6 y _? _ - 48. a. If a

2. Find the truth set for each of the following ineq ualities. a.

+1
0

b. x - 3 > 4


-

j. 4(x + 1) >

9

< 9X f. - 4 X < 12

k. -

e. 18

X -

5

½x - i x + 1 > 3 -

x

l. x ·ll-51 < 12-71

3. Graph the truth sets for the sentences below.

c. x

5

Ix I < 2

1

e. -

3

f. 4 :S 4 + x

g. 6 < 6-x h. - 3 x < - 2

4. For wha t set of numbers is each of the following sentences true?

I+ IS I > I- S I b. I I+ I s I = I - s I c. j x I + 5 I < I - S a. I x X

J

J

262

CHAPTER

6

5. Translate each of the following problems into an inequality. Then find the solution to the problem by solving the inequality,

a. Nancy is shopping for a new skirt and sweater. She is determined that she will spend less than $15 .00 for the outfit. If she buys a sweater that is marked $5.98, how much can she spend for the skirt? Hint. Let x represent the number of dollars Nancy must spend for a skirt. Then x+5.98 15

x

the younger boy, it has been decided that he should save twice as much as the younger boy. How much money must each boy save to meet his father's requirement? c. The sum of a number and 10 is less than 3 times the number. What numbers satisfy this inequality?

d. If one side of a triangle is 2 inches shorter than the base of the triangle and the other side is 3 inches longer than the base, how long is the base if the perimeter is more than 19 inches?

0

EXERCISES

6. If a newspaper carrier had 70 customers week before last and 78 customers last week, how many customers must he have this week to make his average number of customers for the three week period at least 75?

7. Prove the statement: For real numbers a, b, and e, if a+ e > b + e, then

a>

b.

8. Prove the statement: For real numbers a, b, and e, if a > b and e < 0, then ae < be .

9. Prove the statement: For real numbers a, b, and e, if ae then a> b.

> be and e > 0,

10. Graph the solution set of each of the following sentences. a. I x

I< 3 b. I x I < -

2

c. I x I ~ 22

e. I x - 4 I < 2

d. I x + 2 I :s:; 5

f. I x - 5 I > 3

263

The Real Numbers

ESSENTIALS

In this chapter we have given you some of the basic properties of the real numbers and some of the rules for working with them. Since you will be using these properties throughout the remainder of the book you should not leave this chapter until you understand them. Make sure that you

1. Know what we mean by the set of real numbers and can identify the various subsets of the real numbers that we have discussed in this chapter. (Pages 205-206 .)

2. Know what is meant by the real number line and can locate the graph of any real number on it. (Page 207.)

3. Understand the eleven properties that we have assumed true for the real numbers.

(Page 210 .)

4. Understand and can use Theorem 8 and its corollaries: If a ER, then a has only one additive inverse; - (-a)= a; - 0 = 0.

5. Understand the idea of absolute value.

(Pages 212-213 .)

(Page 218 .)

6. Know the rules for adding real numbers, know why the rules are true, can use them quickly and accurately, and can represent addition on the number line. (Pages 214 - 216 and 219- 222.) 7. Know what we mean by subtraction of the real numbers and can perform the operation quickly and accurately. (Page 222.)

8. Understand and can use Theorem 9: If a and b represent real numbers, then a - b =a+ (- b).

(Page 223.)

9. Know the rules for multiplying real numbers and can use them quickly and accurately.

(Page 228 .)

10. Understand and can use the following theorems involving the multiplication of real numbers: Theorem Theorem Theorem Theorem Theorem Theorem Theorem

10: 11: 12: 13: 14: 15: 16:

(- l)a = - a (Page 231.) (- a)b = - (ab) (Page 232.) (- a)(- b) = ab (Page 232.) a - (b + c) = a - b- c (Page 235.) a(b ~ c) = ab - ac (Page 236 .) (a - b)(c - d) = ac - be - ad+ bd (Page 236 .) - (a - b) = b - a (Page 237 .)

11. Know what we mean by a monomial over the integers and a polynomial over the integers.

(Pages 240- 24 1 .)

12. Can multiply a polynomial of two or more terms by a monomial, and a binomial by a binomial.

(Pages 243 - 244 .)

264

CHAPTER

6

13. Can do the kinds of factoring of polynomials discussed in this chapter. (Page 247.)

14. Understand and can use Theorem 17: For real numbers a, b, and e, if a

>

b, then a + e

>

b + e. (Page 256 .)

15. Understand and can use Theorem 18: For real numbers a, b, and e, if a > b and b > e, then a > e. (Page 257 .) 16. Understand and can use Theorem 19: For real numbers a, b, and e, if a > band e > 0, then ae > be. (Page 259 .) 17. Understand and can use Theore m 20: For real numbers a, b, and e, if a > band e < 0, then ae < be. (Page 260 .) 18. Can spell and use the following words correctly : absolute value (Page 218 .) additive inverse (Page 204 .) binomial (Page 240 .) coefficient (Page 240 .) integer (Page 206 .) monomial (Page 240 .)

negative (Page 205 .) opposite (Page 213 .) polynomial (Page 240 .) positive (Page 205 .) rational number (Page 206 .) trinomial (Page 240.) CHAPTER REVIEW

1. Replace each question mark below with a word, a group of words, or a symbol to make the sentence state a truth.

a. A U N = _?_

b. W U _?_ = I

c. F U _? _ = Q

d. The union of the positive numbers, the negative numbers and

_? _ is

equal to the set of real numbers.

2. Arrange the numbers of each of the sets of numbers below in order from greatest t o least.

a. - 4, - 14, o

b. - 3, (- 3)2, (- 3) 3 ,

(-

3) 4

3. Indicate the additive inverse of each of the following numbers. a.-5

b.- (-6)

c.J- 2 1

d. 42

e.-1 4.92

f.- (- x)

4. Indicate - r when r has the values indicated below. a. r=-1

b. r=¾

c. r = 6 + a d. r= a- e

e. r =

- 1 + .2

f. r=

%

5. Indicate which of the following expressions always represents a negative number.

a. - (- 3) b. - (- a)

c. - b

e. (- b) 2

d. - (b) 2

f. (- b) 3

265

T he Real Numbers

6. Indicate which propert y of the real numbers is illustrated by each of the following statements.

a. a + b = b + a b. - 3 x = x(- 3)

d. If a is a real number, - 9 a is a real number.

c. 4(a +b) = 4a + 4b

f. (2 + 3) + (- 7) = 2+ [3 + (- 7) ]

e. - 4

+ 8 = 8 + (-

4)

7. P erfo rm each of t he following additions. a. 15 -1- (- 4) + (- 7) d. (- 3) + 7 + 4 + 4 + (- 8)

b. 18 + (- 1) + (12)

e. - 11.7 + (- 9.3) + 16

c. (- 6) + (- 3) + 7 + (- 9)

f..~ + (- t) + (- l4) +

t

8. Write a single numeral t o express each of t he following indicated sums. a. 1- 5 I+ (- 3) c. - 2 + I - 6 I + I - 4 I b, l- 4 1+ 4 d. I ½I+ I - ¾I+ I - ½I 9. Which of the following sentences are true and which are false? a. I - 3 I = 3 c. I - 3 I > 2 e. 7 I - 5 I d. I - 7 I > I - 1 I f. 9 -,c I - 9 I 10. F ind the truth set fo r each of the following sentences. a. I y I = 3

c. I a I + 3 = - 7

e. 17 + I b I = - 2

b. I x I + 2 = 6

d. 12 = I x I - 3

f. - 9 + I a I = 7

11. Express each of the following subt ractions as an addition. a. 6 - 3 b. 2 - 7

c. 8 - (- 2) d. a - (- b)

e,a -b f. - 9 - (- a)

12. Write a single numeral equivalent to each of the following expressions.

½- ½+ ¾- ¼

a. 9 - 13 + 4 - 7

c. 8 - 6 + 3

e. -

b. - 3 - 6 - 2 - 4

d. - 24 - 13 + 5 - 17

f. 1 - 7 - 9 - 15

13. F ind the solution set for each of t he following equations . a. x + 4 = - 9 c. 3 a + 4 - 2 a = 12 e. 19 - 3 - 2 = x + 5 b. y - 2 = 18 d. 4 c + 15 = - 5 + 3 c f. 6 a - 5 = 11 + 5 a

14. F ind the truth set fo r each of the sentences below. b, Ix- 3 I 2=:: 5 a, Ix I < 5 c. 4 + I x i= 3 15. P erform the following computations. a. (- 3)(- 2)(- 4)

b. (- 1)(7)(2)

½(- 8) + 1 d. - 5(- !) + 7(- 1) c.

e.

I- 3 I. I-

4

f. J C-5)3l-4

J

+2

266

CHAPTER

16. If a= 3, b = express10ns.

6

- 4, and c = - 2, find the value of each of the following

a. a2

c. 4 a+ 5 b

e. abc

b. ab

d. b2 + c2

f. 2 a - b

i. - 3 a - 5 b

g. a+ b- c h. 2 a - 4 b

17. Combine terms.

d.

a. 6 a+ 8 a

b. 2 r - 4 s + 5 r c. 4 m - 5 n + 7 m + 2 n

½a - ½b - ½a + ¼b

e. 7m+m+2m

f. 5.1 a+ 6.2 b - .9 a - 9.0 b

18. Simplify each of the following expressions. a. (5 rs)(- 3 r)

c. (r 2s)(rs)(rs 2 )

e. (- 4 ar 2 ) ( - 5 rs 2 )

b. (6ab)(-2abc)

d.-7 a(a-2b+tc)

f,½abc(4a-6b+10c)

19. Find the truth set for each of the following sentences. a. 4 a+ 7 a= 33

d. 3(x + 5) + 2(7 - x) = 99

b. 6 x + (- 5 x) + 4 = 1 C. 3 t + 7 t - 6 = 8 t + 14

e. - ½(r - 6) = 9 -

f. ¾(x - 4) +

¼r 7 = ½X

20. Write as an equivalent expression with no parentheses. Then combine as many terms as you can.

a. - 3(a - 4 b - c) - (2 a+ 3 b + c)

b. 2(a + 3 b - 4) - ½(6 a - 3 b + 12) 21. Find the truth set for each of the following equations. a. - 2(x + 5) = 0 d. - (6 - 2 y) - 4(3 + ¼y) = 10

b. (2 x + 5) - (x - 7)

= 0

e. 3 x - (x

c. ½(6x-4)-½(9x+ 12) =x

+ 4) =

160

f. x - (x - 3) + (x+ 11) = 2(x+ 6)

22. Which of the following expressions are monomials over the integers? a. 4 a

b. 3 x + 7

c.

½m

d. a(a + c)

e.

¾xy

f. (- 3)6 xy

23. Explain what is meant by a polynomial over the integers.

24. Find: a. x 2 • xy b. (3 be)(- 2 b)

c. (- 4 xy)(- 2 xyz)

e. - 4 a 2 (6 ab 2 )

d. (- 5 a) 2

f. (2 xy)(- 7 xy)(2)

c. (y + 6) (y - 8) d. (ab - 3)(ab - 4)

e. (2

X

f. (3

X -

25. Multiply: a. (x + 7) (x + 3) b. (y-5)(y-4)

+ 5) (2

X -

2 y) 2

5)

267

The Real Numbers

26. Factor:

a. a2 - 7 a+ 12

e. x 2 - 25

b. x 2 - 9 x + 8 c. y 2 - 11 y + 28

f. ay 2 - 36 a g. d2 + 2 d + 1

d. c2 + 6 c - 55

h. 3 x 2 + 24 x + 48

27. Find the truth set for each of the following equations. a. x(x - 3) = 0 d. (x - 3) (x - 4) = 0 b. (x - 4) (x + 5) = 0 e. y 2 - 7 y + 12 = O f. m 2 + 3 m - 10 = 0 C. x 2 + 5 X + 6 = 0

28. Solve the following inequalities. a. x + 4

c. 3(x + 8)

> 12 b. x- 2 < -3

>-

e. 8 - 2 x

16

d. - 4 y < 16

>-

20

f. - ½a< 5 CHAPTER TEST

In the exercises below it is understood that the variables represent real numbers.

1. If a < b < 0, which of the following statements are true? a. ab

126-

51 2·2·3 · 7·3

so

2-3-3-7 -2

336

CHAPTER 8

0

EXERCISES

Add the following fractions.

1 '.± - f. .4 5

9. ~-~-'.! 4 5 2 1 m 6. m+-+8 16

2 ab+!!_ • 6 12

lO. ab+~ -_!_ 12

ll.

3 x+4+x-3 . 5 5

16

4 rs_ rs_ rs

15

8 a+l_a-1 . 12 12

28

21

33

12 _ a + 7 + a - 7 + ~ 20

24

28

Assuming that a, b, and c represent distinct prime numbers, find:

l3.

tb

~+

14. ~ - ...£.. 4a

2a

Replace each ? below with one of the symbols=, true statement.

ls

19. /4?

7 -5 20 · 60? 42 21.

i7s ? l1

22

23 24

1 ? 8 . . 5 y. 35 y' y


, or


0

Multiply each member of each inequality below by the LCD to produce an equivalent inequality.

27.

½>

28.

/4

158

>

1~6

l6 < Th 30. -ls< !s 29,

31.

199 > 1h

32.

1¼0


n,

am an = am -n.

52 For example, 52 = 1.

am (2a) If m = n, - = 1.

a"

(2b) If m

am

5s For example, 52 = 56 •

1

52

1

< n , -an = -, For exa mple, 58 56 an-m

0

EXERCISES

1. Simplify each of the following. 25

a. 22 34

b. 32 75 c. 75

(i)7

52

d. (j) 5

g. 51

(- 2)3 e. (-2)5

h (-

f (.4)4 . (.4)3

.

. (125)10 J· (125)11 .1)3

(- 18)7

. (- .1 )2

k. (- 18)6

11 2

- (7)5

ll3

1. - (7)3

1.--

344

CHAPTER

2. Simplify each of the following. a2 b4 a. 2 e. b4 a 35. a4 ma b.f. ~ m ·a x2 35. a2 c. 7 g. 32. aa X r2s2 rs

. a 2b

3s·3s m.--3s 7 ab 3 n. 7 ab 3

l,~

. 7a J· 7 a2

k 62 (a + b) •

52 · a

d.-

r2 st 3 o. r3st2

6

ab 2c3 d4

I. a2 (r;s)

h. 5. aa

8

p. a4 b3 c2d

a

3. Simplify each of the following. 14 b2 = 7 b · 2 b = '!:J!. = 2 b 7b 7b•1 1

Example.

15 c2 a.~

d. 45 rst

b 15 ab • 30 a2 35 c.~

22 e. 33 a

3 rst

f. 21 m 2n 3 63mn 5

105 a4 b3 g. 21 a5 b2

h.

7 ab 3 21 a2 b4

. 34 a5 b2 c3 17 a4 bc 2

1.---

There are three additional laws governing the use of positive integers, m and n, as exponents with real numbers, a and b, as bases.

(3) (ab)n

= anbn

(a)n

an (4) - = - · b b bn' (5) (am)n = amn

~

O

Each of these is readily verified as shown below.

(ab)n

= (ab)(ab) • • • (ab) =

(3)

n fa ctors (a·a· a··· a)(b·b· b · · · b)

~-- ---~ ----------,,,--------n fa ctors

n fa ctors

= anbn Thus we know that the nth power of the product of two numbers is the product of the nth powers of the numbers. For example, (2 X 3) 4 = (6) X (6) X (6) X (6) and

= 16 X 81 = 1296. 3) 4 = 24 X 34 •

24 X 34

(2 X

= 1296

345

Factors and Exponents

(4)

n factors n factors ,---------A----

a · a· ·· a b· b· · · b

=----= '----v-------'

a" b"

n factors

Thus we know that the nth power of the quotient of two numbers is the quotient of their nth powers. For example,

r-

(1 1-1-1- 1~~ 43 53

and

64 125

-=-·

n addends ~

(am) n = (am)(am) . . . (am)= am+m+· ··+m = amn

(5)

n factors Thus we know that the nth power of the mth power of a number is the mnth power of the number. For example, (23)4 = 23 . 23 . 23 . 23 23·4 = 212_ (23)4 = 23· 4.

and

= 212

0

EXERCISES

1. Write each of the following as an equivalent but simpler expression. d. (-

a. (6 r) 2

b. (- 2 a) c. (-

3

!) 4

e.

!)3

53 •

(¾)2

g. 16.

(4a)3

f. (- 3 b)2

2. Simplify each of the following. a. (4 a 2 ) 3

d. - 6(a 2 ) 2

g. (3 a2)3

b. 4(a 2 ) 3

e. (5 a 2 b) 3

c. (- 6 a 2 ) 2

f. 5 b(a 2 ) 5

h. (3 a3 ) 2 i. (a3b2)3 . (a3b2) 2

346

CHAPTER 8

3. Simplify each of the following. No denominator is zero. 33 x3 2 a. 53 d. 5ab g. (3 a2x) 3

(2

y

2

5 x2 h. (7 xy2) 2

(4 x3)2

b. (j)3

e.7y2

c. (-t)2

f. ( 3 2b)3

!

. (4x) 3 1.~

4. Match each expression on the left with an equivalent expression on the right. No denominator is zero.

a. (- 3 a3 ) 2

3 A. 8a

b.G~Y c. (- 2 a)(- 3 a d. (- 3)(a3 ) 2 3 a2 e. (2 a)3

f.

(32:2)3 (2 a2) 2

B. 2 2)

C. - 3 a6 4 a2 D. 9 b2

E. 9 a6

27 a3 F.8-

g.~

G. 6 a3

h - 2 a3 · (- 1 a)3

i. (- 3 a2)(2 a)

j. (-;

ay

H. - 6 a3

I. ! J.

- 27 a3 8

5. Assuming that no denominator is zero and that the exponents are positive integers, simplify each of the following. a3c a. x 3a. xa ac d.e.2C ac a b. x4b . x Zb

f.

(Xc)3 X

d

c. a• a2 c • ac

6. Indicate the area of a square each of whose sides has the length 3 b. 7. Write the formula for the area of a rectangle the length of whose base is 3 b and whose altitude is b + 4. 8. The area of a rectangle is 16 b2 • If the base has length 8 b, what is the length of the altitude?

347

Factors and Exponents

9. The radius of a circle is 4 r. Write a formula for the area of the circle.

10. The formula s = 16 t 2 shows the relationship between t, the number of seconds that an object falls, and s the number of feet through which the object falls in the t seconds. If two men jump from a jet plane, one falling for twice as many seconds as the other before his parachute opens, which of the following ratios expresses the relationship between the distances that the two men fall before their parachutes open?

a.

½

b.

¼

d. None of these

Laws of Exponents-Zero and Negative Integers as Exponents

We use zero and negative exponents to simplify our work with exponents. a 2 means a· a, a 3 means a· a· a, etc., but certainly a 0 cannot mean a used as a factor zero times. However, unless a= 0, we can give a 0 a meaning that makes it very useful. We reason as follows: If 0 is to be considered as an exponent, we must define a0 in such a way that the previously stated laws of exponents are valid. Thus if a ~ 0, it should be true that am • an = am+n when a e R, me I, and n = 0. That is, it should be true that am• a 0 = am+o = am. Thus it must be true that

Note that a 0 acts as the multiplicative identity in the equation am· a0 = am, so that the definition a 0 = 1 would be consistent with the statement of the multiplication property of one. We have seen that a0 will satisfy the law am . an= am+n if we accept the following definition. When a

E

R and a ~ 0, a 0 = 1.

Note. o0 is not defined; that is, 0° is not a meaningful expression. Likewise, a - 2 , a - 3, etc. have no meaning until we define them. If, when n is a positive integer, a - n is to be interpreted in such a way that our laws remain valid, it should be true that a -n . a" = a< - nl+n = a 0 = 1. Thus it must be true that 1 a-n=an

and

1 a"=-· a -n

348

CHAPTER

8

Note that a -n acts as the multiplicative inverse of an in the equation a -n • an = 1, so that the definition a - n =

_!_ would be consistent with prean

vious definitions and theorems. We have seen that a am. an= am+n if we accept the following definition. When a

E

R, a

;;.£ 0, and n

E /,

a-n =

n

will satisfy the law

1 -;;· a

Now that we have defined a 0 and a -n so that they satisfy the law am· an need no longer say that this law holds for just positive integral exponents. It now holds for any integral exponents-positive, negative, or zero. In fact the other four laws discussed in the preceding section are now true for any integral exponents with the exception that 0° is not defined. Let us summarize by stating the five laws of exponents.

= am+n we

When m, n

and a, b ER,

E /

E1• a m •an = a m+ n (a ;;c 0 when m ::; 0 or n ::; 0) am En. - = a m-n (a ;;.:!c 0) an E111. (ab)n = anbn (ab ;;.£ 0 when n ::; 0)

E,v.

(~r = ::

(b ;;c 0; a ;;c 0 when n ::; 0)

Ev. (a m)n = a mn (a

;;.£ 0 when

m ::; 0 or n ::; 0)

Observe that law En does not have three forms as the corresponding (2) cf the preceding section had. This is because it no longer matters whether m > n, m = n, or m < n. We treat all of these cases alike. Example 1 am an

compares the two ways of treating-·

.

.

a5

as

a2

Example 1.

S1mphfy a. 2 , b. 5 , c. 5 first by using laws 2, 2a, and 2b a a a of the preceding section, then by using En of this section.

Solution.

Using laws 2, 2a, and 2b we have: a5 a. To find 2 we observe that a

5

>

2 so we use the form

Using law En there is only one case to remember. It is:

Factors and Exponents

349 a5

b. To find 5 we observe that a

a5

-

b. 5= a5-o = aD = 1 a

5 = S so we use th e form am a5 = 1. Thus-:= 1. a1n av a2

c. To find 5 we observe that a

a2

c. - =

a5

_

a2 -o

1

= a-3 = - . a3

2 < 5 so we use the form am 1 -m· Thus an= an-

Example 2 .

Perform the indicated operations.

a. (a 4 ) Solution.

b. (a - 4)

- 3

- 5•

a. Applying Ev directly we have (a4)-3

= a C4)(- 3l = a -

12_

b. Applying Ev directly we have (a-4)-5

= a C- 4)( -5) =

a20_

Example 3 . Solution.

0

EXERCISES

In t he following exercises no denominator is zero.

1. Use laws E 1- Ev to find another name for each of the numbers represented below. Do your work as shown in the above examples. a- 2 . a- 4 x2 j. 4-5 . 42 d. 37° g. a O} = {x I (x+ 4)(x- 4) > O} Since the product (x + 4)(x- 4) is positive (the product is greater than zero), we know that both (x+ 4) and (x- 4) are positive or both are negative. Therefore,

16. Graph its truth set.

{x I (x+ 4)(x - 4) > O} = {x I (x 4 > 0 and x - 4 > O) or (x 4 < 0 and x - 4 = {x I (x > - 4 and x > 4) or (x < - 4 and x < 4)} .

+

+


- 4 and x > 4) or (x < - 4 and x < 4)} = {x I x > - 4 and x > 4} U {x I (x < - 4 and x < 4)}. Since the truth set of a conjunctive statement is the intersection of the truth sets of its clauses, we have {x Ix > - 4 and x > 4} U {x Ix< - 4 and x < 4} = [{x Ix> - 4} () {x Ix > 4}] U [{x Ix< - 4} () {x Ix < 4}] = {x Ix > 4} U {x I x < - 4} = {x I x > 4 or x < - 4}. The graph of this truth set is shown below.

-6 -

-4 -3 -2 -


0 and x - 4

< O)}

(4)

=

{x I (x

4) or (x > - 4 and x

< 4)}

(5)

= {x I x < -

4 and x > 4} U {x I x > - 4 and x

< 4}

(6)

= [{x I x < -

4} () {x Ix> 4}] U [ {x I x > - 4} () {x I x

(7)

= 0U

(8)

{x

I-

4

< x < 4} = {x I-

4

< x < 4}.

< 4}]

377

Polynomials and Rational Expressions

(1) Why? (2) Why? (3) If the product of two quantities is negative, the first is negative and the second positive, or the first is positive and the second negative. (4) Why? (5) The truth set of a disjunctive statement is the union of the truth sets of its clauses. (6) The truth set of a conjunctive statement is the intersection of the truth sets of its clauses. (7) Definition of intersection of two sets (8) D efinition of union of two sets The graph of the truth set is shown below.

-6 -5 -4 -3 -2 -1

0

l

2

3

5

4

0 1. Find each of the following products. a. (a + 5)(a - 5) d. (t 2 - 5)(t2 + 5) b. (ax+ l )(ax - 1) e. (t 3 - a)(t 3 + a) c. (3 x + 7)(3 x - 7)

6 EXERCISES

g. (9 b2 + 1) (9 b2 - 1) h. (12 a 2 + 9)(12 a 2

-

9)

i. (3 y - 2 b) (3 y + 2 b)

f. (a 2 + b2 )(a2 - b2 )

2. Factor each of the following polynomials completely. a. a 2

c. 9 a 2

81

-

b. 16 - b2

e. 5 m 2

16

-

d. 9 a 2 - 9

-

g. a4 - b4

20

£. 36 a 2 - 49

h. 16 x 4

-

y4

3. Study each of the following polynomials. If it is possible to factor the polynomial, show the factors; if the polynomial is non-factorable, write "non-factorable."

a. a2 - 5

b.

a2

+ 16

c. a 2 -121

e. 3 a 2 - 12

d. 9 a 2 - 81

f. 5 b2 - 15

4. Factor if possible: a. (x - y) 2 - 4 b. (c + d)2- x 2 c. (r -s) 2 - (a -b) 2

d. a2 + 2 ab+ b2 - 9 e. a2 + 8 a+ 4 - b2 f. b2 -a 2 -4a-4

g. 16 x 2 + 49 h. 81 x 4 - 25 g. x 2 - 2 xy + y 2 - 9 h. 9 - x 2 + 2 xy - y 2 i. y 2 - 2 y - 17 + 4 y 2

5. Solve each of the following equations and inequalities. a. x 2 - 16 = 0

b. 25

y2 -

c. 9 x

2

=

100 = 0 25

=0 36 = 0

d. 4 x 2 - 25

g. 100 a 2 - 400 > 0

e. 16

h.

x2 -

£. r < 25 2

3 a2

i. 2

y2

> 27

< 32

378

CHAPTER

9

6. Graph each of the following sets.

< 0}

a. {x I x 2

> 9}

c. {x I x 2

e. {x / (-1 )x 2 +36

b. {x / x 2

~ 49}

d. {x

f. {x / (- 4)x 2 + 100

> 0} / ½x2 > 8}

~ O}

7. Do each of · the following multiplications mentally and then write the product on your paper. Example: 62 X 58 = (60 + 2)(60 - 2) = 3600 - 4 = 3596. a.21 x19

b. 32 x 28

d. 91 x 89

c. 37 X 43

e. 73 X 67 () EXERCISES

Factor + x3 + y3 = x3 _ xy2 + xy2 + y3 x3

Example 9. Solution.

y3 .

= x(x2 - y2) + y2(x+ y) = x(x + y)(x-y) + y2(x+ y) = (x+ y)[x(x- y) + y2] = (x + y)(x 2 -xy + y2)

8. Factor: a. a3 + b3

c. 8 s3 + 27

9. Find a plan for factoring x3 - y 3 . (Hint. Use the plan of Example 9.) 10. Use the plan you wrote in Ex. 9 to factor the following. a. a3

-

1

b. r3 - s3

c. 8 s3

-

27

Perfect Squares

For real numbers a and b, (a + b)2 = (a + b)(a + b) =(a+ b)a + (a+ b)b

= a2 +ab+ ab + A polynomial such as a 2 + 2 ab 2 is called a perfect square polynomial because it can be written as the product of two identical factors which are poly·nomials over the integers. The figure at the right shows a geometric interpretation of (a+ b) 2 • Study the drawing to identify a 2 , 2 ab, and b2 . Let us apply the relationship (a+ b)2 = a 2 + 2 ab+ b2 to (4 x + 3 y)2.

+b

b2

a

b ,-->----,.

A

b

{

= a 2 + 2 ab+ b2 •

b2

ab

> a+b

a

a

2

ab

. a+b

379

Polynomials and Rational Expressions

We have:

(a + b) 2

t

t

=

a2 + 2 • a • b +

t

tt t

b2

t

(4 x + 3 y) 2 = (4 x) 2 + 2(4 x)(3 y) + (3 y) 2 = 16 x 2 + 24 xy + 9 y 2 In order to use the ideas above in factoring we must be able to recognize a polynomial which is a perfect square. Completing the following statements will help you to recognize such a polynomial. i. (2x+y) 2 =4x 2 + _?_ +y2 (3 a - 5 b) 2 -- _ ?. _ - _ ?. _ + 25 b2 ... ( lll. X + 5 a)2 - _ ? . _ + _ ?. _ + _.? _ iv.( _?_ + _?_) 2 = 16 x 2 +40x+25 V. ( __ ? __)2 =4a 2 - _?_ +49 vi. (7 C + _? _) 2 = _? _ + _? _ + 1 vii. (_? _ + 4 b) 2 = 25 x 2 + _? _ + _? _ .. . ( __ ?. _ _ ) 2 - 36 X 2 + 12 X + _?. _ Vlll. ix. ( __ ?__ ) 2 = _? _ - 28 ab 2 + 49 b4 x. [(2a+b)+c]2= (2a+b) 2 + __ ?__ + _?_ xi. [(1 - 3 b) - 1]2 = __ ?__ - 2 (1 - 3 b) + _?_ .. 11.

You may use the answers below to check the missing terms that you have supplied. 4 x 2 + 4 xy + y 2 9 a2 - 30 ab + 25 b2 x 2 + 10 xa + 25 a2 (4x + S) 2 (2a-7 )2 =4a 2 - 28a + 49 (7 c + 1 ) 2 = 49 c2 + 14 c + 1 (5 x + 4 b) 2 = 25 x 2 + 40 xb + 16 b2 (6x+1 ) 2 =36x 2 +12x+ l (2 a - 7 b2 ) 2 = 4 a2 - 28 ab 2 + 49 b4 [(2a+b)+ c]2=(2a+b) 2 + 2(2a+b)c + c2 xi. [(1- 3 b)- 1]2 = (1-3 b) 2 - 2(1- 3 b) + 1

i. ii. iii. iv. v. vi. vii. viii. ix. x.

Example.

Factor 6 ax 2 - 24 axy + 24 ay 2 •

Discussion.

We observe that the GCF of 6 ax 2 , - 24 axy, and 24 ay 2 is 6 a. We "factor out" this monomial and obtain 6 ax 2

-

24 axy + 24 ay 2

= 6 a(x 2 -

4 xy + 4 y 2 ).

Now we recognize that (x 2 - 4 xy + 4 y2) = (x - 2 y) 2 • Therefore, 6 ax 2 - 24 axy + 24 ay 2 = 6 a(x - 2 y) 2 •

380

CHAPTER

0

9

EXERCISES

1. Which of the following expressions are perfect squares? a. x 2 + 8 x + 16 b. p2 - 2 p + 1

e. 64 a 2 + 8 a + 1

f. 16 x 2 + 24 x - 9 g. (a+ 1) 2 - 2(a + 1) + 1 h. (r + s) 2 + (r + s) + 1

c. r 2 + 2 rs+ s2 d. 9 t2 + 30 t + 25

2. Fill in each missing term so that the result is a perfect square. a. x 2 + _? _ + 25

e. _? _ + 24 x

+ 16 f. _? _ + 8 uv + 16 v2 g. 9 y 2 + 12 yz + _?_ h. (x + 1) _? _ + 25

b. y2 + 16 y + _? _ c. a 2b2 + _? _ + 9

d. a2 + _?_ + 9 b2

2 -

3. Find: a. (x + 5) 2

f. - 2(3 a+ 1) 2

b. (x - 3) 2 (2 d. (3 C.

X

+ 1)

X -

g.(2a-b) 2 h. a(4x- 7) 2

2

i. (5 m - n) 2

5) 2

e. (4 - b) 2

j.(6a-5b) 2

4. Factor each of the following polynomials. a. x 2 + 6 x + 9 f. 3 x 2 + 6 x + 3 b. a2 + 10 a+ 25 g. 5 a2 - 20 a+ 20 c. y 2 - 14 y + 49 h. 11 m 2 + 22 mn + 11 n 2 d. 64 - 16 d + d2 i. a4 + 14 a2 + 49 e. 7 y 2 + 14 y + 1 j. x 4 + 8 x 2 + 16

5. Solve each of the following equations. a. (x + 7) 2 = x 2 + 42

b. (x + 3) 2 - (x - 5) 2 = 0 c. (2 x - 1) 2 - 3 x(x - 2)

=0

d. 4 a + (3 a+ = (2 a - 3) 2 - 29 e. (2 r + 5) 2 - (r - 3) 2 + r(r - 10) = 0 2

4) 2

6. Supply the missing words to make the following statement true . (a+ b) 2 = a2 + 2 ab+ b2 can be expressed in words as follows: The square of the __ ? _ _ of two numbers is the sum of their squares increased by __ ? _ _ their product.

381

Polynomials and Rational Expressions

0

EXERCISES

7. Find: a. (a + b

+ c)

2

b. (x + 3 y - z) 2

c. (x - y - z) 2

8. a. Following the plan that we used for (a + b) 2 , show that (a - b) 2 = a2 - 2 ab+ b2 .

b. Complete: The square of the difference of two real numbers is equal to __ ? __ .

9. Following the hints given below, prove that if a and b are real numbers, a2 + b2 ~ 2 ab. Proof. (a - b) 2 ~ 0 Why? (a -b) 2 =a 2 -2 ab+b 2 Provedin Ex.8 a2 - 2 ab+ b2 ~ 0 Why? The remainder of the proof is left to you.

10. a. Find (a + b)3 •

b. Find (a - b)3 .

Factoring after Completing the Square

Some polynomials can be factored by a process involving completing the square. The following examples will help you to understand the procedure. Do not be concerned that the examples involve polynomials that can be factored more easily by other methods.

+ 16 x + 55 .

Example 1.

Factor x 2

Solution.

We begin by adding a number to x 2 16 x that will produce a perfect square, that is, a number that will give us a polynomial of the form a2 2 ab + b2 . x2 + 16 x + (?) 2 a 2 2 ab+ b2 x 2 2 x(?) (?) 2

+

+

+ +

+

Do you see that the number we must add to x2 + 16 x is (½• 16) 2 , or 64? If we add 64 to x2 + 16 x, we have the perfect squa re number x2 + 16 x + 64. But since adding 64 to x2 + 16 x + 55 increases it by 64, we also subtract 64 so that th e value of the expression remains the same. Thus we have:

x2 + 16 x + 55 = x2 + 16 x + 64 - 64+ 55

= (x 2 + 16 X + 64) = (x +8) 2 - 32 = [(x + 8) + 3][(x + = (x + ll )(x + 5)

9 8)- 3]

382

CHAPTER

9

Example 2.

Factor x 2 - 10 x + 16.

Solution.

We add a number to x2 - 10 x to produce a perfect square. This number is 25. We have: x 2 - 10 x + 16= x2 - 10 x + 25- 25+ 16 = (x 2 - 10 X + 25) - 9 = (x- 5) 2 - 9 = [(x- 5) 3][(x- 5)- 3] = (x- 2)(x- 8)

+

0

EXERCISES

1. Fill each of the following blanks with a monomial which will make the polynomial a perfect square.

a. x 2 + 6 x + _? _

b.

y2 - 8 y

+ _? _

d.

x4

-

8 x 2 + _? _

e. _?_ + 10 rs + 25

f. m 2 - 12

m+ 10+ _?_

2. Factor each of the following polynomials after completing the square. a. x 2 + 12 x + 20 e. m 2 + 16 m + 39 b. y2 - 6 y + 8 f. v2 + 12 v + 35 c. p2 - 14 p + 33 g. t 2 + 8 t + 8 + 4

d.

t2

-

14 t + 45

h. p2 q2 - 1o pq + 16

C,

EXERCISES

3. Fill each blank with a monomial to make the polynomial a perfect square.

a. 9 x 2 + 12 x + _? _

c. 25 x 2 + 30 x + _?_

b. 4 y 2 + - ? - + 9

d. - ? - - 60 X + 25

4. Factor each of the following expressions after completing the square. a. (r + 1) 2 + 8(r + 1) + 15

c. 9 x 2 + 6 x - 15

b. (x + 2) 2

d. 4 y 2 + 12 y + 5

-

lO(x + 2) + 21

Quadratic Polynomials

A polynomial of the second degree is called a quadratic polynomial. Thus 3 x 2 - 5 x + 2 is a quadratic polynomial in the one variable x, and 7 x 2 + 5 y 2 is a quadratic polynomial in the two variables x and y. To be a quadratic polynomial in one variable, the polynomial must have at least one term of the second degree with respect to the variable. Thus 3 x 2 ,

383

Polynomials and Rational Expressions

+ 5 t, 3 z2 - 3, and 4 x 2 + 7 x + 5 are quadratic polynomials with respect to one variable. Since the coefficients of the polynomials in our examples are all integers, we may say that these polynomials are quadratic polynomials over the integers. In general, 3 t2

ax 2 +bx+ c; (a, b, c e I)

is a quadratic polynomial over the integers in the one variable x provided a ~ 0. To be a quadratic polynomial in two variables, the polynomial must have both variables present and must have at least one term of the second degree with respect to one of the variables or both of them. Thus 3 x 2 + 4 y 2 - 5 x + 3, 7 x 2 - 9 + y, y 2 - x + y - 18, and 5 xy + 4 are quadratic polynomials in two variables. Do you agree that all of the polynomials in our examples are polynomials over the integers? In general, ax 2

+ bxy + cy + dx + ey +f; 2

(a, b, c, d, e, J e I)

is a quadratic polynomial over the integers in the two variables x and y provided at least one of the coefficients a, b, and c is not zero. Products Which Can Be Expressed as Quadratic Polynomials

You recognize that many of the products you have been writing have been equivalent to quadratic polynomials. For example (x + 3) (x + 2) is equivalent to the quadratic polynomial x 2 + 5 x + 6. Let us continue our study of such products. First let us examine the product (2 x + 3) ( 12 x - 5). (2

X

+ 3)(12 X -

5)

= 2 x(12 X - 5) + 3(12 X - 5) (0 (2) (3) (4) = 2 x(12 x) + 2 x(- 5) + 3(12 x) + 3(- 5) =

(1) 24 x 2

(2)

(3)

(4)

+ (- 10 x) + 36 x + (- 15)

= 2i12 2

l J1t~3~ W

When we examine the second line of this computation, we observe that the term numbered (1) is the product of 2 x and 12 x, the first terms of the original binomials. The term numbered (2) is the product of 2 x and - 5, the outer terms of the original binomials. The term numbered (3) is the product of 3 and 12 x, the inner terms of the original binomials. The term nu·m bered (4) is the product of 3 and - 5, the last terms of the original binomials. This suggests a way to remember how the product is formed without actually writing all of the steps shown in our computation above. We use the first letters of the words "first," "outer," "inner," and "last" to form the word FOIL.

384

CHAPTER

9

Using the letters F, 0, I, and L as a guide, we can write the product of our two binomials by writing the trinomial whose first term is the product of the two first terms of the given binomials, whose middle term is the sum of the products of their outer terms and inner terms, and whose last term is the product of their last terms. 2 x +3 The algebraic sum of the products of the outer and inner terms is called the cross product term. The reason 12 x - 5 for this is clearly seen when the multiplication is written 24 x 2 + 36 X as shown at the right. _ 10 x _ 15 A scheme used to help us remember something (as in the case of the word FOIL) is called a mnemonic 24 x 2 + 26 x - 15 cross product (ne mon'ik) device. Can you think of other mnemonic devices? Let us see how this one is applied.

>


0.

X

Solution.

E xamination of the inequality reveals that x -,6- 0. To solve the give n inequality we must multiply both members by x . Since the result of multipli cation by x depends upon th e sign of x (Theorems 19 and 20), we must treat each case separately.

> 0, - + 2 > 0 and x > 0 +-+ 3 + 2 x > 0 and x > 0 ~ X 2 x > - 3 and x > 0 +-+ x > - i and x > 0 +-+ x > 0 C ASE 2. If x < 0, } + 2 > 0 and x < 0 +-+ 3 + 2 x < 0 and x < 0 +-+ X 2 x < - 3 and x < 0 +-+ x < - i and x < 0 +-+ x< - l The solution set is {x I x > 0} U {x I x < - i} . Thus

C AS E

1. If x 3

{x l ~+2

>

o}= {x i x < -

!or x >

o}-

409

Polynomials and Rational Expressions

0

EXERCISES

Solve each of th e following equations and inequalities. 3

1.-+1= 0 X

8

2

12 _ x - S = x + 1

3

2. -+-= y 3 y

x- 6

5 8 3. - + 3 > m m

13. __i!_ + _1_ a2 - 1

14. - 6 a2

5.3-=_ 6_ x+ l x +6 6.

1 9 -+ a a

2

5 2a

7 + _2_ 3a

_s_ = _J_Q_ a -S

lO.

X -

1=

X -

2

a- 3 X

X

+3 + 10

x- 3

-

a+ 1

+ _1_ = 0 a- 1

-+ ___g_ +-6 - = 4 4 a+ 2 a - 2

15 __ c _= c - 2 c +2

c +l

16 _x + 10 = _3_ x +4 x- 6 17.

l= t+ 3 t2

t2

+t

18. _ 4 _ _ _1_= 4 p - 2 p - 4 p2 - 6 p + 8 19 _

6a + 8 S a - 1 3 - 15 a

20 . X + 2 + X

+3

X

1 6

+ 3=

X -

S

13 x2

-

2X

-

15

21. Th e sum of a n umber and its multipli cative inverse is \ 3 . Wha t a re the numbers? 22. T he sum of a num be r and three times its multiplicative inverse is What a re th e num be rs?

1 9• 4

23. One numbe r is 2 grea t er tha n another. If three times the smaller is clividecl by th e la rger, the quotient is 2 and the remainder is 1. Wha t are the numbers?

24. 1f th e sum of a number a nd its multipli ca tive inverse is cliviclecl by the diffe rence of th e number and its mu ltiplicative inverse, the quotient is i g_ What is th e number?

25. On the first clay of their trip last summer th e J enk ins fa mily had a n ave rage rate of travel 5 miles per hour grea ter tha n on the second clay, a nd they traveled 360 mil es in 1 hour less than on t he second clay . What was their rate of travel the first day?

410

CHAPTER 9

a- arn

26. Solve the formula S = -1- - for a. -r rl- a

27. Solve the formula S = - - l for l. r-

kL

28. Solve the formula R = D 2 for L. 29. Mrs. Cox took her husband to the airport and then started home at the same time his plane departed. She arrived at their home, which is 15 miles from the airport, at the same d r time that her husband was landing at the Kansas City airport, which 15 ? Mrs. Cox X is 144 miles from the hometown Mr. Cox 10 X - 10 144 ? airport. If the plane's rate was 10 miles per hour less than ten times the rate of Mrs. Cox's automobile, what were the rates of the automobile and of the plane? Meaning of Division of One Polynomial by Another Polynomial

In arithmetic a proper fraction is one whose numerator is less than its denominator. For example,¾, 1\ , and 1°3 are proper fractions. An improper fraction is one whose numerator is greater than or equal to its denominator. For example, !, ¼, and ~ ~ are improper fractions. You will recall that a rational expression is an algebraic expression that can be put into the form of a fraction whose numerator and denominator are polynomials over the integers. A rational expression involving one or more variables with non-zero exponents is a proper fraction if the degree of the numerator is less than the degree of the denominator; otherwise it is an im1 and x 2 + 2 xy + y 2 are proper fractions; x 5 , proper fraction . Thus 2x , -, 2 3 52 xx x+xy x x4 x 2 +4 x+ 7 . . 4 , and ----'--- are improper fract10ns. x x+l In arithmetic you learned to change an improper fraction to a mixed number, that is, to an integer plus a proper fraction. For example, you learned that 3 Quotient Divisor 2;f Dividend 6

1 Remainder In this case, 7 is the dividend, 2 the divisor, 3 the quotient, and 1 the remainder.

411

Polynomials and Rational Expressions

. ·1ar proce d ure on t he improper . f ract10n · x3 + · L et us use a s1m1 -l • T h"1s time X

we have x2 Quotient Divisor x)x 3 + 1 Dividend

xa 1 Remainder

In this case x 3 + 1 is the dividend, x the divisor, x 2 the quotient, and 1 the remainder. If we let N represent the dividend, D the divisor, Q the quotient, and R the remainder when Dis of lower degree than N, we can express the pattern of these examples by the statement

(degree of R less than the degree of D) or the equivalent statement

N=QD+R

(degree of R less than the degree of D).

These statements are really the definition of division of one polynomial by another. The statements really say: To divide a polynomial N by a polynomial D (when D is of lower degree than N) means to find a polynomial Q and a polynomial R such that the sum of Rand Q X Dis N. In our work the polynomials N, D, Q, and R will be polynomials over the rational numbers. However, the quotient of two polynomials over the rational numbers can be expressed as the quotient of two polynomials over the integers. Consequently it will be sufficient for us to consider only the case where N and D have integral coefficients. In arithmetic you learned that a division is not complete as long as the remainder is greater than or equal to the divisor. Thus in the 2 3 two divisions at the right the one with the remainder 3 is incom- 2)7 2)7 plete because 3 > 2. The division with the remainder 1 has been 4 6 carried as far as possible. Now you should learn that when one 3 1 polynomial is divided by another of lower degree, the division is incomplete as long as the degree of the remainder is greater than or equal to the degree of the divisor. Thus in the divisions below the one with the remainder

x 5 - x 4 + x 2 could have been carried further. The division with the remainder x 2 is complete. xz X

x5 - x4

+ xz

xz

412

C H APTER

9

How to Divide a Polynomial by a Polynomial

We need a systematic way t o divide a polynomial by another polynomial of lower degree . Study the method illustrated by the following example. 6 x2 + 5 X

13

-

Let us find - - - - - -· We choose the first term of the quotient so 2x - 3 that the degree of the first remainder is less than the degree of the dividend , 6 X 2 + 5 X - 13 . 3x 2 x - 3)6 x 2 + 5 x - 13 (dividend) 6 X2 - 9 X 14 x - 13

(first remainder)

The first remainder, 14 x - 13, is of lower degree than 6 x 2 + 5 x - 13. H owever, the degree of 14 x - 13 is not less than the degree of 2 x - 3, so we must continue the division, using 14 x - 13 as the new dividend and using the original divisor.

2X

-

3X +7 3)6 x 2 5 X

+

6

X2 -

-

13 .

9X 14 x - 13 14 X - 21

8

(new dividend) (final remainder)

We have come to a remainder, 8, whose degree is less than the degree of the divisor, 2 x - 3, and our division is complete. H ence

2+ 5 X 6X

-

2x- 3 Example

l.

13

= 3X +7+

8

2x - 3

+

. d'1cated d'1v1s1 . .on 5 x 3 - 2 x 7· P erform t h e m 3 X2 + x - 5

Solution.

3 x2 + x - 5)5 x3 +0 x2 - 2 x + 7 5x3.+ !x 2 - ¥ x - ½x2 + \ 9 x + 7

- ½x

2-

½ x+¥ \2x + ¥

Notice th a t the leading term ½ x was obtained by d ivid ing 5 x 3 by 3 x 2 . Simila rly, th e - ½in th e quotient was obtained by d ivid ing - ½x 2 by 3 x2 .

S x3 - 2 x +7 3:r 2

+ x- 5

5

5

¥ x+ ¥

3 x - 9 + 3 x2 + x -5

413

Polynomials and Rational Expressions

Example 2. E xpress 3 Y3 - y 77 + 5y

+ 13

· l a nd a as th e sum o f a po1y n om1a

proper fraction. Solution.

We arrange our work as follows:

3 y 2 - 15 y y + 5)3 y3 + 0 y 2 3 y3 15 y2 - 15 y 2 - 15 y2 -

2 77 y + 13

+

77 y + 13 75 y 2 y+ 13 2 y - 10 23

Note that we obtain 3 y 2 , the leading term in the quotient, by dividing 3 y3 , the leading term in the dividend, by y, the lea~ing term in the divisor. How do we obtain - 15 y, the second term in the quotient?

• 3 y3 - 77 y + 13 = 3 2 - 15 I - 2 + _?l_ .. Y+ 5 y ) y+ 5 Observe the advantage of arranging both the d ivisor and the d ividend in descending powers of y and of "hold ing a place" for the m issing power of y by writing O y 2 . Check.

We can use the relation N ?

3y 77 y+ 13 ='= (3 3 y 2 - 15 y - 2 y+5 3 y 3 - 15 y 2 2y 15 y2 - 75 y - 10 3y3 - 77y - 10

y2 -

3 -

= QD + R fo r checking our work. 15 y- 2)(y + 5) + 23

?

3 y3 - 77 y + 13 ='= 3 y3 - 77 y- 10 + 23 ✓ 3 y 3 - 77 y + 13 = 3 y3 - 77 y + 13 Example 3 .

If 2 x + 3 is a factor of 6 x3 + x 2 - 10 x + 3, fi nd the pri m e fac t ors.

Solution.

First we d ivide 6 .x3 + x 2 - 10 x + 3 by 2 x + 3, t hen we try to facto r the quotient so obtained.

+

3 x2 - 4 X 1 2x + 3)6x3 + x 2 - 10x+3 6 x3

+9x

2

- 8

2 -

x

- 8 x2 -

10 x+ 3 12 X 2x+ 3 2 x+ 3

. ·. 6 x 3 + x 2 - 10 x + 3 = (2 x + 3) (3 x2 - 4 x + 1) = (2 x + 3)(3 x - l )(x - 1)

414

CHAPTER

9

We may check by substituting 2 for x in both 6 x3 + x 2 - 10 x + 3 and (2 x + 3)(3 x - l )(x - 1). We have: ? 6 · 23 + 22 - 10 · 2+3 == (2 · 2+3) (3 · 2 -1 )(2-1) ? 48+ 4- 20+ 3 == 7 · 5 · 1 35 i:_ 35

Check.

0

EXERCISES

1. Express each of the following fractions as the sum of a polynomial and a proper fraction.

a.

x 2 + 4x + 5 3 x+

f. 6 y3 + 11 y 2 + 15 y + 7 3

b. x 2 - 5 x + 9

x3

g.

x -2 2a

2

c.

+ 11 a + 10 a+

y+ 1

+ x2 -

7 x - 1S x- 3

h. x 2 + 5 x+ 4

4

x+ 3

6 a2 + 5 a - 3 d. -----'---

. a3 + 6 a2 + 2 a - 1 a+2 . 6 m4 + 5 m3 + 2 m 2 - 5 m + 5 J· 2m- 1

l.--'---------'--- - -

2 a+ 1 x 3 - 2 x 2 - 11 x - 19 e. ------x - 5

2. Perform each of the following divisions. a. Divide 9 x 4 - 6 x 3 + 21 x 2 - 17 x + 2 by 3 x - 2.

b. Divide 2 a4 - a3 + a 2 - 3 a + 3 by 2 a - 1. c. Divide x 3

-

2 x + 1 by x + 1.

d. D ivide y3

-

8 by y - 2.

e. Divide 12 b3 + 11 b2 - 9 b + 2 by 5 + 3 b.

f. Divide 2 x 3 + x 2 + x 4 - 4 x + 1 by x - 1.

3. Determine k so that (3 x 2 + 7 x + k) 3 x3

-

x2

-

+

(x + 1) has no remainder.

kx - 1

4. Determine k so that - - - - - - - is equivalent to a polynomial . x -1 over the mtegers .

5. Determine k so that 10 x 3 - x 2 + kx - 3 is exactly divisible by 2 x - 1. 6. Determine k so that 2 x - 1 is a factor of 2 x 2 + x - k. 7. One facto r of 6 x 2 - x - 117 is known to be 2 x - 9. Use division to find the other.

8. Use division to find the other factor of a3 to be a - 3.

-

27 when one factor is known

415

Polynomials and Rational Expressions

9. Give the complete factorization of each of the following polynomials. One of the fac tors of each polynomial is shown.

a. a3 + a2 - 16 a - 16 = (a+ 1) X ? X ?

b. a3 + 9 c. x 3

-

a2

125

+ 26 a+ 24 = (a + 4) X =

(x 2

?X ?

+ 5 x + 25) X ?

d. 6 x 3 + 17 x 2 + 14 x + 3 = (2

x

+ 3) X ? X ?

10. Perform the indicated divisions and check your answers.

+-x -+-3-x -+-x-+-2 ax. 4

3

2

+1 + 5 x3 + 5 x2 - x - 2 b. ---'------'-----x2

x4

x2

+2 +1 X

3 x4 + 17-x 2+-15-x c. - 7-x -+x2 + 3 X

d.

x 5 - 3 x4

+ x3 + 4 x 2 X

3

+

4

12 x

+4

ESSENTIALS

Before you leave Chapter 9 make sure that you

1. Can identify the degree of a polynomial. (Pages 362- 363 .) 2. Can add polynomials. (Page 363.) 3. Know what we mean by a "factorable" polynomial and a " non-factorable" polynomial. (Page 365 .) 4. Can multiply polynomials . (Pages 366-368 .) 5. Can solve equations involving the products of polynomials. (Page 370 .) 6. Can factor a polynomial over the integers by each of the methods listed . (Page 392 .)

7. Can solve equations involving factoring of polynomials.

(Pages 389

and 393.)

8. Can factor polynomials over the rational numbers. (Page 395 .) 9. Can simplify rational expressions. (Page 399 .)

10. Can add, subtract, multiply, and divide rational express10ns. (Pages 401-403 .)

11. Can solve fra ctional equations and inequalities . (Page 407.) 12. Can divide polynomials. (Page 4 10 .) 13. Know the meaning of the following expressions and can spell the words in them. degree of a polynomial (Pages 362- 363 .) polynomial over the rational numbers (Page 395 .) quadratic equation (Page 389 .) rational expression (Page 397.) root of an equation (Page 389 .)

416

CHAPTER

9

CHAPTER REVIEW

1. If xy = 0 and x > 0, what can you say about y? 2. Write the indicated product of the sum and difference of 6 x and 7 y. 3. Write the expression indicated by: a. The square of the sum of a and b.

b.

The sum of the squares of a and b.

4. State the degree of each of the following polynomials. a. 4 x 5

e. 5 x 3 + 6 x 2 + 3 x + 1

c. abc

f. a 8 - 1

b. -3 y

5.

What is the degree of 5 x 4 y

+ 7 xy + z with respect to y? 3

6. Add the polynomials in each exercise below. a. a2 + 3 ab - b2 ; 4 a2 + 6 b2 ; 6 ab - 4 b2 b. x 2 + y 2 ; 3 x 2 - 4 xy + 2 y 2 ; 6 xy c. 7 y 3

4 y2

-

-

2 y; 6 y 3

y2

-

-

y - 1; 2 y3

3 y2 + 1

-

d. 2 ab + a2 - b2 ; 3 b2 - 4 a2 ; 6 ab - 1; b2 e. 6 x 3 - 3 x 2 - 3 x f. 6 m 2 - 4 mn - 2 n 2 - 8 x2 + 6 x - 4 - 4 x2 - 2 x

- 7 m2 +

m2

mn

+ 3 mn + 2 n 2

7. Find the products of the following polynomials. a. 6 x 2 + 7 x + 3 x -1

f. 6 x - 4 y + z 2x -5

b. (4 r 2

-

g. (t - u - v)(t+u- v)

c. a(4 a

+ 1) (2 a -

5 rs+ s 2 )(3 r - 1)

h. (2 a+4)(5 - 3 a)

3)

i. 4 rs(t + 5)(t - 5)

d. (x + y)(x - y)x e. (x

+ 4)(x -

5)(x + 2)

j. (3 a+ b)(2 a - b)(a - 3 b)

8. Use the FOIL method to write products for each of the following pairs of binomials.

a. (x + 7) (x - 3)

b. (3

y+ 1)(2 y -1 )

c. (a+ 2 b) (a + 3 b)

e. (t - 5 a) (t - a)

d. (4r+s)(2r -s)

f. (x 2 -

y)(3 x 2

+ y)

9. Simplify: a. 4(a - 6) - 5(a + 3)

d. (4 x + 1) 2 + 16 (x - 3)

b. (r + s) (r - s) + s (r + s)

e. (a + l )(a + 5) - (a + 5)2

c. t(t - 5) - (t + 2) (t - 3)

f. (3

X -

4y

+ 5)(2 X -

3 y) - (x

+ y)

2

417

Polynomials and Ratio11a/ Expressio11s

10. Find th e roots of each of th e foll ow ing eq ua ti ons. a. (x + 1) (x + 3) - x(x + 3) = 8 b. (x + S) (x - 4) + (x - l )(x - 2) - 6 = 0 c. (x + 6) 2

(x - 1)2= 7(3x+S)

-

d. (4x+7 )(2x+3)-8=7x+2 11. Graph {x i (x - l ) (x +4)-(x+2) - x 2

~

10}.

12. Find th e soluti on set for each of the following se ntences . a. (t - 3)(t + 2) + (t - 2)(1 - 1) - 2 t 2 ~ 27

b. (x + 4) 2 - (x +6) 2 < (x - 3) (x + 8) -x 2 -5

13. Find the compl ete factorization of each of the foll owing polynomials. a. ax 2 - 9 a j. 2 abx - y + x - 2 aby b. 3 y 2 C.

x2

-

k. 4 - a 2 - 2 ab - b2

75

-

5X

-

l.

14

r4

-

16

d. x 2 -22 x+121

m. 3 x 2 + 18 x

e. ab + 3 b + ac + 3 c

n. 3 s 2 - 147 o. 6 y 2 - 19 y - 7

+

f. t3 - t 2 - t 1 g. a2 + 2 ab + b2 - c2 h. s m 2 - 180

+ 27

p. 6 X 2 - 11 X + 3 q. 9 12 - 24 t + 16 r. 9 b2 + 30 b + 25

i.6 n 2 +1 4n +4

14. Factor: a.

J x2 + S x + 6 x2+ 3 x + ~

b . 2

2

2

d.

¼x 2 -

2x+ 3

15. Find the roots of th e following equations. a. (x - 4) (x - 3) (x + 2) = 0

b.

r2

-

9r

+ 18 = o

c. s2 - s = 30 d. 3 x 2 = 27

e. xz = "(6

f. 2 t 2 - 15 t + 25 = 0 g. (2 X + 3) (3 X - 4) (x - 5) = 0 h. 6 t 2 = 13 t - 5

16. Whi ch of the following a re pairs of eq uivalent expressions? 4 a2b a. 5 ab 3 ; rst

b. 22; r t

a

Sb s rt

c. x(x + 5)(x - 3); (x 2 + 5 x)(x 2

d. x2 - 4 x + 3; x - 4 _ X

1 X

-

3 x)

418

CHAPTER 9

17. Simplify: 27 r 2t a.-9 rst 2

d 3x+6y •

r2 - s2 e.-r-s

a3 b2c

b.

g.

9X

4 ab 2 c2

+

x(x 3) c. x(x - 3)

f. a2 -

4(x + y) - 3(x + y) x+y

h. m 2 -m-20 m 2 -25

+ + 5 n- 2

2 . _6n____;_ 14n+4 1. _ _...;__

3 a+ 2

a-2

3 n2

18. Simplify:

r 2st t r rs

f.a 2 -a-6 __!!:!:_ a a 2 -4

a.-·-

x+3 x-3 x+2 g .-•-·-

b. m2n2 + .l_ 4

c.

x-3 x+3 x-2

mn

a(a + 4) . a 6 -.-- 3 a2 + 12

d. ~

(y y - 4 15

+

t+4

5 t - 15 t+4

+ 9 y + 20)

i. ( - 1 - 1)(-1 + 2) x+4 x+3

2 . 7 X + 35

j. ( - +

2

3 x+ 15

e.

h. 5 t2 - 8 t - 21

2 1) (--=-2_ - 1) x+l

x +l

+

19. Express in simplest form: a.

a

a

f._a _ _ _ b_ a+ b b- a

b+~

b .6 - - 7-

7 +-2g. x 2 + 5 X + 6 X + 2

9 3 c.-+y y2

h.

abc

abc

.

6

d. 14 t - St 7

4

x2 - 2 x - 3 2

+

1

x2 - 4 x + 3 3

1--------. t2 - 3 t 2 t 2 - 2 t 1

+

3

+

. _r_+ 6r J• r2 - 4 r2 + 4 r + 4

e.----a+4 a-4

20. Simplify each of the following. r2

~+1

a.!.

b. _b_

r s2

~-1

b

3 -1) 2 )+(c.(3-d. (x - 3) (x + 5 - ~4) 1-m +

m-1

419

Polynomials and Rational Expressions

21. Find the solution set for each of the following equations.

a. x + 2 = _3_ x +6 x +4

d. x + 7 + 2 x + 4 = 5

b. r + 1 _ _2_ = 1

e. _7__ _!Q_ = 2 x + 3 x -1 1-x 6

r

r+7

c. s + 1 + s - 1 = § s-1 s s

x +3

x

f. a + 3 + _4_ = ~ a

a+l

3a

22. Express each of the following fractions as the sum of a polynomial and a proper fraction. 2 + 8 X + 12 ax. ----

x

+2

b. x 2 + x -

x+-5 -x -+-3 f. 4

12

x2 + x + 1

x +l

x 3 x + x- 3 c. - - - - - x - 3 d. (x3 + 4 x + 7) + (x - 1) 3 -

2

2

g.

x

3 -

6 x2 x2

-

-

7x - 1 1

h. (x3 - 64) + (x - 4)

23. If x - 2 is a factor of x3 - 3 x 2 - 10 x + 24, find the other factors. 24. Determine k so that x 2 - 14 x + k is a perfect square polynomial. 25. Determine k so that x 2 + kx + 36 is a perfect square polynomial. CHAPTER TEST

1.Add 7a 2 +4a-3; 6a 2 -7; 4a+3. Study each of the polynomials in Ex. 2-5. If the polynomial is factorable, write the factors. If it is non-factorable, write "non-factorable."

2. x 2 - 9

4. m 2 + 16

3. y 2 - 8 y + 16

5. ar 2 - 4 ar - 3 a

6. Simplify (3 r + S)(r - 1) - 2 r(r - 3). 7. Find the solution set for 3 x 2 + 23 x - 8 = 0.

8. Graph the solution set for x 2 - 6 x = 7. 9. Solve (x + 3)(x - 4) - x(x - 3) :2:: 6.

1+! 10. Simplify

X

3+~ X

11. Find the complete factorization of x 2 + x + a - a 2 •

420

CHAPTER

12. Find the roots of

!- X

13. Simplify

2-

X -

1

=

9

6 . 2 x2 + X

- 1)(-1 + 2)(_1 a+ a -2 3

2 10 14. Simplify a +5 3 a a2 + 5 a+ 6 15. Graph the solution set for {x [ x 2 - 24 > I}.

16. Find x3 - 5 x2 + 2 x -+- 8_ x -2

17. The difference of the squares of two consecutive integers is 35. What are the integers? Which of th e st atements in Ex. 18- 21 a re always true?

18.

20 _6 rst = 3 r2 s2 t

i = 4+a 5+a

5

r2t

21 a + 3 _ (a + 3)(a - 2) - a2 + 9 · a2 - a - (a + l )(a -1 ) - (a -1 )

1

1

½rs 2t

19.-=--l- x x -1

22. Find k so that x 2 - 18 x + k is a perfect sq uare polynomial. 1 - = ~23. Solve - 1-+-

t+3

24. If x

t-3

7

+ 6 is a factor of x 3 -

25. Simplify _ x _ - _ l_ x -3

3- x

13 x 2

-

30 x

+ 504, find

the other factors.

+ 2. CUMULATIVE REVIEW [Chapters 1-9]

1. Assuming tha t a e R , be R, and a > b, state whether each number represented below is positive or nega tive or cannot be determined as either positive or negative. a. a- b

b. I a 1-1 b I c. I a - b I

f. ab

1

d.-b-

-a

e. a 2

a2

h. --;;; b b-

+b

2

2. Which of the following sets contain a smallest number? a. the set of natural numbers

c. the set of integers

b. the set of whole numbers

d. the set of real numbers

~ O

421

Polynomials and Rational E xpressions

3. If x 5 - 1 = Q(x + 1) + R , then what is the degree of Q, a polynomial in x?

4. Determine whether x - 3 is a factor of x 3

-

7 x 2 + 15 x - 9.

· 5. If x represents a real number, which of the following expressions represent real numbers for every value of x?

a. (x - 4) (x + 3)

c. x 2

b x 2 - 7 x + 14 • 7

-

14

5

f. -~

e.-x2+ 6

4

X

d.-5 x+

6. Assuming tha t a, b, x, and y are non-zero real numbers, simplify 2 a 22b. ab 6x2y2

7. Complete: The integer a is a factor of integer b if and only if there is __ ? __ ,

8. Find the truth set for ~ S - __i__5 = O; x ER. xx9. Find the truth set for (x - S)(x + 3)(x + 1) = 0. 10. Solve x(x - 4) = 3(x - 4); x ER. 11. F ind the simplest expression equivalent to each of the following. 1

3

d.-7-+_4_+_1_

a.--a2 4a

a2 - 9

e.

b.3 -+_1_ x+l

x-1

a- 3

4a + 8 + 6a2 - 5 a + 6 a - 3

2 7 f.---

4 5 c.------x2 2 X 3 X 6

+

a+ 3

+

a -b

b-a

3 2x+l 12. Write a simpler name for { x /;;=Sx ; X ER } ·

13. Graph the solution set for x I = 7. J

14. When are two open sentences equivalent ? 15. When is a sentence a statement? 16. Write the converse, inverse, and contrapositive of the state ment: If r, then s.

17. Write a contradiction of each of the following statements: a. This A is not a B.

c. No A's are B 's.

b. All A's are B's.

d. Some A's are B's.

422

CHAPTER

9

18. If n > - 2 and - 2 > b, what conclusion can you draw about n and b if they are real numbers?

19. Insert one of the symbols >, s and a > 0, then ar _? _ as. b. If r > sand a < 0, then ar _? _ as. c. If r > s and a= 0, then ar _? _ as. a. If r

20. Which of the following statements are true and which false if all the variables represent real numbers?

d. rs= (- r)(- s)

m -m a.-=-; n~0 n -n m -n

-m n

b . - = - ; n~0 m n

c. -

= -mr ; nr

n ~ 0, r ~ 0

f. n is odd -+ 2 n is odd

~ 0, c ~ 0, ands ~ - c, under what cir. . h r r+c? cumstances 1s 1t true t at - = - - .

21. If r, s, and care real numbers, s S

s+ C

22. Find the solution set for (x - l )(x - 5) - x(x + 4) = - (x 2

23. Solve X

2

5

+ 2)

2.

4

= -- - - · + 3 X x+3 X

24. Simplify: r4 _ s4

a.-sz _ y2

C,

1 (6 X 2 + 17 X 2x+ 5

25. Divide: a. (x 3 - 1) by (x - 1)

b. (3 x 3 - 7 x - 5 - ,10 x 2 ) by (x - 4) c. (5x 2 + 2x + 4) by (2x + 1)

26. Factor the following polynomials over the rational numbers. a. x 2 - ¼

27. Express as the sum of a polynomial and a proper fraction: 2 x3

-

11 x 2

-

x+2

6 x - 10

+ 5)

423

Polynomials and Rational Expressions

28. a. Assuming that p and q are simple statements, complete the following truth t able. p

q

(p -+q) ,...._., q (p and,...._., q) ,..._, (p and,...._., q)

T T T F -F T F

j

F

-

,~

b. What does the truth table tell us about the truth or falsity of the statement (p-+ q) and the statement ,....., (p and,....., q)? c. Replace the following question mark with a symbol to form a true statement : (p-+ q) _?_ ,....., (p and ,....., q).

29. Use a truth t able t o find a simpler statement fo r [,....., (,....., p or q) and p]. 30. In the Venn diagrams below R represents the set of real numbers, Q the set of rational numbers, and I the set of integers. Whi ch of the diagrams correctly pictures the relationship among R , Q, and I?

b.

a.

d.

c.

e.

CUMULATIVE TEST [Chapters 1-9]

1. Supply a reason for each step of the following proof: (1) a( pq +pr) = a(pq)

(2)

(3)

+ a( pr) = (a p)q + (ap )r = ap (q + r).

2. ·w rite the converse, inverse, and cont rapositive of the statement : If a is a negative real number, then a

~

0.

3. Graph the solution set of each of the following sentences.

IxI>

2 b. x > 2 or x < - 2 a,

424

CHAPTER

9

4. Copy the following table on your paper. In each space write the word "yes" if the property listed in the column on the left holds for the set of numbers indicated above the space in which you are writing. If the property is not tru~, write "no_. " Property

Set of natural numbers

Set of rati onal numbers

Set of real numbers

Multiplication is associative Additi on is commutative 0 is the identity element for addition I is the identity element for multiplication The set is closed under subtraction

5. If ax 2 + bx+ c is a polynomial ove r th e integers, which of the following statements are true and which false?

a. x must be an integer.

b. a,

b, and c must be integers.

c. a, b, and c are positive numbers.

6. Which of the sentences below have the null set as their truth set? a. 3 x - 1 = 5; x e C

C. X

> !;

b. 4 x - 3 = 6; x e C

d. x

► 3; x e W

X f

W

7. a. Complete the following truth table. (Assume that p and q are simple statements.)

p q ""i> (p-+q) ("" p or q) T T T F F T

F F

b. What does the truth table tell us about the truth sets of the two st at ements (p --+ q) and ("" p

or q)?

c. Replace the following question mark with a symbol which will make the statement true: (p --+ q) _? _ (,.._, p or q).

425

Polynomials and Rational Expressions

8. Whi ch of the following statements are true and which false if a, b, c, and d are real numbers?

a

a.

- a b

e. If a

b= -

b. (a - b)(c - d) = (b - a)(d - c) c. a · 0 = a

d. - (a - b) = -

a+ b

> > >

b, then a + c

>

b

+ c.

f. If a b, then b > a . g. If a b and c < 0, then ac < be. h. If k is a real numbe r such t hat a+ k

= 0, then k = -

a.

129. Solve - 6 - - ~ = - x

+2

X

X2

+ 2X

1-~- l_ X

10. Simplify

X2

1+lX 11. Express the product (2 x - 3) (4 x 3 - 5 x + 1) as an indicated sum. 12. Factor x 2

-

f x + ¾-

More about Theory of Numbers When we examine the first thirty counting numbers, 1, ~'

1, 4, ~'

6,

2,

8, 9,

10, .!_!, 12, _!1, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, we

find that those underlined are prime numbers. Recall that a prime number is a counting number greater than 1 that has no positive factors other than itself and 1. Remember also that a non-prime counting number other than 1 is called a composite number. One of the important theorems in the theory of numbers states that when the complete set of counting numbers is considered, the number of primes is infinite. We know that 17, 23, 29, and many other prime numbers are greater than the prime number 13, but even if we did not know this, we could prove that there is a prime number greater than 13. We could arrive at this conclusion by reasoning as follows: Since 13 is the sixth prime number, we shall let N6 = 2 • 3 • 5 • 7 • 11 • 13 + 1; that is, we shall let N 6 represent the number which is 1 greater than the product of the first six primes. Now N6 is either prime or composite. If N 6 is prime, then it is a prime number greater than 13 and our statement is proved. If N 6 is composite, it must have prime factors, one of which we shall call p. pis not equal to any of the numbers 2, 3, 5, 7, 11, or 13 because none of these is a factor of N 6 • Since p is a prime number different from any of the first six prime numbers, it must be a prime number greater than 13. Thus whether or not N6 is prime, we must conclude that there is a prime number greater than 13. Let us now prove the theorem: The number of primes in C is infinite. We begin by assuming that the number of primes is finite. We let n represent this finite number. We can arrange these n primes from least to greatest as follows: 2, 3, 5, · • ·, Pn• By reasoning similar to that of the preceding paragraph, we let Nn represent the number that is obtained by adding 1 to the product of these n primes. Thus Nn = 2 · 3 · 5 · 7 · 11 · · · Pn + l. Now Nn is either prime or composite. If Nn is prime, it is not one of the primes 2, 3, 5, 7, 11, • • •, Pn, and we have a contradiction since we are assuming these to be all of the primes. (Why is Nn not one of the primes mentioned?) If Nn is composite, it must be divisible by some prime p. It is evident that p cannot be 2, because if we divide Nn by 2, we get the remainder I. Similarly, p cannot be 3, 5, or any of the other primes 7, 11, · · ·, Pn• Since we are assuming that {2, 3, 5, 7, 11, · · ·, Pn} contains all of the primes, and since we have just shown that none of these can be a factor of Nn, we see that if Nn is composite we have another contradiction. So whether Nn is prime or composite, we have a contradiction of our original assumption that there is a finite number of primes. Therefore this assumption is false, and we have proved that there are infinitely many primes. Some subsets of C contain infinitely many primes. In fact there are infinitely many primes of the form 4 k - l when k e C. This means that the sequence 3, 7, 11, 15, • • • contains infinitely many primes. If you attempt to prove this, you should observe that any integer of the form 4 k - l when k e C is either 426

prime or has a factor of the form 4 k - 1 when k E C. Using this you can show that the supposition that there is a finite number n of primes of the form 4 k - 1 leads to a contradiction. There are many fascinating questions about prime numbers. The answers to some of these questions are not yet known. For example, in 1742 Christian Goldbach proposed that every even counting number except 2 can be expressed as the sum of two primes. Thus, 4 = 2 + 2, 8 = 5 + 3, 100 = 97 + 3, and so on. To this day no one has been able to prove or disprove this conjecture. Sometimes conjectures that seem very difficult to prove actually have simple proofs. For example, is it true that there are arbitrarily long sequences of consecutive composite counting numbers? Could we prove that there are as many as twenty-five consecutive numbers all of which are composite? (Observe that when we wrote the first thirty counting numbers the longest sequence of consecutive composite numbers we encountered consisted of the five numbers 24, 25, 26, 27, 28 .) To prove that a sequence of twenty-five composite counting numbers can be found, weconsiderthesequence26! + 2, 26! + 3, 26! + 4, • • •, 26 ! + 26 where 26 !, read as "26 factorial," means 1 • 2 • 3 • 4 • 5 • • • 26. There are twenty-five consecutive counting numbers in this sequence. The first, 26 ! + 2, is divisible by 2, since both 26 ! and 2 are divisible by 2. (Recall that, according to Theorem 31, if a is a factor of b and a is a factor of c, then a is a factor of b + c when a, b, c E /.) Similarly, 26 ! + 3 is divisible by 3, 26 ! + 4 is divisible by 4, and so on to and including 26 ! + 26, which is divisible by 26. Therefore, every integer in this sequence is composite. By similar reasoning we can find sequences containing any number of consecutive composite counting numbers. Isn't it remarkable that although there are infinitely many primes, there are arbitrarily long sequences in which no primes are found? Having considered twin primes 3 and 5, 5 and 7, 11 and 13, and so on, we know that consecutive primes may also be close together. Are there infinitely many such primes? No one knows, although some very distinguished mathematicians have tried to find the answer. What do you think? Finally, we might ask about prime triples such as 3, 5, and 7. Are there other prime triples, that is, three prime numbers whose common difference is 2?

427

Chap~

10 The Real Number Plane In this chapter we establish a relationship between pairs of real numbers and the points of a plane.

In previous chapters you studied sets of real numbers and their graphs in the number line. In this chapter you will study sets of pairs of numbers which you will graph in a plane . An example of such a set is

{(1, 3), (2, 6), (3, 9)} . Ordered Pairs of Numbers

The open sentences

(1) X - y = 3 (2) X + 2 y > 11 (3) y = x 2 are examples of open sentences having two variables. If we let x = 5 and y = 2, sentence (1) above becomes the true numeri cal sentence 5 - 2 = 3. If we let x = 2 and y = 5, sentence (1) becomes the false numerical sentence

2 - 5 = 3. Since interchanging the numbers represented by x and y can affect the truth or falsity of a sentence having these variables, we must be careful to specify which number of a pair of numbers is to replace x and which is to replace y. T o do thi s we write the members of the replacement set of an open sentence havin g the two variables x and y in the form (x, y). Note that we write x before y . By writing the numbers to replace x and y in the same order as x 428

429 and y in (x, y), we know which number is t o replace x and which is t o replace y. Thus (5, 2) means that 5 is t o replace x and that 2 is to replace y. (2, 5)

means that 2 is t o replace x and that 5 is to replace y . Since we attach importance t o the order in which the membe rs of such pairs appear, we call the pairs ordered pairs, and we read (x, y) as " the ordered pair x, y." In (x , y) we call x the first member and y the second m ember. We may use variables other than x and y t o represent an ordered pair of numbers provided we are careful to st ate whi ch is the first member and which the second member. Thus if we stat e that we are dealing with t he ordered p airs represented by (c, d), then (6, 3) indicates that c = 6 a nd d = 3. Open Sentences in Two Variables

N ow that we know what is meant by an ordered pair of numbers, we can examine the relationship between an open sentence with t wo variables and the set of all ordered pairs of real numbers. Consider the open sentence x - y = 5. If its replacement set is the set of all ordered pairs of real numbers, (x, y), then (7, 2) is a member of its replacement set. Substituting, we have the true numerical sentence

7 - 2 = 5. There are many other ordered pairs of real numbers which make this sent ence true. Among these are (5, 0), (6, 1) , (11 , 6), (3, - 2), (0, - 5), (- 3, - 8), and (7½, 2½) . Can you think of some other ordered pairs which make t he sentence true? There are also many pairs which make the sentence false . Among these are (1, 1), (17, 3) , (2, - 6), (0, 5), (1, 6), (6, 11 ), and (- 8, - 3). Similarly, a study of the open sentence y = x 2 reveals tha t (0, 0), (1, 1) , . (2, 4), (3, 9), (- 3, 9), (a, a 2), and many other ordered pairs make the sentence true, while (2, 2), (9, 3), (- 5, - 2), (¼, ½), and many other ordered pairs make it false. An open sentence with two varia bles acts as a sorter- it sorts any given set of ordered pairs of real numbers into two subsets: (1) The subset S of all ordered pairs in the given set which make the se nt ence true, and (2) The subset S' of all ordered pairs in the given set which make the se n~ t ence false . We call the first subset (S) the truth set or solution set of the sentence . An y ordered pair whi ch belongs to the solution set of a sentence wit h t wo va riables is called a solution of the sentence, and this ordered pair is said t o satisfy the sentence. Not e that S' , the subset of members fo r which t he sentence is false, is the complement of the truth set .

430

CHAPTER

10

The following examples show how an open sentence can be used to sort a given set of ordered pairs into the two subsets described on page 429. Example 1.

Use the open sentence x + y = - 7 to sort {(2, - 6), (5, - 12), (4½, - 2½), (- 7, 0), (- 8, 2), (3, - 5)} into subsets Sand S'.

Solution.

We test each ordered pair by substituting its first member for its second member for y in x + y = - 7.

(x, y)

x+y= -7

x

and

True or False

(2, - 6) 2+ (- 6)=-7 (5, - 12) 5 + (-12) =- 7 (- 4½, - 2½) - 4½ + (- 2½) = - 7 (- 7, O) -7+ 0 =- 7 (- 8, 2) -8+2=-7 (3, - 5) 3 + (- 5) =- 7

False True True True False False

.·. S= {(5 , -12), (- 4½, - 2½), (- 7, O)} S' = {(2, - 6), (- 8, 2), (3, - 5)}

We may use the set-builder notation to denote the truth set of an open sentence in two variables. For example, {(x, y) Ix + y = - 7} means "the set of all ordered pairs, (x, y), such that x + y = - 7." Example 2.

Find { (x, y) I y

>

2 \ - 5 } when (x, y) e {(l, - 1), (4, 1),

(- 2, - 1), (6, 1), (0, 0), (11, 6)} . Solution.

We test each ordered pair by substituting its first member for

x

5 . secon d member f or y my> . fi d its -2 x3- - We n :

2x-5 y> - 3

(x, y)

2 -1- 5 or -1 > -1 3 2 · 4- 5 (4, 1) 1> or 1 > 1 3 2(- 2)- 5 (- 2, - 1) -1 > or-1 > - 3 3 2-6-5 7 (6, 1) 1> or 1 > 3 3 (1, - 1)

-1

>

(0, 0)

0

>

(11, 6)

6

>

2·0-5 5 or 0 > -3 3 2 -11 -5 17 or 6 > 3 3

True or False False False True False True True

and

431

The Real Number Plane

The ordered pairs in the given replacement set which satisfy 2 x- 5 y > - 3are (- 2, - 1), (0, 0), and (11, 6). Thus { (x, y) \ y > 2 \ - 5 } =

{(-

2, -1), (0, 0), (11 , 6)}.

0 1. Given that P = {(1, 2), (2, 2), (3, 6), (6, - 2), (-

EXERCISES

1, - 2), (- 2, O)}, use

each of the sentences below to sort P into two subsets-the one whose members satisfy the sentence and the one whose members do not. a. x + y = 4

e. 2 x + y

>x d. y < x c. y

b. y = 2 x

=-

4

f. x < y + 2

2. Find the members of {(l, 2), (2, 4), (- 3, 9), (- 4, 1), (½, ¼), (- 4, 3)} which satisfy each of the following sentences.

> x+1

a. y = x 2

c. y

b. y = x + 1

d. I x I > j y j

= {(O, 3), (½, 1), (2, 4), (lowing subsets of P.

3. If P

a. {(x, y) / y

< 2 x; (x, y)

E

b. {(x,y) I y=x +3; (x,y)

t

e. (3 x + 1) < y

f. x +

½y = 4

3), (- 3, - 6), (- 4, 1)}, find the fol-

P} EP}

c. {(x,y) \ 2x+y=3; (x,y) EP}

d. {(x,y) \ ½x + ½y = ¾; (x,y) EP} Cartesian Products

Let us consider the two sets X = {l, 2, 3, 4} and Y = {1, 2, 3}. If we write all the possible ordered pairs of numbers having a first member from set X and a second member from set Y, we have the following pairs:

(1, (2, (3, (4,

1) 1) 1) 1)

(1, (2, (3, (4,

2) 2) 2) 2)

(1, (2, (3, (4,

3) 3) 3) 3)

that is, we have the set of ordered pairs

= {(l, 1), (1, 2), (1, 3), (2, 1), (2, 2), (2, 3), (3, 1), (3, 2), (3, 3), (4, 1), (4, 2), (4, 3)}.

P

We call this set the Cartesian product of X and Y. Instead of referring to the set as set P, we might refer to it as set X X Y. We read X X Y as "X cross Y."

432

CHAPTER

10

In the Cartesian product on the preceding page X and Y were not equal, but we may find the Cartesian product of equal sets. For example, the Cartesian product EXE when E = {1, 2, 3} is EXE= {(l, 1), (1, 2), (1, 3), (2, 1), (2, 2), (2, 3), (3, 1), (3, 2), (3, 3)}.

The set EXE is the set of all ordered pairs that may be formed from the elements of E.

Q

EXERCISES

1. If B = {l, 2, 3} and D = {l, 2}, find: a. B XD

b.DXB

Is the operation of finding the Cartesian product commutative?

2. Find M X N when M = {l, 2, 3, 4} and N = {5, 6}. 3. Find Y X Y when Y= {O, ½, 1, t 2}. 4. Find TX T if T = {- 2, - 1, 0, 1, 2}. 5. Find the set of all ordered pairs in BX B when B = {l, 2, 3, 4}. 6. If set P has six members, how many members has set P X P?

7. How many sets of ordered pairs are there in C X C when C is the set of all counting numbers? How many sets of ordered pairs are there in RX R when R is the set of all real numbers? Graphs of Sets of Ordered Pairs of Numbers

To show the graph of a set of ordered pairs we start with two number 1 lines. We place these number lines so that they have a common Y origin as shown in the diagram at 3 the right. As usual, we represent positive numbers to the right of the origin and negative numbers to the left of the origin on the X 0 horizontal number line, the line -2 -1 -3 3 parallel to the bottom of this -1 page. We represent positive numbers above the origin and -2 negative numbers below the ori-3 gin on the vertical number line, the line parallel to the left edge of this page. (Two lines are said to be parallel if they lie in the same plane and have no point in common.)

The Real Number Plane

433

We label the horizontal number line with an X, as shown in the diagram, and call it the x-axis . We label the vertical number line with a Y and call it the y-axis. These two lines are referred to as the coordinate axes . The surface on which they are drawn is called the coordinate plane and their intersection O is called the origin of the coordinate system we are studying. We now show that there is a one-to-one correspondence betwee n ordered pairs of numbers and points in the coordinate plane. Given any ordered pair of numbers, we assign a unique point in the plane to correspond to the pair. We illustrate the process by assigning a point to (- 5, 3). First we draw a vertical line through the point S whose coordinate is - 5 on the x-axis. Then we draw a horizontal line through the point T whose coordinate is 3 on the y-axis. The intersection of the two lines we have just drawn -is the point P which cory responds to the ordered pair (- 5, 3). 4 Thus P is the graph of the ordered pair P( - 5,3) T -------3 +'-----(- 5, 3). The first number, - 5, is the x2 £ coordinate of P and the second number, 3, is the y-coordinate of P. S A 1 2 3 4 5 - 4 -3 -2 -1 Observe th at the graph of the ordered - 1 pair (- 5, 0) is the point Sin the x-axis. -2 Indeed each point in the x-axis is th e - 3 F graph of an ordered pair whose second member (y-coord inate) is zero. A is the graph of (- 4, 0), Bis the graph of (- 2, 0), Dis the graph of (4, 0), and the origin O is the graph of (0, 0). Similarly each point on the y-axis is the graph of an ordered pair whose first member (x-coordinate) is zero. Eis the grap h of (0, 2), F is the graph of (O, - 3), etc. Given any point in the plane, we now assign a unique ordered pair of numbers to correspond to the point. We illustrate this process by assigning a n ordered pair of numbers to the point Q in y the diagram. First we draw a vertical line 4 through Q and read the x-coordina te of the 3 point K where the line we have drawn intersects the x-ax is. This is found to be 7. 1 Next we draw a horizontal line through Q 0 K X and read the y-coordinate of the point W 1 2 3 4 5 6 78 -1 where this horizontal line intersects the -2 y-ax is. This is found to be - 4. We assign -3 (7, - 4) to the point Q. The numbers of --4~w'--------~Q__ this ordered pair a re called the coordinates of Q. As indicated above, the first number,

434

CHAPTER

10

7, is called the x-coordinate and the second number, - 4, is called they-coordinate. Sometimes the x-coordinate of a point is called the abscissa of the point and they-coordinate is called the ordinate of the point. Thus we may say: The abscissa of the point Q is 7 and the ordinate of the point is - 4. Points are in the same vertical line if and only if they have the same abscissa, and points are in the same horizontal line if and only if they have the same ordinate. Thus all the points in the vertical line QK have the abscissa 7 and all the points in the horizontal line WQ have the ordinate - 4. What is the abscissa of each point in the y-axis? What is the ordinate of each point in the x-axis? Given an ordered pair of numbers, we have shown how to find a unique point in the coordinate plane which corresponds to it. Also, given a point in the coordinate plane, we have shown how to find a unique ordered pair of numbers which corresponds to it. Thus we have established a one-to-one correspondence between ordered pairs of numbers and the points in the coordinate plane. Let us now draw the graph of the Cartesian product X X Y where X = {0, 1, 2, 3} and Y= {l, 2, 3}. We see that X X Y= {(0, 1), (0, 2), (0, 3), (1, 1), (1, 2), (1, 3), (2, 1), (2, 2), (2, 3), (3, 1), (3, 2), (3, 3)}. We first draw the x-axis and the y-axis as previously described. On the x-axis we locate the points whose x-coordina tes are O, 1, 2, and 3, and we draw a vertical line through each point. On the y-axis we locate the points whose y-coordinates are 1, 2, and 3, and we draw a horizontal line through each of these points. y

-

(0,3) (1,3) (2,3) (3,3) (0,2) (1,2) (2,2) (3,2)

• T

.

w

(0,1) (1,1) (2,1) (3,1)

0

N

X

These seven lines form a lattice. The four vertical lines intersect the three horizontal lines in twelve points whose coordinates are shown in the diagram. For each of these points the coordinates were assigned by the procedure previously described. For example, point W is assigned the coordinates (3, 2) because a vertical line through W intersects the x-axis at the point N whose x-coordinate is 3 and a horizontal line through W intersects the y-axis at a point T whose y-coordinate is 2. Accordingly point W has the abscissa 3 and ordinate 2. The twelve points shown in the diagram are the graph of X X Y.

435

The Real Number Plane

It is important to notice that there is a point of intersection to correspond to each of the number pairs in X X Y and that there is a number pair in XX Y to correspond to each of the points of intersection which we are at present considering. The graph of the set BX Bin which B = {- 3, - 2, - 1, 0, 1, 2, 3} is shown at the right. This set consists of the following number pairs: y

(- 3, - 3), (- 3, - 2), (- 3, - 1), (- 3, 0), (- 3, 1), (- 3, 2), (- 3, 3), (- 2, - 3), (- 2, - 2), (- 2, - 1), (- 2, 0), (- 2, 1), (- 2, 2), (- 2, 3), (- 1, - 3), (- 1, - 2), (-1, -1) , (- 1, 0), (-1, 1) , (- 1, 2), (- 1, 3), (0, - 3), (0, - 2), (0, - 1), (0, 0), (0, 1), (0, 2), (0, 3), (1, - 3), (1, - 2), (1, - 1), (1, 0), (1, 1), (1, 2), (1, 3), (2, - 3), (2, - 2), (2 , - 1), (2, 0), (2, 1), (2, 2), (2, 3), (3, - 3), (3, - 2), (3, - 1), (3, 0), (3, 1), (3, 2), (3, 3).

V

-

.

-,

X

0

-

-

'

:

-

The graph of the set consists of the forty-nine points which are the graphs of the forty-nine number pairs. We are frequently interested in a subset of a set of ordered pairs. For example, we might be interested in the subset E of the number pairs in B X B for which y = x. Study of the number pairs in B X B shows us that E = {(- 3, - 3), (- 2, - 2), (- 1, - 1), (0, 0), (1, 1), (2, 2), (3, 3)}. In the figure at the left below we have placed heavy black dots at the points whose coordinates are elements of E to distinguish them from the points whose coordinates are in B X B but not in E . To the right of the graph of E we have shown the graphs of two other subsets of BX B: the subset for which y > x and the subset for which y = 2 x + 1. y

y

. .

--

_, -

.

: -

V

V

:

.

X

0

'

:

:

Graph of E, the subset of number pairs in B X B for which y = x

-

-

-

.

.

X

D

'

: :

Graph of the subset of number pairs in B X B for which y > x

-

_,

y

')

X

:

:

Graph of the subset of number pairs in BX B for which y = 2 x + 1

436

CHAPTER

0 1. Copy the la ttice at the right and on it indi-

10

EXERCISES y

cat e the following points:

2

a. The point A whose abscissa is - 2 and

:

whose ordinate is 3

b. The point B whose abscissa is 3 and

-

-

0

-

X

whose ordinate is - 2

c. The point C whose abscissa is - 3 and whose ordinate is - 3

0

2

d. The point D whose coordinates a re (- 1, 3)

e. The point E whose coord ina tes are (2, 3)

f. F , the point whose coo rdinates a re (0, 2)

g. G, the p oint whose coordinates are (2, 0) 2. F ind the Cart esian product P of each of the pairs of sets X and Y indicated below.

a. X= {1, 2, 3}, Y= {l , 2}

b. X= {l , 2}, Y= {3, 4, 5} c. X = {- 3, - 2, - 1}, Y = {- 2, - l }

d. X= {-2 , -1, 0}, Y= {0, 1, 2} e. X

f•

= {-

4, - 3, - 1}, Y

= {-

X -- {.12 , 1.12 , 2'-J, Y -- tr1, 2 ,

2, 0}

31J-

3. Graph each of the Cartesia n products that you wrote in Ex. 2.

4. a. Graph the set of ordered pairs: P = {(1, 1), (1, 2), (1, 3), (1, 4) , (2, 1), (2, 2), (2, 3), (2, 4), (3, 1), (3, 2), (3, 3), (3, 4), (4, 1), (4, 2), (4, 3), (4, 4)} .

b. Using the graph that you have made for part a, place a small cross on the points of the graph whose coordina tes satisfy the sentence y when (x, y) represents a member of set P.

=x

c. Again use the graph that you made for part a, this time placing a heavy dot a t the points whose coordinates satisfy the inequality y > x when (x, y) represents a member of set P.

d. Describe the graph of the solution se t for the sentence y 2: x when (x, y) represents an element of set Pin part a.

437

The Real Number Plane

5. Graph the set

P of Ex. 4 and on it place heavy dots at the points whose coordinates satisfy the equation y = 2 x when (x, y) represents any member of set P.

The Graph of IX I

Before we study the graph of I X I, let us examine the graphs of two other Cartesian products. First let us look briefly at the Cartesian product B X B where B = {l , 2, 3}. Do you agree that B X Bis {(1, 1), (1, 2) , (1, 3), (2, 1) , (2, 2), (2, 3), (3, 1), (3, 2), (3, 3)}? The graph of B X Bis shown as the set of nine red dots at the left below. At the right we have shown the graphs of two subsets of B X B by means of black dots. Of course, there are many other subsets of B X B . y

y

y

~

n

,

-

n

0

0

X

X

0

Graph of B X B where Subset of the points in the graph of B X EB = {l , 2, 3} the points whose abscissas are 3 (Shown as black dots)

0

X

Subset of the points in the graph of B X B- the points whose ordinates are 1 (Shown as black dots)

Now let us look at the Cartesian product T X T where T = {- 5, - 4, - 3, - 2, - 1, 0, 1, 2, 3, 4, 5}. Do you ag ree that T X T consists of 121 number pairs? The graphs of these number pairs are the 121 points of intersection in the lattice shown in red below. - y

u

438

CHAPTER

10

The following are graphs of four subsets of T X T. y

}'

0 ·! - I

-

X

-

0

-s -

IX

:

' Subset of the points m the graph of T X T - the points having positive integers as coordinates

Subset of the points in the graph of T X T - the points having non-negative integers as coordinates

y

y ?

-~

s-

u

rA

u

A

-~

1

p

_,

.,

Subset of the points m the graph of T X T - the points the sum of whose coordinates is 5

Subset of the the graph of T points having scissas equal to nates

points in the their abtheir ordiX T-

Now let us consider the graph of I X I where I is the set of all integers. Since I is an infinite set , do you agree that if we graph the elements of I in the horizontal axis and in the vertical axis, then the graphs will extend infinitely far in both directions along each axis? Accordingly, the graph of I XI will consist of the intersections formed when an infinite set of vertical lines is crossed by an infinite set of horizontal lines. Of course we cannot possibly draw all of these lines. We draw as many as necessary and include the origin, understanding that when we do so, we are graphing only a small part of the set I X I .

439

The Real Number Plane Example.

Use a graph to show the subset of all ordered pairs of numbers in IX I such that the sum of each pair of numbers is 2.

Solution

We draw a lattice of convenient size (or use a piece of graph paper). We choose one of the intersection points y as the origin 0 and draw and label the : x-axis and the y-axis. Since the sum of the two numbers in each pair is to be 2, we write: : x +y=2.

u

X

Solving for y, y= 2- x.

Now we replace x by any integer. Let us use - 4. Then y = 2 - (- 4) = 2

+4=

6.

We place a black dot at the point representing (- 4, 6). (Note. The black dots help us to distinguish the graph of the subset from the portion of the graph of I X I which is not in the subset. Remember, the intersection of any two lattice lines is a point of the graph of I XI .) Next we replace x by another integer to find the corresponding value of y. Proceeding in this way, we draw the graph shown above. The set of black dots is the graph of a portion of the required subset.

Q

EXERCISES

1. a. Graph the set BX B when B = {0, 1, 2, 3, 4}.

b. Using black dots, mark the points which represent the subset of B

XB

in which the second member of each ordered pair is one greater than the first member.

c. How many members are there in the subset of part

b?

2. a. GraphthesetSXSinwhichS= {-5,-4, - 3, -2 ,-1,0,1,2,3,4,5}.

b. How many members are there in set S

X S?

c. Using black dots, indicate the subset of S X S each of whose ordered pairs of numbers has its second member the negative of its first member, that is, the subset in which y = - x .

d. How many members are there in the subset of part c? 3. a. Graph the set TX T when T= {-3, - 2, -1, 0, 1, 2, 3}. b. How many members are there in the set T X T? c. Using black dots, indicate the subset of TX Tin which each ordered

440

CHAPTER

10

pair has its second member one greater than its first member, that is, the subset in which y = x + 1.

d. How many members are there in the subset of part c? 4. a. Graph the set TX T of Ex. 3 once more. This time use black dots to indicate the subset of T X T in which the second member of each ordered pair of numbers is less than or equal to the first member, that is, the subset in which y ~ x.

b. How many elements are there in the subset of part a? c. On the same graph use small crosses to indicate the subset of T X T in which y ~ - x.

d. How many points are there in the subset of part c? e. How many points are there in the intersection of the subset of part a and the subset of part c?

5.

a. Graph the set I X I where I is the set of all integers.

b. How many elements are there in I

X I?

c. Using black dots, indicate A, the subset of I X I in which y = x.

d. How many elements are there in set A? e. Using crosses, indicate B, the subset of IX I in which y = - x.

f. How many elements are there in set B? g. Describe the graph of A U B. h. Describe the graph of A n B. 6. a. Graph the set IX I. Then use black dots to indicate C, the subset of I X I in which y > x. b. On the same graph use small crosses to indicate D, the subset of IX I in which y < x. c. How many points are there in CUD?

d. How many points are there in C n D?

7. a. Graph the set I XI and then use black dots to indicate E, the subset of I X I in which y

= 2 x.

b. On the same graph use crosses to indicate F, the subset of I XI in which y is always 2, that is, the subset in which y = 0 x + 2. Note. Replace x with various integers. The subset will have such elements as (- 5, 2), (- 4, 2), and (6, 2). c. Describe the graph of the set E U F.

d. Describe the graph of the set En F.

The Real Number Plane

441

8. a. Graph the set IX I. Then use black dots to indicate G, the subset of IX I in which y = 0 x + 3 . .

b. On the same graph use small crosses to indicate H, the subset of IX I in which x > 0 y + 3. c. Describe the graph of the set GU H.

d. Describe the graph of the set G n H. The Graphs of Q X Q and RX R

Do you recall that although the rational numbers (the members of set Q) seem to fill the number line, actually there are holes to be filled with such irrational numbers as 1r and V3? When we draw the lattice lines through points representing the rational numbers, there are spaces in which there are no lattice lines. This in turn means that there are points in the plane not represented by ordered pairs of rational numbers. The graph of Q X Q does not completely fill the coordinate plane. When we consider the set of real numbers (set R), however, every point on the number line corresponds to a member of the set and every member of the,set is represented by a point on the line. The graph of R X R, therefore, is the set of points formed by the intersections of lattice lines through all the points in the coordinate axes. In this case the points of intersection of the lattice lines completely fill the plane and there is no point in the plane not represented by an ordered pair of real numbers, and no pair of real numbers not represented by a point in the plane. We call the plane the real number plane. To graph R X R we draw a pair of coordinate axes and use our imagination to supply the points since every point in the plane is a point of the graph. Let us now graph some subsets of R X R. First let us graph the set of all ordered pairs of numbers in R X R in which y x > 0. The open sentence x > 0 acts as a sorter for the ordered pairs of numbers in R X R and hence for the points of the real number plane. We wish to graph all points (x, y) such that X 0 x > 0. Note that the given sentence is equivalent to the sentence x + 0 y > 0. Thus no restrictions have been placed on the second member, y. This means that we wish to graph all number pairs whose first member is a positive real number and whose second member is any real number.

442

CHAPTER

10

Every point having a positive x-coordinate (abscissa) lies to the right of the y-axis, and every point which lies to the y ij right of the y-axis has a positive x-coordinate. ' Therefore, the graph of the subset of R X R in ' , X •U which x > 0 is the entire portion of the plane to the right of the y-axis. Since we cannot possibly A IU draw all of the points, we shade the area to be included in the graph. The y-axis itself represents those points of the plane for which x = 0 and hence is not part of our graph. We indicate that these points are excluded by making the axis a dashed line as in the figure at the right. - ,- ';' Y Let us now graph the subset of R X R in which y l y > 1, that is, the set of points (x, y) for which I . 0 x + y > 1. This is the set of ordered pairs in I 0 IJ which xis any real number and y is a real number greater than 1. The graph is the shaded area indicated in the figure at the right. Note that the points having 1 as the ordinate are not included in the graph and that a dashed line has ' been drawn through these points.

r

.

Q X R in which y

> 0.

2. Graph the subset of R X R in which x

< 0.

1. Graph the subset of R

3. a. Graph A, the subset of' R

X R in which y

>-

EXERCISES

1. (Use vertical lines for

the shading.)

b. To the figure of part a now add the graph of B, the subset of R X R in which x

>-

1. (Use horizontal lines for the shading.)

c. Indicate the graph of A U B.

d. Indicate the graph of A

n

B.

4. a. Graph C, the subset of RX R in which y > - 2. (Use vertical lines for the shading.)

b. To the figure of part a now add the graph of D, the subset of RX R in which x < 3. (Use horizontal lines for the shading.) c. Indicate the graph of CUD.

d. Indicate the graph of C n D.

443

The Real Number Plane

5. a. Using one set of axes, graph the two subsets of RX R : E = and F = {x I x > 2}.

{x / x

< - 2}

b. Indicate the graph of E U F. c. Indicate the graph of En F.

6. a. Using one set of axes, graph the two subse ts of RX R : G = {x / x > - 3} and H = {x / x < 2}.

b. Indicate the graph of G U

H.

c. Indicate the graph of G n H . 7. Graph the subset of RX R in which y :s; 1.

8. Graph the subset of R

XR

~ -

in whi ch x

1.

9. Graph the subset of R X R in which x < - 3. Distance between Two Points

If two points have the same ordinate, it is easy to find the distance between them. By counting squares it is easy to see that the distance between B (3, 2) and S (8, 2) in the figure below is 5 units. Similarly, the distance between K (- 2, 2) and S (8, 2) is 10 units and the distance between W (- 8, 2) and S (8, 2) is 16 units. What is the distance between Kand B? between Wand B? ,- Y ..

n ,~

-

"

\

~ 1.. ,

.,,

..

- , ·6 -5 ·4 ·3 · 2 - 1

l

\

, ,

, \O bL

DjVJL,

L.,L.)

IA

') I

2

'

'

5

f

!

9

'

_,

Let us agree that the distance between two points will always be a nonnegative real number. After considering the examples above, it seems reasonable to think that the rule for finding the distance between two points with the same ordinates is: Take the absolute value of the difference of their abscissas. Thus the distance between B and S is equal to I abscissa of S abscissa of B I= I 8 - 3 I= 5. Observe that interchanging the abscissas does not affect the results. Thus I abscissa of B - abscissa of S I = I 3 - 8 I = I - 5 I= 5. Similarly, the distance between Wand Bis equal to I abscissa of B - abscissa of WI= I 3 - (- 8) I= 11.

444

CHAPTER

10

Generalizing this procedure, we let P and Q be two points having the same ordinate y1 (read "y sub one"). We let P have the abscissa x 1 and Q have the abscissa x2. Following the rule, we y have: The distance between P and Q = I x2 - x1 I- To shorten the writing of I::'. ,... , ,.Y I I' I the phrase "the distance between P and iA 0 Q" we write I PQ 1- Thus for points P and Q having the same ordinate, ~

J

I PQ I= I X2 -

X1

I-

We have learned that order makes no difference; that is, I QP I= I PQ IFor points P and Q having the same abscissa x 1 but different ordinates y 1 and y2,

I PQ I= I Y2 -

\•

YI

1-

y

/C I ~

\ · 1' 21

Observe that our symbol for the distance between two points is the familiar symbol for \A , .1 JI IA 0 absolute value. This emphasizes that such a distance is always positive or zero. The distance is positive if the points are distinct. Let us now consider a more difficult y question-how to find the distance beD17 ') _,.,.,,. tween two points P and Q which have / ~V" different abscissas and different ordi.... , / nates. Our method depends upon the V . Pythagorean Theorem, a theorem which p h ,I I() pertains to right triangles. -I p One of the angles of a right triangle is a . right angle (an angle such as that formed by the bottom edge and the left edge of this page). In the given triangle ABC, angle C is a right angle. The side opposite the right angle is called the hypotenuse B and the other two sides are called the legs of the right triangle. In right triangle ABC shown here, segment AB (denoted by AB) is the hypotenuse and AC and CB are legs. The A-------- c Pythagorean Theorem is: In a right triangle the square of the length of the hypotenuse is equal to the sum of the squares of the lengths of the legs. Let us return to the problem of finding I PQ / when P and Q have different abscissas and different ordinates. Suppose that Q has coordinates (7, 7) and that P has coordinates (- 5, 2). n

J

445

The Real Number Plane

We first draw the vertical lattice line through Q and the horizontal lattice line through P. It can be proved that these lines intersect at T to form a right angle PTQ. Thus triangle PTQ is a right triangle and by the Pythagorean h1 r/ 1\ - V L--- .... , theorem we know that I PQ

12

= I PT

12

+ I TQ

12 -

The coordinates of Tare (7, 2) because T has the same abscissa as Q and the same ordinate as P. We have

_..,.. bf

'/ of 3 x + y - 7 = 0 and 2 x - y - 8 = 0. I X V I - - Recalling that \ ,'

~,

7=

~

ab

= 0 +-+ a = 0 or b = 0

- - ,_

-

+ y - 7) (2 X - y - 8) = 0. To write 2 x + y = 4 or 5 x = 3 y - 1 in

I I

4

we see another way to write the compound sentence above. It is

(3

1/

~ ...2

I

6

~

-,

I

I/

X

-

--

\

I

\,

I

--

\

similar form, we first perform the operations necessary to make the left member of each equation equal to zero . Thus we have 2 x + y - 4 = 0 or 5 x - 3 y + 1 = 0 +-+ (2 x + y- 4)(5 x - 3 y + 1)

0

:::;=

0.

EXERCISES

1. Use ab= 0 +-+a= 0 orb= 0 to write each of the following sentences in another form .

a. x + y = 7 or x = 2 y- 4 d. (x - 3 y)(2 x + y - 1) = 0 b. (2 x - 3 y- 4)(3 x - y- 5) = 0 , e. y = 20 + x or 3 x = y f. J x + y = 0 or 2 y = - x - 2 c. x = - y or 2 x - 3 y = 50

465

The Real Number Plane

2. Make the graph of the truth se t of each of the following sentences.

+ y = 5 or x + y = 2 b. x - y = 1 or x + 2 y- 13 = 0 c. 4 x + 3 y = 8 or y = 4 - 2 x

g. (x + y - 5) (x - y - 1) = 0

d. 2 x + y = 4 or 2 y - x = 12

h. (y - 2x)(3x -y - 6)=0

a. 2 x

+ 3 or y = 4 x - 2 f. (2 x + 3 y - 6)(4 x + 6 y- 12) = 0 e. y = 4 x

Systems of Equations

Now let us consider conjunctive sentences whose clauses a re equa tions. You recall that th e compound ' sentence "a and b" is true only when both a and b are true. \ y I Let us consider th e compound sen\ tence I

3x + y - 7=0wd2x-y-8=0

I

, \ ,µ -

I/

I/

=3

V

I

'

I'\ lu "-

There are many other systems equivalent to these systems . Two of these are: 2 x+ y

>--

I/

\

I'- "

,_,l·2

j

\

'

I

I ')

'

' I"-

I

~

+

"'\: X-2,

' I"-."/

\

I/

----

l"-~0 1

I

N

/ I The system "x = 3 and y = - 2" is I \ I-" I I I"the simplest system equivalent to these y systems. In the graph at the right we I ';? I/ \ have added the graph of " x = 3 and 0 / ~ ' y = - 2" to the graphs of the two equiv, 1.:s.J ~ \ I alent systems shown in the figure above >->I /~ . 1 it. We have used dashed black lines for ✓-~ I'\. I \ ~ the graphs of x = 3 and y = - 2. We I / may interpret x = 3 to mean x + 0 y = 3. /\ u I'\. ' V I ID / Its graph is a line parallel to the y-axis. \ ' N I/ . Similarly y = - 2 may be interpreted as v , ') IA ~ - ,12 meaning Ox+ y = - 2. Its graph is a / I \ ~ / j line parallel to the x-axis. I + ~ " 1'-x-2I~ The system "x = 3 and y = - 2" is so I'-' / i/ \ ~o I / simple that we can " see" the solution, / J (3, - 2). Had we been able to find thi s \ .... I I II/ simple system, which is equivalent to the systems above, we would have known the solution of the given system without graphing or guessing . In the following paragraphs we will show first how to find a system equivalent to a given system and then how to find the simplest equivalent system. You have seen above that the graphs of the individual equations of equivalent systems all pass through the same point, the point whose coordinates are the ordered pair which is the solution of each of the systems. That is, the I

II

Q~z

'

' "" '

"

~

'

--""

468

CHAPTER

10

individual equations are all satisfied by the same ordered pair. To find another system equivalent to a given system, we must first find other equations whose graphs are straight lines passing through the point common to the graphs of the equations of the given system. The following discussion leads to a method for finding such equations. Let us begin by re-examining the system below, whose solution set we already know to be {(3, - 2)}. (1) {3 x+ y - 7 = 0 (2) 2x- y -8=0 Suppose we choose any two numbers as multipliers, for example, 5 and 2, and use equations (1) and (2) to form the new equation

5(3 X + y - 7) + 2(2 X - y - 8) = 0.

(3)

Simplifying this equation, we obtain the equivalent equation

19 x+3y-Sl =0. This last equation has the form ax+ by+ c = 0. Since any equation of this form has a graph which is a straight line, equation (3) is the equation of a straight line. Moreover, you may verify for yourself that the ordered pair (3, - 2) satisfies equation (3). Thus we have found what we were seeking- the equation of a straight line passing through the point whose coordinates are the solution of the given system. To convince you that this method will always enable you to find the equation of a straight line passing through the point whose coordinates are the solution of any system of two equations having exactly one solution, we offer the following argument. Let {

ax + by+c= 0 dx+ey+J=0

(1) (2)

be a system whose equations have graphs intersecting in the single point P (x 1, y1). We choose any two real numbers, k1 and k 2 , which are not both 0 and then form the new equation

k1 (ax+ by+ c) + k2(dx + ey + J) = 0.

(3)

Simplifying this equation, we have the equivalent equation

+ k1by + k1c + k2dx + k2ey + k2J = 0 (k1a + k2d)x + (k1 b + k2e)y + (k1c + k2J) = 0. k1ax

or

This last equation is a first-degree equation in two variables in a form which you should recognize as th e equation of a straight line, although its coefficients are more complicated than those with which you have been working. Since this last equation is a form of equation (3), the graph of equation (3)

469

The Real Number Plane

must be a straight line. W e wish t o know whether this straight line passes through P (x 1, y1), whose coordina tes satisfy the given syst em. Substituting (x1, y1) in (3) we have

k1 (ax 1 + by1

+ c) + k2(dx1 + ey1 + f) = 0.

Since (x 1 , y 1 ) satisfies the given syst em, we know tha t the expressions in parentheses h ave the value O; tha t is, ax1 + by1 + c = 0 a nd dx1 + ey1 + f = 0. H en ce the a bove equation is equivalent to the true st a tement

k1(O) + k2(0) = 0 or

0=0.

Thus we see tha t the number pair (x 1 , y 1 ) satisfies equa tion (3) as well as equa tions (1) a nd (2). We may now conclude tha t equa tion (3) is the equa tion of a st raight line through the point of intersection of the graphs of the two equa tions of the given system. Summarizing: If ax+ by+ c = 0 and dx

+ ey + f = 0 are the given equations of two lines

which intersect in exactly one point and if both

k1 and k2 are real numbers, not

0, then ki(ax + by+ c)

+ k2(dx + ey + f) = 0

is the equation of a line which passes through the point of intersection of the lines of the given equations.

Now the question a rises : H ow do we choose the multipliers k1 and k2 so as to find the simplest equation of a line through the intersection of two given lines? L et us return t o a study of the syst em we were originally considering, n amely,

{3 x + y - 7 = 0 2 x - y - 8 = 0. Y ou were t old tha t y = - 2 and x = 3, tha t is, 0 x + y = - 2 a nd x + 0 y = 3, are the simplest equa tions of st raight lines through the intersection of the given lines. How could you have arrived at these equa tions by yourself? Study of the coeffi cients of x in the original system shows us tha t if we multiply both members of the first equa tion by 2 and both members of the second equa tion by - 3, we shall have Ox when we add the new equations. Using 2 and - 3 as multipliers, we fo rm the new equation 2 (3

Thus

X

6

+yX

+

7)

+ (-

3)(2 2 y - 14 - 6 X

y - 8) = 0. + 3 y + 24 = 0 Ox +S y =-10 y =-2 X -

470

CHAPTER

10

Having found the equation y = - 2 which we want, we study the coefficients of y in the original system to see which multipliers will give us Oy in a new equation . The numbers we need are 1 and 1. We form the equation

1(3 Thus

X

+y-

7) + 1(2 X - y - 8) = 0. 3x+y - 7+2x-y - 8 =0 5X

+ 0 y = 15 x =3

Thus, using the equations of the given systems, we have found two simple equations, y = - 2 and x = 3, whose graphs are parallel to the x-axis and the y-axis respectively and which intersect in the point (3, - 2), whose coordinates are the solution of the given system of equations. The system formed of these simple equations is equivalent to the given system. {

X

=3

{3 x + y - 7 = 0 2 x - y - 8 =0

y =-2

Now we have a method by which we can find the simplest system equivalent to a given system. This is called the addition method. Since the solution of this simple system is obvious, once we have found this simple system, we can easily state the solution set for the given system. Example 1. Solution.

Solve : {3 x + 7 y - 23 = 0 5 x -11 y +75=0

(1)

(2)

To get an equation having the term Ox we choose the numbers 5 and - 3 to form 5(3 x + 7 y - 23) + (- 3)(5 x -11 y + 75) = 0. Thus 15 x + 35 y - 115 - 15 x + 33 y - 225 = 0 0 X 68 y - 340 = 0

+

y =S To get an equation havi ng the term Oy we use the multipliers 11 and 7 to form 11 (3 x + 7 y - 23) + 7(5 x - 11 y + 75) = 0. Thus 33 x + 77 y - 253+ 35 x- 77 y + 525= 0 68

X

+ 0 y + 272 = 0

x =-4 4 is equivalent to the original system. Its

The system {x =-=-y=::; solution set is {(- 4, S)}, so the solution set of the original system is {(- 4,5)}. We could have found the value of x in another way. H aving found that y = 5, we could have substituted 5 for y in (1); 3 x + 7(5) - 23 = 0 or x = - 4. That is, we could have solved the given system by solving {

3X

+7y -

y = 5.

23

=0

471

T he Real Number Plane

As a third way we could have substituted 5 for yin eq uation (2), 5 x 11 y 75 = 0. This would have given us 5 x - 11 (5) 75 = 0 or x = - 4. That is, we could have solved the given system by solving

+

+

{

5 x - l l y +75=0 y = 5.

The reason why these alternate methods of fi nding the value of x are acceptable is important. We state it as a theorem. Theorem 36 : If S

= {ax + by + c = 0

T_ -

and where

ey + f = 0 { ax + by + c = 0 k1(ax + by + c) + k2(dx + ey + f) = 0 dx +

k2 ~ 0, then the systems S and T a re equival ent.

Proof of Theorem 36: If S = {ax + by + c = 0 dx + ey + J =0

-

where k2

~

+ +

by c = 0 k1(ax + by+ c)

T _ {ax

and

+ k2(dx + ey + J) = 0

0, then the systems S and Tare equivalent .

To prove this theorem we must prove two statements:

(1) Any ordered pair that sat isfi es S also satisfies T . (2) Any ordered pair that sati sfies T also satisfies S. Proof of (1 ). If (x 1 , y1 ) satisfies the system S, we wish t o show that (x1 , y1) satisfies T . Since (x 1 , y1 ) satisfies S , we know that (x 1 , y 1 ) satisfies each of the equations in S, ax + by + c = 0 and dx + ey + f = 0. Now ax + by + c = 0 is also the first equation of T. We showed previously (page 469) that if (xi, y 1) sati sfies S, it satisfies h1 (ax + by + c) k2(dx + ey + J) = 0, the second equation in T. Thus since (x 1 , y1 ) satisfies both equati ons in T, it satisfies T. Proof of (2). If (x2 , y2) satisfies the system T , we wish to show that (x2, y2) sati sfies S. Since (x 2, y 2 ) sati sfies T , we know that (x2, y2) satisfies both equation s of T. It follows that the second equation of T becomes 0 + k2(dx2 ey2 + J) = 0, and since h2 ~ 0, dx 2 ey2 + c = 0. Thus both equations are satisfied. Since any ordered pair that satisfies S also satisfies T and any ordered pai r that satisfies T also sati sfies S , T and S are equivalent syst ems.

+

+

+

472

CHAPTER

10

Let us now turn our attention to some examples in which we solve systems by using the methods we have discussed. Example2. Solution.

.

Fmdthetruthsetof PART

{35 x+ 27y=3 _ x-

y-

100

21·

1. We write the equations so that the right member is zero.

{3x+7y+3=0

5 x - 2 y - 1,l--f- = 0

To form a simple equation of a line passing through the point whose coordinates are the solution set of the given system, we choose the multipliers 5 and- 3 in order to eliminate the variable x. We form the equation

5(3 x + 7 y + 3) + (- 3)(5 x - 2 y -1ff-) = 0. Simplifying, we have: 15 X + 35 y + 15 - 15

X

+ 6 y+

~o= Q 41 y= 1

2

~.2.

y=-t According to Theorem 36 we may obtain x by substituting - ,} for yin either of the given equations. Using the first equation, we have:

3 x +7 (- f) +3=0 3 x= 2

x=

¾

The truth set is{(¾,--})}. 2. We make sure that both equations of the original system are satisfied when x = ¾and y = - ,}. For 3 x + 7 y =- 3, we have:

PART

3(¾)

+

7(-t) J:_ 3 ? 2+ (- 5) ==- 3

-3i-3 For 5 x

-

2 y = ¼_0f-, we have: ?

5(¾) - 2(- t) ='= ¥f? 130 + ¥ == ¼_Of? H+H==¥f100 ::!_ 100 -2r-2r Example 3. Solve: {· 3 x+. 2 y- 6 =0

.2

Solution.

X -

.5 y - 23

=0

1. While we can solve this system without changing the form of its equations, we may find it easier first to write each equation as an equivalent equation without decimal fractions as coefficients.

PART

473

The Real Number Plane

Multiplying both members of each equation by 10, we have

f 3 X + 2 y - 60 = 0 l2 x - 5 y- 230= o. Using the multipliers - 2 and 3, we write: (-2)(3 x +2 y -60) +3 (2 x -5y-230)=0 - 6 X - 4 y + 120 + 6 X - 15 y- 690 = 0 -19y=570

y= -30 To find x we can substitute - 30 for y in either equation of either system. Let us use the first equation of the first system. We have: .3 x + .2 (- 30)- 6= 0 .3 x - 6 - 6 = 0 .3 X = 12 x =40 PART 2. We substitute 40 for x and - 30 for yin each of the original equations. ? ? .2 (40) - .5 (- 30) - 23 == 0 .3 (40) + .2 (- 30) - 6 == 0 ? ? 12- 6 - 6 == 0 8 +15-23 = 0

oio

oio

The solution set is {(40, - 30)}.

Q

EXERCISES

1. Find the equation of: a. The line through the intersection of the graphs of the equations in 6 . an d para11 el to t he x-axis. { x +y+6= O 2x-y+ = 0

b. The line through the intersection of the graphs of the equations in part a and parallel to the y-axis.

c. The horizontal line through the intersection of the graphs of the equa. . { x+y=2 tions m 2 x+y= 5 . d. The vertical line through the intersection of the graphs of the equations in part c.

2. Find the simplest system equivalent to each of the following systems of equations. a. {

x+y-9= 0 x-y-5=0

b. {X + 2 y + 2 = 0 2x - y-11=0

x- 3 y- 2 = 0 c. { 2x - y+6 =0

d.

{4 x + y + 6= 0 x - 2y - 3=0

3X +y = - 5 e. { -3 x -2 y =7

+ 3 y = 15 3x-4y =-3

f. { 2 X

474

CHAPTER

10

3. Solve each of the following systems of equations.

a. {2 x + y - 11 3X

b.

{3 x

-

=0 2y- 6= 0

h.

+4 y - 2 = 0 =0

{X+ 4 y - 6 = 0 • 2 X - 3 y + 21 = 0

d

.8 x

.9 Y - .8

I. {t x +2y=14 4

x- 3 y = - 13

2x = m. { i

f. {2 X - 3 y - 23 = 0 3 x - 4 y - 32 = 0

3X

.3 X + .6 y = - 9 .8 X - .6 y = 86 X

5 y = - 12

k. {.4x~ .9y '.:2.8

7x+4y-8=0 e. { 4x-2y+4=0

{

-

. {6 x =-6y+S J• x =-5 y+i-

2x+y-11=0

g.

+ 3 y = 11

i. {9 x = 6 y + 4 6x=-3y+5

2x-y+6=0 x- 4 y + 17 c. {

{4 x 6X

J y + 28

= 3 y+ 26

9 X - 7 y - 40 = 0 n. { 5 x+ 4 y + 38 = 0

= - 2 y - 11 = 4 y + 37

Example 4.

Solve: { 3 x

Solution.

This solution shows a varia tion of the addition method which we have been using to solve systems of equations . First we write each equation in the equivalent fo rm shown below.

{x +z y+ ll=O

3x -4 y - 37 =0

Now we multiply each member of the first equation by - 3 and each member of the second equation by 1, and add the equations .

(- 3) (x + 2 y + 11) = (- 3)0 1(3x -4y-37) = 1 · 0 - 3X3x -

0X-

6 y - 33 = 0 4 y - 37 =0 10 y - 70 = 0 - 10 y = 70

y =-7 Substituting - 7 for y in eith er equation of either system, we have

x = 3. T h e solution set is {(3, - 7)}. Many people find it convenient to set down their work in solving this problem in the following manner.

{x +Z y= -11 3 x - 4 y = 37

475

The Real Number Plane

Multiplying both members of the first equation by- 3 and adding the resulting equation to second equation, we have: - 3 x - 6 y= 33 3 x - 4y = 37 - 10 y = 70 y =-7

Substituting - 7 for yin x + 2 y = - 11, we have : x +2 (-7) =-11 x -14 =-11 x=3 The solu tion set is {(3, - 7)}.

4. Solve the following systems of equations by the addition method as shown in Example 4.

a. {x+ y = 3 -x+2y=-6

b

{3 X

{X + y = 7

• 2

X

h.

+ 3 y = 16

e.

{X + 7 y- 16 = 0 {

+ .2 y = 5.4 + .1 y = - .2 {130 = 2 X + 3 y

. { .3 .1

1.

_x-2y=6

- 2 x + y-13

{½ x = 6-y

- x =- y -12

3 x +y=32 c. {

d.

+2y = 9

g. - x + y =-V

j.

=0

X

X

8 = .1x+.2y

+3 y = - 5 x-2y=7

X

f. {2x+ y -L=0 6x= y

5. Using the addition method as shown in Example 4, find the solution set for each of the following compound sentences. a. x

+ y = 190 and 3 x -

2y

=-

5

b. 2 x - 3 y = - 17 and 4 x - 2 y = 18 c. 3 x + 4 y = 42 and 2 x - y = 6

d. 6 x + 4 y = 48 and ½x - y = 4

6. Using any of the methods shown in the above examples, solve each of the following systems of equations. a. 3 x

=

b. 4 x -

+3 y = 5 7 + 4 y = 0 and y + 2 x = 2

2 y - 2.25 and 2 x

c. 5 x - 3 =

- 3 y and 10 x - 3 y = 0

476

CHAPTER

Example 5. Solve:

i +~ = 12 y { ~X _ Z = _ 11 X

Solution.

10

y

Here again, we seek multipliers which will eliminate x or y. Let us try to eliminate y . To do this we choose 7 as the multiplier for the first equation and 2 as the multiplier for the second equation.

31

Adding, we have

= 62 31 = 62 X

X

½= x To find y we may elimin ate x in a similar manner or we may substitute ½ for x in either of the given equations. By either method we obtain y = ½, The solution set is{(½,½)}.

Note. In solving equations of this type it would not help to eliminate fractions by multiplying both members by xy . If we do this, we encounter difficulties.

7. Solve each of the following systems of equations. 10 X

c. { 1

+ y~ = 23 3

---=1 X

10

d { .

y

+ 12 + 4 = 0 y

X

S

4

X

y

----3=0

0

EXERCISE

8. Prove: If S= {ax +by + c =O and dx+ey+J=0 T = {k 1 (ax +by+ c) + k2(dx + ey + J) = 0 where ka(ax +by + c) + k4(dx + ey + j) = 0 k 1 k 2kak4 ~ 0 and k1 k4 ~ k 2ka, then the systems S and Tare equivalent.

477

T he Real Number Plane

Solving Systems of Equations by the Substitution Method

{

Consider the system

3y-x=-5 + 5 X = - 23.

y

Writing they-form of each equation, we have {

X 5 y=- - 3 3 y=-5 x - 23.

(1) (2)

The y-value in the solution will be the y-value common to the truth sets for (1) and (2). When the value of y in (1) is equal t o the value of yin (2), we may say

because of the transitive property of equality. Multiplying both members of this equation by 3, we have: X -

5=

16 X

- 15 X - 69 = - 64

x =- 4 This is the value of x when equations (1) and (2) have the same y-value. To find that y-value, we substitute - 4 for x in (2) . y=(-5)(-4)-23 y=-3 The solution set is {(- 4, - 3)}. Instead of writing both equations in y-form, we may shorten our work by writing only one in y-form as shown below. 3 y{ y

X

=-

5

= - 5 x - 23

Substituting - 5 x - 23 for yin 3 y - x = - 5, we have:

3(-5 x -23) - x =-5 - 15 X - 69 - X = - 5

= 64 x= - 4

- 16 X

Substituting - 4 for x in y = - 5 x - 23, we have: y = (-5)(-4) -23 y=-3 The soluti on set is {(- 4, - 3)}. We call this method of solving an equation the substitution method.

478

CHAPTER

0

10

EXERCISES

1. Use the substitution method to solve each of the following systems. X + 2 y = 14 y =2x-7

f. {3 X + 6 y = 13

a. {3

b.

9x+y=5

{2 x + y = 2 y=

c. {

X

{4 x -

+8

y= -

5

g. 3 X + y = - 9

3 x - y =-5 y = x +3

h.

{4 x+ y = -

5 -6x+ y =10

. {12 x - y =11 4x+2y=-1

d {x + y = - 7 • 2 X + y = - 10

1.

, {2 + 4y = 9

4X- y= 5 e. {

X

J• 3 X + 8 y

8x+y=1

= 14

2. Solve each of the following systems by expressing one equation in

x-form

and substituting the indi cated value of x in the other equation. P a rts a, b, and e a re already expressed in this form.

a. {3 x + 2 y = 16 x =3 y -2

d. {4 x - 3 y + 14 = 0

b. {4 x + 7 y = - 10

e. {3 x

x+6y+ 17=0

x=y+3 C,

= 5/ + 63

x =-

{3 X + 2 y = 22 x -4 y =5

3

y +3

f. {X+ 4 y = 5

-S x -12 y = - 19

Summary of Ways to Solve Systems of Equations

W e now have three ways t o solve a system such as

+

{3 X 7 y - 10 = 0 Sx - 3y=2. 1. We may draw the graphs of the equa t ions, estima te the coordina t es of the

point of intersection , and then check our estimat e by substituting the coordinat es in both equations of the system. 2. W e may ch oose k1 .and k2 so that k1 (3 x + 7 y - 10) + k2(S x - 3 y - 2) = 0 is the equation of a line through the intersecti on of the graphs of the two given equations and parallel to one of the axes. Simila rly , we find the equation for the line parallel to the other axis. Fro m these eq ua tions we may form th e simplest syst em which is equivalent to the given system. In this case we form {x y

=1 = 1.

This is the addition method. Theorem 36 may

be used as a variation of this method.

479

The Real Number Plane

3 . We may write the equa tions in they-form a nd equate values of y. This is what we are really doing when we use the substitution method. Note that it is sometimes simpler to write the equations in x-form and equa t e the values of x.

0

EXERCISES

Solve th e following systems of equations by any method that you wish to u se.

8 {3x+y=15 - (2x + y) · 12x-2 y =45-(3x+y)

1. {X + 3 y = 34 2x-5y=-31 2 . {4x+19=-3y x - 3y -14=0

9 _ {2 (x + y) = 3(x + y) - 5 3 X + 4 y = 18

3. {2

X - 3 y + 24 = 0 3 x - 5 y - 22 = 0

4 _ {½ x+ ¼y = 5 2 X + y = 22

5 {6x=19-2y · 3(x + y)

= 25½

6 _ {.4x+.ly:8 .2 x - .3 y- 11

7 {x-i=~

• 3X 5 - y =2

0 l3.

{2 + y = 7

17 {3(x+y)-4(y-x) = 6x + 4

X

· 2(2x+3y) - (x - 2)=4x +3 y

2 x= y y

14.

{

+3 X =

EXERCISES

l

2(x + y) _ y-x = y + 1

8

18. {

2x - 3y =l

3

2

4x - 3y - 2 x+y =-11

9

2

+ 1 _ y - 3 = 12 4 3 x + 3+y=l1 2

X

15.

16

{

{¼(x + 1) -

· ½(x + 2 y)

½(y + 3)

=

11

19

=5

{X + y = 9000 • .03 X + .04 y = .32(y -

X)

20 _ {3x+2y-(y-x)=5(x+l) 4x-(x - y)=Sx +l

480

CHAPTER

10

Types of Systems of Equations The graphs of two linear equations in two variables may have none of their points in common, one of their points in common, or all of. their points in common. When the graphs of two such equations have no point in common, the equations are called inconsistent ; otherwise they are called consistent. The following examples illust rate these relationships among the equations of a system. y

1. Inconsistent equations: T heir graphs are

-

pa rallel lines. For example, the two equations of {

h

:

3x ---7 y = 21

3x --- 7y =0 _.....

-

v

'"

0

,,, / ~

-

I-'""

:_.....'

-~

_.... )- y V

have graphs that are parallel. In this case there ✓ II Y is no point which is on both lines and conse- _..... .,,., quently there is no ordered pair which satisfies both equations. If S 1 is the solution set of the first equation and S2 is the solution set of the second equation, then S1 and S2 have no common elements; that is, S1 n S2, the solution set of the system, is the null set. The null set is the solution set fo r all systems of inconsistent equations. ;•

2a. Consistent equations: T heir graphs may

l's YI I '\J+I

I

have exactly one point in common. For example, the two eq uati ons of {

l'\J ,.

: 1---f'

+

y=5 x --- y = 3

X

I

0

-

/

0

I/

/ !1 / J'

/ '\ ~ ,1 /

V

~v

" '\

I

A

'"

I, / have graphs that intersect in exactly one point. I/. "\. If S1 is the solution set of x + y = 5 and S2 is the I/ solution set of x --- y = 3, then S1 n S2 has ex/ actly one member. It is (4, 1). Consistent equations such as these which have exactly one point in common are sometimes referred t o as simultaneous or indepeny dent equations.

2b. Consistent equations:

Their graphs may be the same line. In this case the solution sets of the two equations are exactly the same. The equati ons of 3 { 9

X --X ---

7 y = 13 21 y = 39

0

1..--

- 1

-"

h\

\l ' ,_ _......

9 :,:~ i.::- ' \

.,,.,

"✓

J

-

_,,,,.. -

0 I/

,.,,. .....

,,,,.. 11

481

The Real Number Plane

have the same line as their graph. If Sis the solution se t of 3 x - 7 y = 13, then S is the solution set of 9 x - 21 y = 39, and Sn S, the solution set of the system, is S. We have already noted that such equations are called equivalent equations. Consistent equations such as these are also referred t o as dependent equations. We have learned (page 460) that equations whose graphs are the same line can be readily recognized if we write the equations in y-form. It is also possible to recognize two equivalent equations by considering the ratios of corresponding coefficients when the equations are written in the form ax+ by + c = 0. Consider the following system of linear equations in two variables when none of the coefficients, a, b, c, d, e, and f, are zero.

{ax+ by+ c = 0 dx+ey+J=0 Writing the equations of this system in y-form, we have

These y-forms are the same if and only if

~ = ~ and £. = [. b

e

b

e

But these proportions are equivalent to the proportions a b c b - = - and - = -. d e f e

Thus the equations of the given system under the given conditions are equivalent if and only if

~ = ~ = ];

that is, if and only if all corresponding

coefficients are proportional. Thus either equation can be obtained by multi-

plying the members of the other by some real number. For example, in {3 X - 7 y = 13 9 X - 21 y = 39 we can obtain the second equation by multiplying both members of the first equation by 3, and we can obtain the first equation from the second by using ½as the multiplier. Note that we have discussed only those systems of linear equations in two variables having non-zero coefficients. You may be interested in making an independent study of systems in which some of the coefficients are zero to learn to recognize equivalent equations without first writing the y-forms.

482

CHAPTER

0

10

EXERCISES

State whether the equa tions within each system below are a consistent or an inconsistent pair. If the equations are consistent, state whether they are dependent (equivalent) or independent . Be ready to support your answers.

1. {x+y=6

4. {3x+y=-6

y= 7

3 X- y = - 6

X -

2. {3 X + 2 y = 4 3x+2 y =8 3, {2

X

4

X

5.

10 X = - 2 y + 20

{2 + 7y = 1 X

4

+ 5 y = 12 + 10 y = 24

7. {sx+y=10

X

8

+ 14 y = 2

{x- 3 y -

• 6

X -

1= 0

18 y - 6

=0

9 {4 X - 3 = 2 y . x=¾ +½ Y

6. {6 x - y = 1 6x=y-1

Problems Involving Systems of Equations

As you solve problems, you will find that some require the use of two or more variables, while others can be done using either one v a riable or two or more va riables. The problem in Example 1 below can be solved by using either one variable or two variables. Example l.

Find two numbers whose sum is 16 and whose difference is 6.

Solution. Using one variable.

1.

PART

Let 16 -

x

PART

x

Solution. Using two variables.

1. Let l represent the larger number and s represent the smaller number. {t+ s =16 l- s = 6 2 l= 22 l= 11 11+ s = 16 s= 5 PART

represent one number. Then represents the other number. x - (16- :r) = 6 x -16+ :r =6 2 x = 22 x = 11 16- :r= 16-11=5

2.

PART

2.

?

11- (16- 11 ) == 6

6~6

?

11

+ 5 == 16

16 ~ 16 ?

11-5==6

6~6 The solution set is {11}. The numbers are 11 and 5. Example 2.

The solution set is {(l, s) and l- s = 6} = {(11, 5)} . The numbers are 11 and 5.

I z+ s =

16

If a pen a nd 3 pencils cost 21 cents while 2 p ens a nd 4 p encils cost 36 cents , what is the cost of one pen and one pencil?

483

The Real Number Plane Solution.

PART

1.

Let n represent the number of cents in the cost of one pen and p represent the cost of one pencil. Then n + 3 p = 21 and 2 n + 4 p = 36. We may multiply both members of the first equation by - 2 and leave the second unchanged. Thus we have: {- 2 n- 6 p = - 42 2 n+ 4 p= 36 - 2 p=- 6

Substituting 3 for

PART

p=3 pin n+ 3 p= 21, we have: n+ 3(3) = 21 n= 12

2. ?

- 2(12) - 6(3) == - 42 ? - 24- 18 =- 42 ✓ -42 = - 42 ?

2(12) + 4(3) == 36 ? 24+12=36 36 i36 The solution set is {(p , n) I - 2 n - 6 p = - 42 and 2 n + 4 p = 36} = {(3, 12)}. Therefore each pen costs 12 cents and each pencil costs 3 cents. Example 3.

Steve can row upstream at a rate of 4 miles per hour and downstream at a rate of 6 miles per hour. At what rate can he row in still water and what is the rate of the current?

Solution.

Let x represent the number of miles that Steve can row in 1 hour in still water and y represent the number of miles that the current flows in 1 hour. Going upstream the current will hold Steve back so that x - y = 4 and going downstream the current will increase Steve's rate so that x y = 6. Thus we have the system

+

{x - y = 4 x+ y= 6. Solving this system, we have: 2 x = 10 x=S Substituting 5 for

x in x+ y= 6,

y= 1. The solution is (5, 1). Therefore Steve can row in still water at the rate of 5 miles per hour. The current flows at 1 mile per hour.

484

CHAPTER

0

10

EXERCISES

1. The sum of two numbers is 110. Their difference is 68. What are the numbers? 2. The sum of one number and four times another is - 7, while the sum of three times the first and twice the second is 9. What are the numbers? 3. At the fair Chris paid a total of 80 cents for two rides and three candy apples, while Chuck paid 85 cents for three rides and one candy apple. What was the cost of one ride if all the rides are the same price? of one candy apple if all apples are the same price?

4. Jack Selby works at a filling station. His first customer this morning bought 16 gallons of gasoline and 2 quarts of oil at a total cost of $5 .84. His second customer bought 12 gallons of gasoline and 1 quart of oil at a total cost of $4.22. What was the price of one gallon of gasoline and of one quart of oil if both customers bought the same kind of gasoline and the same kind of oil? 5. Linda's club has been selling magazine subscriptions. On Tuesday Linda sold three subscriptions for magazine A and four subscriptions for magazine B for a total of $28.50. On Wednesday she sold one subscription for magazine A and five subscriptions for magazine B for a total of $20.50 . What was the subscription price for each magazine?

6. The average of two numbers is 64. Their difference is 66. What are the numbers? 7. The average of two numbers is What are the numbers?

U.

One third of their difference is

3\ .

8. According to an important physical principle, two children on a seesaw will balance each other when the weight of one child times his distance from the fulcrum (the fL_ 3 ft . 5 ft. ~ point at which the board is supported) equals the Joe-45 lb. Don- 75 lb. weight of the other child times his distance from the fulcrum. Thus in the figure at the right Joe, who weighs 45 pounds and is sitting 5 feet from the fulcrum, balances Don, who weighs 75 pounds and is sitting 3 feet from the fulcrum , because 45 X 5 = 75 X 3.

·w--!i----$··

If Mary, who weighs 70 pounds, and Jill, who weighs 40 pounds, balance the seesaw when they sit 8 feet apart, how far from the fulcrum is each girl?

The Real Number Plane

485

9. The combined weight of two boys on a seesaw is 180 pounds. T he boys are balanced on the board when one sits 5 feet from the fulcrum and the other sits 4 feet from the fulcrum and on the other side of it. What are the weights of the boys?

10. A lever works on the same principle as a seesaw. If a weight of 6 pounds placed at one end of a twelve-inch lever is to balance a weight of 4 pounds at the other end of the lever, at what point on the lever must the fulcrum be placed?

11. At the grocery Mrs. Young spent $1.97 for three dozen eggs and two quarts of milk while Mrs. Adams spent $1.63 for two dozen eggs and three quarts of milk. What was the cost of one quart of milk? of one dozen eggs?

12. The music department of Eastern High School bought 12 band uniforms and 3 hats at a total cost of $615. Later the department bought 3 uniforms and 2 hats at a total cost of $165. If the two orders were for identical items, what was the cost of each uniform and each hat ?

13. The athletic department of Eastern High School bought 6 bats and 4 balls for $22. Later they bought 3 more bats and 2 balls for $11. Why is it that you cannot find the cost of one ball and one bat in this case although you could find the cost of one uniform and one hat in Exercise 12? What descriptive word applies to the system of equations in this exercise and what to the system in Exercise 12 ?

14. On its opening day 1348 persons attended the Madison County hobby show. If the admission price for an adult was 75 cents and the admission price for a child was 50 cents, how many adults and how many children attended that day if a total of $879.75 was collected from admissions?

15. Mr. Sanders recently deposited $8000, part of it in a bank paying 4% interest per year and part in a bank paying 3% interest per year. If the money is so divided that his total interest for the year will be $305, how much was deposited in each bank?

16. Mr. Sellers has invested one amount of money at 6% and another at 4% so that his total investment yields $370 per year. Were he to reverse the amounts, the yield would be only $330 per year. How much is invested at each rate?

17. A plane flying against the wind travels 270 miles per hour and flying with the wind (in the same direction as the wind) travels 360 miles per hour. What is the rate of the plane in still air? What is the rate of the wind?

486

CHAPTER

10

18. On a 480-mile trip a plane flying with the wind was able to cover the distance in 2 hours. On the return flight , flying against the wind , the trip required 2½ hours. What was the rate of the plane in still air? What was the rate of the wind?

Q

EXERCISES

19. Two weights balance on a lever when they are respectively 10 inches and 8 inches from the fulcrum and on opposite sides of it . If one pound is added t o each weight, the fulcrum must be adjusted so that fT of an inch is subtracted from the 10-inch part of the lever and / 1 of an inch added to the 8-inch part. What were the original weights? N ote. The lever referred to operates on the principle of a seesaw. The ful[wJ~ w_,_ _ _d_1_ _ _ _ _ _d _2_ _ _ _ [wJ _, crum is the point about which the lever turns. If W t and w2 fulcrum represent the number of units of W1 d 1 = W2 d 2 weight at the two ends of the lever and dt and d2 represent the respective distances of W t and w2 from the fulcrum , then any one of the fo ur quantities Wt , w 2 , dt , and d 2 can be found by the formula w 1d 1 = w2d2 when the other three are known.

/i

20. A company selling gasoline wishes t o mix some gasoline worth 32 cents per gallon with some worth 27 cents per gallon to make 1500 gallons of a

1500 gal. @ 30¢

\32x+ 27 y=? I x + y=? mixture which it can sell fo r 30 cents per gallon . How many gallons of each kind should the company use?

21. A syrup manufacturer wishes to mix two kinds of syrup, one selling at $6.00 per gallon and the other at $2.75 per gallon to make a mixture of 500 gallons that he can sell for $4. 75 per gallon. How much of each kind of syrup should he use?

487

The Real Number Plane

Graphs of Open Sentences Involving Inequalities

If we graph the set of numbe r pairs {· · ·, (- 2, - 4), (- 1, - 2), (0, O), (1, 2), (2, 4), (3, 6), ··•}, we have a set of points whose ordinat es a re equal to twice their a bscissas. \Ye have indicated y the points by the reel cl ots in the figure at the right. L et us first conside r th e point having the coordinat es (- 2, - 4) . When we locate other points having the a bscissa - 2 but having ordinates that are intege rs greater than - 4, we obtain a se ries of points above the A 0 point (- 2, - 4). These points are shown in , - I ~ black a bove the point (- 2, - 4) . \Ye have shown only a few of these points, but were we I to loca te all of them we would have an infinite i I set of black clots extending upwa rd from the I I I ! I reel clot at (- 2, - 4) . Similarly we can locate points a bove each of the points indicat ed by the other reel dots . You recognize th e points shown in reel as the graph of the truth set of y = 2 x; x, y f I. The points shown in black are points of the graph of the inequality y > 2 x; x , y f I. Let us now graph the truth set of the equation y = 2 x ; x , y f R. This time we have the set of points fo rmin g the line indi cated by the dashed line in the fi gure at the right. vVhen we locat e any J I ,{ I I I I point whose a bscissa is x but whose ord inate is a I ! I I I real number greater tha n 2 x , we find that it lies ,J , )y in the area a bove the dashed line. For exa mple, ._ ~ the point (O, 1), whose y-value is more than / 1 I i twice its x-value, is located a bove the clashed .A ~ line. When for each a bscissa of a point in the ',12 4 -~ I line indicated by the clashed lin e all of the points having ordina tes greater th an twi ce this - f-- I a bscissa are locat ed , then the space abo ve the I/ dashed line is complet ely filled with th ese points. .' , These points, indi cated by the shaded area in I the figure, are the graph of the inequality y > 2 x; x, y ER. Obse rve tha t the line represe nted by the clashed line is not part of the graph of the inequ ality. When we are asked t o graph an inequality, we first graph the equation which results when the symbol > or < is replaced by the symbol = . We then shade the portion of the plane whose points are the graph of the inequality . 0


x e. y < 3 x h. y > - 3 x y ~

,r

0

0

C.

I

}

>

0

.

EXERCISES

j. y < x+4 k. y > 3 x+4 l. y~Jx-1

2. Write an open sentence and draw a graph t o fit each of the sets of ordered pairs of real numbers described below. a. The set for which the ordinate is half as great as the abscissa

b. The set for which the ordinate is S less than the abscissa c. The set for which the ordinate, y, is greater than 2

+3 x

where x is

the abscissa

d. The set for which the ordinate is the opposite of the abscissa e. The set for which the ordinate is the opposite of the absolute value of

the abscissa

489

T he Real Number Plane

Graphs of Sentences Involving Absolute Value

The sentence / x / = 3 is equivalent to the compound sentence "x = 3 or x = - 3." The graph of the clause y "x = 3" is a line through the point (3, 0) and parallel to the y-axis. The graph of M I II II the clause "x = - 3" is a line through the point (- 3, 0) and parallel to the A 0 -p y-axis. Since a compound sentence with the connective or is true fo r the values which satisfy either clause, the graph of I x I = 3 consists of all points on both lines. To graph the truth set of the sentence Ix I > 3 we recall tha t "Ix I > 3" + + "x > 3 or - x > 3" ++ "x > 3 or y x < - 3. " Accordingly, the graph of , I x I > 3 consists of the points to the X , , .) ' X> right of the line whose equation is x = 3 A and to the left of the line whose equation 0 -I I ) I is x = - 3. The lines whose equati ons are x = 3 and x = - 3 have been made ! -i dashed lines to indicate that they are 1--~ - ·· -· not part of the graph of Ix I > 3. If our I I I I I I I I original equation had been Ix I 2: 3, then the graph of the sentence would include the graphs of x = 3 and x = - 3. We would have drawn the graph of Ix I 2: 3 the same as the graph of Ix I > 3 except that we would have made the graphs of x = 3 an'd x = - 3 solid lines to indicate that they were a part of the graph. n

n

0

n

,-f--~

L et us study the graph of y = Ix 1 - \,Ve make a t able of some values as shown below. When we graph each of y / rr S"'I/ - 3 -2 -1 0 1 2 3 X V

-

- -

y

3

- -- --

2

1

-

0

-

1

"1"'-

-

2

3

""

V

0

V

I'-. I/

/{

0 the ordered pairs shown in the table and I draw the lines connecting them in order, we have the graph shown at the right. The lines appear to form a right angle. If x 2: 0, then the equation is y = x and the graph is the ray OT. If x ~ 0, the equation is y = - x and the graph is the ray OS. The slope of ray OT is 1 and the slope of ray OS is - 1. Since the product of the slopes is - 1, we know tha t ray OS is perpendicular to ray OT. n

490

CHAPTER

0

10

EXERCISES

1. Draw the graph of each of the following equations.

+1 /= 3 d. I x + 4 I = 3

a. I x b.

I= 2 / x - 5 I= 2

c. I x

e. / y / = 7

f. I y + 1 / = 4

2. Draw the graph of each of the following sentences.

a. I x I > 4

c. I x

I< 2 d. I x I ~ 2

b. I x I ~ 4

e. I y I > 1 f. I y + 1 I ~ 2

3. Draw the graph of each of the following equations.

= 2 Ix I b. y = ½I x I a. y

= - Ix I d. y = - 2 I x I

I- 2 x I f. y = I - x I

e. y =

c. y

g. x = h. X =

I YI 1-y I

4. Draw the graph of each of the following equations.

I x I+ 1 b. y = I x I - 3 c. y = 3 I x I+ 2 d. y = - I x I+ 4

e. y = 2 I x

a. y =

f. x

I-

5

= I y I+ 2

g. x = 3 I y I - 4

h. x = - 2 I y I+ 3

5. Draw the graph of each of the following equations. a. y = I x - 1 I

b. y = I x

+4 I

c. y = I x - 5 I

e. y = 2 I x - 1 I

d. y = I 5 - x

f. y =

I

½I x -

Q

1I EXERCISES

6. Draw the graph of each of the following sentences.

a. I x

I+ I Y I = 5 b. I x I+ I y I < 5

c. I x

I + IY I ~ 4 d. I x I+ I y I 2 d. I x I - I Y I ~ 2

Systems Involving Inequalities

To write the conjunctive sentence "y + 3 x may write

{y+ 3 X 3 y-

~

X ~

-4 18.

~

- 4 and 3 y - x ~ 18" we

491

T he Real Number Plane

If we let S1 represent the truth se t of y

+3 x ~ -

4 and S2 represent the

truth set of 3 y - x ::::; 18, then S1 n S2 represents the truth set of the system. In the figure at the right we have made the graph of the sentence y + 3 x ~ - 4 and the graph of the sentence 3 y - x ::::; 18. The portion of the plane which appears in both graphs is the graph of the truth set for the system. You will observe that this is the most heavily shaded portion- the portion enclosed by rays FE and FG and including these rays.

0

EXERCISES

1. Draw a graph of the truth set for each of the following systems.

+

3X y- 6 < 0 e. { x - 3y -1>0

O

f.

x

X

2x-y>4

{2 + y > 4 X

g.

3x-y > 1

d.

{5 + 2y = 0

{3 x + 2y ~ 1

{

3X X

-

h. {- 3 y >

2x- y =4

>6

c. 2 x - y


1 or 3 x - 4 y < 12

> 2

10 and 4 x - 9 y 12 and x - y

1 or 3 x

+y < 1

=4

Q

EXERCISES

3. Consider the sentence (x + y - 3) (x - y + 2) > 0. a. Since the product of (x + y - 3) and (x - y + 2) is greater than zero (that is, the product is a positive number), you know it is true either that both (x + y - 3) and (x - y 2) are positive or that both are negative. Why?

+

492

CHAPTER

b. According to part a, we have (x + y - 3 > 0 and x (x+y-3

y

+ 2 > 0)

10

or

< 0 andx-y+2 < 0).

Complete: While each clause of this compound sentence is itself a compound sentence with the connective - -, the sentence as a whole is a compound sentence with the connective - -·

c. Draw a graph of the clause "x + y - 3 > 0 and x - y + 2 > 0." Then, using the same set of axes, graph the clause "x + y - 3 < 0 and x - y

+ 2 < 0."

d. In the graph you have just made point out the solution set of the original sentence.

4. Draw the graph of the truth set for each of the following sentences. a. (x + y - 2) (x - y + 4) > 0

d. (x - y - 5) (2 x - 2 y - 4) < 0

b. (x - y- l)(x + y + 3) < 0

e. (x - 2 y - 4) (2 x

C.

(x

+ 3 y + 2)(3

X -

y - 1)

>0

f. (x - 3 y - 6) (3

X

+ y + 4) + y + 3)

>0

y examining a given pair of equations can determine which of these words applies to it. (Page 480 .)

14. Can graph open sentences involving inequalities and absolute values. (Pages 487-489 .)

15. Can use a graph to find the solution set for a system involving inequalities. (Page 490 .)

16. Know the meaning of the following words and can spell them. abscissa (Page 434 .) axis (Page 433 .) Cartesian product (Page 431 .) consistent (Page 480 .) coordinates (Page 433 .) directed distance (Page 446 .)

inconsistent (Page 480 .) lattice (Page 454 .) ordinate (Page 434 .) system (Page 465 .) slope (Page 454 .) y-intercept (Page 453 .) CHAPTER REVIEW

1. Write six members of the solution set for the equation 2 x + y = 12. How many other members are there? 2. Which of the following ordered pairs of numbers satisfy the equation y = x 2 ? (- 2, 4), (4, - 2), (- 1, 1), (1, - 1), (2, 4), (4, 2) 3. Find the members of {(- 3, 1), (- 1, 2), (- 2, 2), (1, 2), (1, 3), (2, 3)} which satisfy the sentence 2 x < y. 4. If A= {1, 2, 3, 4} and B

5. Graph the set A

=

{1, 2, 3}, write the set AX B.

X B of Ex. 4.

6. Within the set of points of Ex. 5, make heavy dots to indicate the subset of A X B for which y = x + 1.

494

CHAPTER

10

7. If B = {1, 2, 3, 4, 5}, graph the set BX B. 8. Use heavy dot s to indicate the points within the graph of BX B of Ex. 7 for which y

< x.

9. Use a drawing to indicate the difference between the graphs of y =

x

when (x, y) EI XI and y = x when (x, y) ER X R.

10. a. Within the graph of IX I use heavy dots to indicate the subset A = {(x, y) I y = 2 x}. b. Use small crosses to indicate the subset B = {(x, y) I y = - 2 x}. c. Describe An B. d. Describe AU B.

11. Find the distance between each of the following pairs of points. a. (- 1, 7) and (5, 7)

c. (2, O) and (8, O)

b. (9, - 1) and (9, 3)

d. (1, 3) and (5, 6)

12. Draw the graph of each of the following equations as a subset of R X R .

a. y =

X

+3

f. 2 X + 3 y - 12

b. 3 x + y = 6 c. x - 2 y

=

=0

g. 4 x = 5 y + 20

h. y = x 2 + 1

3

! x- ½y = 6

d. x = - 4

i.

e. y = 3

j. y = 2 X - 3

13. Which of the following equations have graphs that are straight lines, that is, have the form ax+ by + c = O? a. y = V2 x + 7 b. (3 + a)x+(7 + b)y = c

3

2

x

y

d. ~ x e. y

c. -=-

=

1+ 1 =

O

Y 4 x2

f. 2 x 2 + 3 y 2 + 4

=0

14. State the slope and the y-intercept of the graph of each of the following equations. a. y

= 4x+

1

b. y= - 2 x - 3

= 6X- 9 d. 2 x + 3 y = 5 C,

3y

e. y - 2 X

=0

f. x = 3y+ l

15. Find the slope of the line through each of the follow ing pairs of points. a. (1, - 1) and (6, 3)

c. (2, 4) and (- 3, 4)

b. (6, 2) and (3, - 4)

d. (4, 2) and (4, 7)

16. Use the slope-intercept method to graph each of the following sentences. a. y = 2 x - 3

b. y =

- ½x + 3

495

The Real Number Plane

17. Write the equation of the line which passes through the point (- 4, 1) and has the slope

f.

18. Write the equation of the line passing through the points (- 3, 5) and (4, - 2).

19. Write the equation of the line whose slope is -

¾and whose y-intercept

is - 2.

20. What is they-intercept of the line passing through the points (- 3, - 1) and (6, 2)?

21. Which of the following lines have graphs that are parallel to the graph of 2 x+ y = 6?

a. y =-½

b. 6 x

= - 2 x+ 6 d. ½x + lo y = - 3

x+ 6

+3 y -

c. 3 y

18 = 0

22. What is the slope of a line perpendicular to a line whose slope is - ½? 23. Which of the following equations have graphs that are perpendicular to the graph of y = 3 x - 4?

a. y=-3 x-2

c. 3 y+ x= 2

b. - 4 x + 12 y- 1 = 0

d. 6 x + 2 y - 9 = 0

24. Express in another way, "2 x + y - 4 = 0 and x + y = 1." 25. Write the equation of the line parallel to the x-axis and through the intersection of the graph of 2 x

+ y - 4 = 0 and the graph of x + y = 1.

26. Find the simplest system of equations equivalent to

{! ~: ~=:_

9_ 3

27. Solve each of the systems of equations below. a,

{X + y = - 7

{½ X- J y = 0

f

x - y= l

·

+y = - 7 · 2x+3y=9

b {4 x

{1

1

5

;+;;=6

6 x - 2 y- 6 = 0 c. { 3x-y-3=0

d.

¼x+½y=4

g.

~_ ~=~ X

y

2

{4 x = y+ 2 2X

e. {

=-

3 y - 20

2x-3 y=5 x-2y=-3

28. Graph each of the following inequalities. a. y > 2 x + 1

b. 3 x + 2 y < 6

c. - y > - 3

d.

y

< ix -

1

496

CHAPTER

10

29. Solve each of the following using graphs. {

X -

y

>5

a. 2 X < 7

b {4 x + 3 y > 2 • 6X

-

5 y > - 30

30. Two angles are complementary. If the measure of the supplement of one angle exceeds the measure of the supplement of the other by 70°, how many degrees are there in the measure of each angle?

31. Where must the fulcrum be placed so that a 5-pound weight and a 9pound weight will balance each other when they are placed at the ends of an 18-inch lever?

32. Two loaves of bread and one quart of milk cost 54 cents. One loaf of bread and 2 quarts of milk cost 57 cents. What is the cost of one loaf of bread if all loaves cost the same amount? of one quart of milk? CHAPTER TEST

1. If A = {l, 2, 3} and B = {1, 2}, write the set A X B. 2. Draw the graph of {(x, y) I x > 0 and y > O; x, y EI}. 3. In the graph you made for Ex. 2 use heavy dots to indicate the subset of points such that the abscissa of each is greater ,y than its ordinate.

4. State the coordinates of each of the points A,

._,

~

....n

B, and C in the lattice shown at the right.

5. Make a graph of the truth set of the equation y = 2 x + 1 when x and y are real numbers.

_,

. 0

-

n.'

X

'

:

6. State the slope of the line through the points (5, 7) and (- 1, 3) . 7. State the slope and the y-intercept of the graph of the equation y =- ¾x+ 5.

8. Replace k by a number so that the graph of y = kx + 5 is parallel to the graph of 3 x

+ y + 4 = 0.

9. Replace k by a number so that the graph of y to the graph of 3 x + y + 4 = 0.

= kx + 5 is perpendicular

10. Find the equation of the line parallel to the x-axis and passing through . {x+ 3 y- 9 = 0 the graph of the solution set of 3 x _ 2 y _ 5 = O.

11. Draw the graph of the solution set of y = 4 Ix I . 12. Draw the graph of {(x, y) 11x 1- 2 2: y}.

497

The Real Number Plane

13. Use a graph to find the solution set for the system of inequalities {-3 x+ y < -4

2 X + y < 3. 14. Solve each of the following systems of equations.

a. {2 X + 2 y = 3 2x+3y=4

- 5 y = 0 Sx + 3y=0

c. {3 X

2 3 b. { ; + y= 1

~+1=1 X

y

15. The sum of two numbers is 36. If the smaller number is increased by 1 and the larger is decreased by S, the resulting numbers are equal. What are the original numbers?

Linear Programing Linear programing is a new field of mathematics which helps a businessman to make decisions on a sound mathematical basis which will minimize his costs and maximize his profits. The example which follows shows how linear programing can be used to solve a simple problem .

Example. A manufacturer wants to produce a compound which contains at least 200 pounds of ingredient Mand at least 200 pounds of ingredient N. He must do this by mixing brands X and Y. 20% of brand X is M, 80% is N. Brand X costs 8 cents per pound. 60% of brand Y is M, 40% is N. Brand Y costs 10 cents per pound. What amounts of brands X and Y are needed to produce the required compound and keep the cost at a minimum?

Solution. Let x represent the number of pounds of brand X needed, and y the number of pounds of brand Y needed. The cost C of the required compound can be expressed by the equation

+

(1) C = .08 x .l0 y. Amounts of brands X and Y needed to obtain at least 200 pounds of ingredient M can be expressed by the open sentence

+

(2) .2 X .6 y 2: 200. Amounts of brands X and Y needed to obtain at least 200 pounds of ingredient N can be expressed by the open sentence

+

(3) .8 X .4 y 2: 200. Since the number of pounds of brands Mand N cannot be a negative number, we have (4)x 2:0 and (5) y 2: 0 . We seek a va lue of x and of y which will minimize C and at the same time satisfy the conditions stated by the open sentences (2), (3), (4), and (5). In linear programing such sentences are ca lled constraints beca use they impose 1 restrictions upon the I variables. I'I 's V . I First we graph the ,I ' 'I ~ I'solution set of the sys, •' s I tem of constraints. :1 , KI f I" - '-' The graph of the solu' '" ......' " " ..... ~ '- 1"- I'

0 , then

j=J· The proof of this theorem is left to you as an exercise. Let us first explain by examples what is meant by simplifying the quotient of two radicals and simplifying a radical whose radicand is a fraction .

512

C H APTER

Example 1.

Simplify v

Solution.

'\J 49 =

Exa mple 2.

Simplify

✓

Example 3.

. 1·fy S1mp1

V8 V9.

Solution.

✓s V2G ✓'fl . ✓2 ✓9= ✓3 2 = V32

/1s

11

g.

✓ is ✓-is ✓ 49 = -7-

:

2 ·

Solution.

2 ✓ 2 , or _g_vz 3

3

In the examples above, the denominators are perfect squares or the square roots of perfect squa res. If we are treating a fraction whose denominator is not in such a convenient form, we may change the denominator to a perfect square or to the square root of a perfect square and then proceed as we did above. Example 4. Solution 1.

Solution 2 .

Since in Example 4 we have made the irrational number into a fraction whose denomin ator is a rational number, we say that we have rationalized the denominator. We could have found an equivalent expression for~ by rationalizing the numerator as shown in Example 5. Example 5.

Rationalize the numerator of ~

Solution 1.

✓l = ✓~ ·1= ✓~ ·i= ✓~: = ~~: = J21

Solution 2 .

-

/3- V3 - ✓3 . 1 - V3 . ✓3 - ✓:V -

'\j 7- ✓7 - ✓7

-

✓7

V3- ✓ 21 -

3 ✓ 21

To find an equivalent expression for the square root of a fraction whose denomin ator is not a perfect square we usually prefe r the method shown in Example 4 to that shown in Example 5. We can see a reason for this pref-

513

Radicals

erence when we try to express yt as an approximately equal rational number. To do this we may proceed in several ways, as shown below.

/3 -- vii 7

. E xamp 1e 4 : "\) U smg 7

~~ 4 ·5826 ~~ .6547 7

. {3 3 Usmg Example 5: "\) 7 = vii

~

{3 = ~ ~

A third method:

"\J1

A fourth method:

yt

v7

3

4 _5826 ~ .6547

1.7321 2.6458

~

.6547

~ V .42857143 ~ .6547

Of these, the proced ure using Example 4 is the easiest to carry out, so that . a fract10n . wh ose d enommator . . a rat10na . 1 num b er 1s . t he form vii - 7- wh"1Ch 1s 1s considered the simplest form of yt. We now have another condition which a radical must meet to be in simple form; namely, there must be no fractions in its radicand. Now let us agree to the following definition of a radical in simple form. A square root or cube root radical is in simple form if and only if its radicand is an algebraic expression in simple form not involving fractions and not having any perfect square factor when the radical is a square root, nor perfect cube factor when the radical is a cube root. A fraction whose numerator, denominator, or both are radicals is said to be simplified when there is no radical in the denominator, when any radical in the numerator is in simple form, and when numerator and denominator have no common factors other than ± 1. Example 6.

V33

Solution.

3lxl

0 1. Simplify ea.ch of the following.

a.yIJ. b. -V459 c. v~ 36

d.vNi

'144 e • -V25

f

/15 . "\)~

g. - ~

h

ITTa2

·"\)7

EXERCISES

514

CHAPTER 11

2. Simplify each of the following expressions. a.

b

✓9y a

£. ✓1+i

~

g.- ✓f+t

y

h. V¾+-vt

. -V7

. -V3m VS-

1. - - · -

~~

. ✓5 x J,

-

y

2

>

; m

0

2

3. Simplify :

a.

vt

b ~125

3 -

c.

. ~1000

~

«H 81

d. ~16

4. Simplify:

a. v1 b.

V7

h. -Vr\

.n

- "Vt

1.

c. v?; d. -Vi a e. -VJ

4

' J ,-

k

r. ✓l

m. ✓25

q.

3 -v16 n ---

r.

o. 3- ~

s.

ff

t.

54

1 ✓ 9 3 50

.

V6 ' vi3

1~

' ✓s ;x> O

g. ✓I

p.

4 ""V27

~ 2-5a ; a > 3a ✓2 ; a 2: V7 +VJ

0 0

vs

V2 - ✓s

V3

5. Simplify: a. ✓i - ✓1s

e.

b. ✓J --vt C _

J;

b 2: 0

/9+ 12

.

\)

d.

✓j ;

i.

3

x

2: 0

h.

V10 ✓2 g.-2 - · -v:s

j.

5--V¾·Wo

· i ✓! ·✓~; ✓! ✓I;

a

> X

0

>

0

0, y

>

0, b

>

EXERCISE

6. Prove Theorem 40: If a and b are real numbers such that a 2: 0 and b ✓-

then

r

.Ji=\Ji·

0

> 0,

515

Radicals

Sums Involving Radicals

By means of the distributive property, algebraic sums involving radicals may often be expressed in equivalent forms which are more convenient to use. For example, 4VS + 7V5 = (4 + 7)VS = 11 VS. The latter form would be simpler to use in most situations. Similarly, xVS+ yVS = (x + y)VS, but here it would be difficult to say which form would prove to be the more convenient in general. You should be alert to the possibility of expressing a sum involving radicals in an equivalent form which might be better suited to the solution of a given problem. Example. Solution.

Express "V24 -

v'54 as a single radical.

✓24-V54=~-~= ✓4V6-V9V6 = 2✓6- 3V6= (2- 3)V6= (- 1)✓6=-V6 ORAL EXERCISES

Use the distributive property to express each of the following in an equivalent form.

1. V2 +SV2

2. 3VS+ 4vs SV7-V7 6V3 + SV3 - 2V3

3. 4.

5. 6V71- 4V71 -V71

3V13 - 4 V13 - 7V13 7. 13--vl7 - SW+ 4W 8. 18V21 - 1SV21 - 2V21 -vii 9. avis - bVlS

6.

10. 7Va- 2-Y~;

0

a~

0

EXERCISES

1. Use the distributive property to express each of the following in an equivalent form.

V2+V3+3V2-V3 b. SV7 - 4V3 - 2V7 + 6V3 c. 9VS + 2V2S - 6VS d. 2V3-4V9-2V9+V3 e. V3 +vu f. vs+V45 g. vs+v'i6 h. v'18-V27 ·-IT 1-r:: I , v 2 + 2 v2 j. 2-v'so + 4V75 a.

v'24 - vu 1. ¼v'32 - ½v'so m. ½v32 - ½v'2 n. ¾v2s + !v'63 k. V8 -

o. ½V2+3V8+V½ p.

3V25 -VSO+ V½

q.V½+ Jv t r. ½ vis+v1 s.

V6 + 2V3 + 12-vt

t.

vs+v1-v1-v't

516

CHAPTER

11

2. Use the distributive property to express each of the following in an equivalent form.

v'½Cv'½ + V2)

a. V2(V8+V2)

e.

b. V3(V3 + v12)

f. !(~ + 3Vl0 + 6)

c.

Jvs(vs - 2)

½(V288 + V2 + vs) h. V½(v?;- v+, + v'12) g.

d. V8(2V2 - 3V8)

Multiplication and Division of Binomials Containing Radicals A polynomial over the real numbers is a polynomial whose coefficients are real numbers. Consider the following products of polynomials:

(SV3 +

2V2)(7 +VS).

Example 1.

Find

Solution.

We may use the distributive property directly or we may use the FOIL method, which is based upon it. Using the latter, we have

(5V3 + 2V2)(7 + VS) = 7. 5 -V3+5 ·V3 -vs+ 1. 2-vz+2 .vz .vs = 35v'3 + 5-vis + 14vz + 2vw.

+ 2V3).

Example 2.

Find the square of (7

Solution.

We may use the distributive property directly or recall that as a result of using the distributive property we have the pattern (a + b)2= a 2 + 2 ab+ b2 . Using the latter, we have

(7 + 2V3) 2 = 72 + 2 . 7 . 2V3 + (2V3) 2 = 49+ 28V3+ 12 = 61 + 28V3.

+ VLl)(S -VLl).

Example 3.

Simplify (5

Solution.

We may use the distributive property or recall that as a result of using the distributive property we have the pattern (a+ b)(a - b) = a2 - b2 . Using the latter, we have

(5 + V13)(5-V13) = 52 - (V13) 2 = 25- 13 = 12.

v;

Two expressions of the forms + bVc and Va - bV~are said to be conjugates of each other provided a, b, and care non-zero, b is rational, and c is not a perfect square. Since (Va+ bVc)(Va - bVc) = (Va) 2 - (bVc) 2 = a - b2c, we see that the product of two conjugate rational expressions does not involve radicals. Observe that this was true in Example 3. To rationalize the denominator of a fraction when the denominator is a binomial containing a radical, we multiply both the numerator and the denominator of the fraction by the conjugate of the denominator as in Example 4.

517

Radicals

Js

Example 4.

Rationalize the denominator of

Solution.

The denominator is - 7 2-VS. According to our definition, the conjugate of the denominator is - 7 - 2~. Multiplying the numerator and the denominator by- 7 - 2-VS, we have:

2

5-7

+

3

_ 3(-7- 2-VS) _ -21-6-VS - 1 + 2✓s- (- 1 + 2-vs) (- 1 - 2-vs) - (- 1) 2 - (2vs) 2 - 21- 6-vs - 21- 6-vs 21 + 6-vs 29 49- 20 29 Note. We could have obtained this result by multiplying the numerator and the denominator by the negative of the conjugate of the denomina tor, namely 2-VS + 7. Do you see why?

We can now conclude our discussion of the simplified form of an expression involving radicals with the agreement t hat when we say "simplify" such an expression , we expect the following conditions to be met . Any fractions, radicals, and products of radicals appearing in the expression should be simplified; no fractions or radicals should appear in any denominator; and the expression should be written with the fewest number of terms possible, combining t erms by means of the distributive property.

0 1. Simplify: a. (2 + vs)(2 -vs) b. (4 -v5)(4 + VJ) c. (2 + VJ)(2 + VJ) d. (3 -V7) 2 e. (a+ V2)(a -V2) f. (7-v~)(7 +Va);

I. (v J+v'z)(v 3+4v'z) m. (3 V2 - 7)(2V2 - 9) n. (3 V6 + 1)(6V6 - 7)

a> o

g. (vs -VJ)(VJ + vs) h. (2VJ - V6) (2VJ + V6) i. (3 + VS)(2 + vs) j. (7 -vJ)(s - VJ) k. (8 + Y2)(3

-vz)

EXERCISES

o. c2 vs + v3)(3v s + VJ) p. (6V7 + VJ)2 q. (SVJ - V7)2

V7)(SYS + V7) s. (3 + VJ)(s -vs) t. (3V2 + V7)(V7 - 3V2) u. C½VJ + vs) r. (SYS -

2

v. c2v1 + 3vs)(v28 -V45)

2. Simplify: a. (V6 + 8)7

c. vs (6Y2 + 1)

b. VJ (2VJ + 1)

d. 3V2(4V2 - 5)

518

CHAPTER

11

3. Rationalize the denominator of each fraction below.

2 1-vs b 3 ·V6-1

4V3+2 . VJ-2 2-vs g. 3-vs f

a.

5

3+2V7 . V7-1 . 1-VJ l.VJ3- 1 . 2+-vs

h

c.vs+ 6 3 d. 2V6-3 V2-1 e. V2 + 1

J• 5-VS

0 4. Simplify:

(V18 + VJ)(v'2 + v 6) b. (3v'2 + viz)(vs + V3) c. (V32 - VJ) (V2 + V48) a.

(J2+ Js)(J2 -~) e. (J27 - ~)(Ju- J3)

f.

(v714 + 1)(-1 - 1) V28

d.

h. (

l

1 1 + V2) -v2 + 4)(2 - 1-v'2

5. Find the square of each of the following expressions. a. 3 v ; + Vy;

x

~ 0, y ~ 0

d. 2(5 - 3 V7)

-

h.v½-v~ C.

Vx-Vy

Vx

; X > 0, y

/i

e. Va - \J~; a> ~ 0

f.

v'2(v3 -vs)

6. Simplify:

a.

(V½+ v ½)(6 -v'iz)

e.

V½-V3 f v'27-vi2 . V3+2

b. v s + (V125 + vs) c. (3

d.

V18 + 1)(6 V2 - 7)

(Va+Yb)(2Va-3Yb);

~3

a~

0, b ~ 0

0

EXERCISES

519

Radicals

Square Root

In this section we consider ways to find a rational expression which is approximately equal to the sq uare root of a given positive number. The square root table on page 608 gives the sq uare root of each perfect square integer from 1 to 99, inclusive. It gives an approximate squa re root for each integer between 1 and 100 which is not a perfect sq uare. Such a table can also be used to find the square root or an approximate sq uare root of a number not shown in the table. Three possibilities for doing thi s are discussed below. I. Sometimes the radicand can be expressed in the form t2 k where t is an

integer and k is a number whose squa re root (or approximate square root) is given in the table . For example, an approximate square root of 17 5 can be found as follows: vm = ✓25. 1 =

sV7 = s x 2.6458 = 13.229

2. We can "read between the lines" to find square roots of numbers that do not appear in the table. For example, the square root of 37 .5 is not given in the table on page 608. To find a rational number approximately equal to Vlli we use the table to find that -v37 = 6.0828 and V38 = 6.1644 . Since V 37.5 lies between V37 and V38, we proceed as follows:

-v37 = Vlli : V38 -

6.0828} d

I

X

r

6.1644

)

.0816

The d represents the difference between 6.0828 and x while .0816 is the difference between 6.0828 and 6.1644. (We call the latter difference a tabular di.fference because it is the difference between two numbers given in the table.) Since 37.5 is halfway between 37 and 38, V3Ll should be approximately halfway between 6.0828 and 6.1644. Thus d = ½of .0816, or d = .0408. Thus V37 .5 = 6.0828 .0408 = 6.1236. As a second example, let us find V51.7 . From the table we find:

+

V51 = 7.1414} VSl.7 = ? v52 = 1.2 111

I d

r

.0697

)

Here we reason that d should be approximately .7 of the tabular difference. Why? d = .7 X .0697 = .0488 VSl.7 = 7.1414 + .0488 = 7.1902 Check: (7.1902)2 = 51.699 The process of "reading between the lines" of a mathematical table is called interpolation .

520

CHAPTER

11

v;

3. We can use our square root table to find by first expressing n as the product of a positive number between 1 and 100 and an even power of 10.

The examples below show this procedure.

V7300 = V73 X

102 =

V73 X "'V1Q2 ""' 8.5440 X

10 = 85.440

V730 = V7 .3 X 102 = V73 X 10. By interpolation, V73 ""' .·. V730 ""' 2.7006 X 10 = 27 .006

vs

2.7006 .

vs

V .05 = vs X .01 = vs X 10 - 2 = X vio=2 = X V (l0-1) 2 ""'2.2361 X 10 - 1 = 2.2361 X .1 = .22361 V.000078 = V78 X 10 - 6 = V 78 X VlQ-6 = V78 x V (10- 3 ) 2 ""' 8.8318 X 10 - 3 = 8.8318 X .001 = .0088318 V .00078 = V 7.8 X 10 - 4 = V7.8 X ~ ""' 2.7919 X 10 - 2 ""' .027919

0

EXERCISES

Use the square root table on page 608 to find an approximation for each of the irrational numbers below.

1. v s.4

5. v45 .3

V24.6 3. V31.2 4. V9.5

6. V66:7

2.

7. V45.1 8. V84.7

9. V65.3 10. V11.8 ll.V4500 12. v'no

13.V65000 14. v 8oo 15. v74o 16. "Y.06

17.Y.0014 18. v.000011 19. v.000066 20.V:OOOSZ

Instead of using a table to find the square root or an approximate square root of a number, we may use a method based upon the formula for the -square of a binomial. While an explanation of this method is lengthy, actual computation by the method is carried out in a short, compact form , as you will see in the examples on pages 523-524. For those interested in the reasoning behind this method , we include the following discussion. Suppose we wish to calculate an approximation for the principal square root of 753. We may say that 753 is about equal to the square of the twodigit number (10 a+ b), where a is the tens digit and b the units digit. Thus we have 753 ""' (10a+b) 2 = 100 a2 +20ab+b 2 = 100a 2 +b(20 a +b ). First we seek the largest a such that 100 a2 ::I> 753 . Evidently a= 2. (We observe that 2 is the largest integer whose square is not greater than 7.) Writing 2 for a, we have

Thus we have

753 ""' 100·2 2 +b (20·2+b) ""'400 + b(40 + b). 353 ""' b(40 + b) .

Now we seek the largest integer b such that b(40 + b) ::I> 353. By trial we find that b = 7. We have 7(40 + 7) = 329.

521

Radicals

Now we know that we have found the largest possible integral values a and b such that (10 a+ b) 2 ::I> 753 . In fact, we have (10 • 2 + 7) 2 = 27 2 = 729. Evidently we have the best approximation to V753 that we can obtain if we wish the square root to be an integer whose square is not greater than 753. (We know that the square root of 753 is somewhat larger than 27 although not so large as 28.) Let us now find the digit that should be in the tenths place in our square root . If c is this digit, then

753 "" (21 +

toY,

and we work to determine c so that the right member will not exceed 753. We have

753

""(21 +tor= 729 + 2 · 27 ·to+ t;o "" 729 + to (2 · 27 + to}

Therefore

24 ""

_£_ (54 + _£_) • 10

10

We get an estimate of what c should be by dividing 240 by 54. This suggests that we try 4 or 5. We find the latter too large, but when we substitute 4 for c in the expression immediately above, we find that the right member is not greater than the left.

24 "" 140(54 + 140) = .4 (54.4) = 21.76 Thus 4 is the value for c, the digit that should be in the tenths place in our square root . Now we know that 27 .4 is the largest three-digit number whose square is not greater than 753. Carrying the process further,

753 "" ( 27.4 +

l~Or d

d2

"" (27.4)2 + 2(27.4) 100 + 10000 "" 750.76 + 1i 0 [ 2(27 .4) + 1i 0} Therefore

2.24 "" 1io (54 .8 + 1io}

We get an estimate of the value of d by dividing 224 by 54.8. We obtain 4, which does not make the right member of the above expression exceed the left, while the next integer, 5, does make the right member exceed the left.

2.24 "" 160(54.8 + 1!0) = .04 (54.84) = 2.1936. We are now certain that 27.44 is the largest four-digit number whose square is not greater than 753.

522

CHAPTER

11

It is worthwhile to observe that the work we have done in finding the square root of 753 can be arranged in a systematic manner. You will see that the system shown below involves repeating a certain sequence of operations.

2 ✓ 753 400

(10 a+ b)2 (10 a) 2

353

Find the largest positive integer a such that (10 a) 2 )> 753. That is, find the largest a whose square is not greater than 7. We have found th at a= 2. We write 2 above 7 in our computation. We then subtract (10 X 2) 2 from 753. The remainder, 353, is

b(20X2 +b).

20 X 2

27

V753 -Mr 40+7=47

400 353 329 24

b(20X2+b) 7(40+ 7)

27 X 2

to (2 x 27 +

2 7. 4

✓ 753.00

..&-t

54+ .4 = 54.4

400 353 329 24.00 21.76 2.24

We now seek the largest integer b such that b(40+ b) :i> 353. We estima te this by dividing 353 by 40. We write the quotient, 7, beside 2 in the first line to indicate 20 + 7 and add 7 to the divisor, 40. We now find 7(40+ 7) = 329 and we write 329 below 353. Subtractin g 329 from 353 gives 24. We now seek an integer c (the tenths digit) such that

to (27 x2 + to) .4(54 + .4)

; 0)

> 24.

We estimate this integer by dividing 240 by 2 X 27, th a t is, by 54. We write the quotient 4 beside 27 in the tenths place in the first line to indicate 27 .4 and add .4 to the divisor 54. We now find

+

.4(54+ .4) = 21.76.

523

Radicals

We write 21.76 below 2'±.00 and subtract. The difference is 2.24. We now seek an integer d such that

l~O (2X 27.4+ l~o) ► 2.24.

27.4 X 2 2 7. 4 4 ✓ 753.00 00 400

~

54.8 + .04= 5'±.84

353 329 24.00 21.76 2.2400 2.1936 .046'±

d ( d ) 100 2 U X 2 + 100 .0'±(5'±.8 + .0'±)

We estimate this integer by dividing 224 by 54.8, that is, by 2 X 27.4. We write the quotient 4 beside 27 .4 in the hundredths place in the first line to indicate 27 .4 .04 and acid .04 to the divisor 54.8. We now find .04(54.8 + .04) and write the product 2.1936 below 2.2400 and subtract.

+

Example 1.

Find the largest three-digit number whose square is less than 59.

Solution.

Since 59


0 l. (27 a)½

a. 36½

e. 100½

i. (- 125)!

m. (n)¾

b. 8~

f. (- 27)!

j. 32!

n. (.027)!

c. s1½

g. s1¾

o. 25½

d. (.001)!

h. - (U) ¾

k. (!)½ l. (1.44)½

1+

4. Simplify:

529

Radicals

0

EXERCISES

5. Some of the following expressions have not been defined and consequently cannot be simplified at this time. Others can be simplified. For each expression write a simplified form if one is possible; otherwise write "undefined." a.~

g. (25)½

b.~ c.~

h. (- 49)½ i. (- 8)1

d.~

j. (- 2)4

e. (V9)2

9 / 2 m. ( TT

n. (- 64)¾ o. v (- 27)(- 3) p. v (- 12)(6)(- 2)

k. ~ 1. (- 125)1

f. Va; a 2 0

q. ~ r.~

More about Rational Exponents

Having used positive rational numbers as exponents, it is natural to inquire if it is possible to use negative rational numbers in the same way. We can show that it is. We have learned that for any integer n, a -

n

= _!_ if a is real an

and a ~ 0. Let us extend this idea to the case in which n is a rational number. Let us try to define 8 -¼so that our previous laws of exponents will remain true for the new situation. We have

s -½= (s!) - 1 =.l=!· s¼

2

Similarly,

(- 27) - ! =

(- 27!) - l = - 1- = 2

- 27 3

!. 9

It can be shown that the definition _m

a

n

=

1

m;

n

>0

an

enables us to retain all the laws that we have studied. We must, of course, be careful to insist that a be positive when n is even. We shall restate the five laws of exponents for rational numbers m and n . Er. am. a"= am+n am En. - = am-n a" Enr- (ab)"= a"b"

= ~: Ev. (am)n = amn Erv- (~)"

530

CHAPTER

Example l.

Simplify V2

Solution .

✓2. ~2= 2½ . 2½= 2½+½= 2¾ = ~2.5= ~32

11

· ~2.

Example 2. Solution.

0

EXERCISES

Simplify each of the following expressions. _;i_ 1. 3-½ 13. 3 >:· yz 2. 36 -½

14. (m!)½

3. (¼) -½

4. (49 a 2 ) -½

5. (9 x 2 y 2 ) -½

15.

vs ~s

6. (3 2 a 5 b10 ) -½

16. (163)¼

7. - (8 a 3 b6 )½

17. (VSl) - 2

8. (.0001 x4 )½

16½)½ 18. ( ----:i" 83

1

1

9. a2 • a4 ; a> 0 10. bL b - ½

11. 14½ · 14 - ½ 12. 25½· 36 - ½

19. 25° + 25 -½ 20. ~~ · V~; x > O 21. (x!) 3

Equations Containing Radicals

The truth set of the equation x = 5 is {5}. If we square both members of x = 5, we have the equation x 2 = 25. The truth set of the equation x 2 = 25 is {- 5, 5}. Since x = 5 and x 2 = 25 do not have the same truth set, we know that the equations are not equivalent. On the other hand the equations x = 0 and x 2 = 0 2 both have {O} as the truth set and are consequently equivalent. The fact remains, however, that squaring both members of the equation x = a produces a new equation x 2 = a 2 which may not be equivalent to the original equation. Sometimes we square both members of an equation t o simplify the equation. The procedure is very useful , but we must keep in mind that in the process of squarin g we may obtain an equation which is not equivalent to the original equation.

531

Radicals

Example Solution.

1. Solve the equation Vx + 5 = 3. 1.

PART

PART

V4+s =13

✓x +S= 3

V9 J 3

Squaring both members, we have

3i3 The solu t ion set is {4}.

x +S=9 x=4 Example 2. Solve the equation x Solution.

2.

V~= 6.

1.

PART

We want an equation free of radicals, but if we square each member of x - Vx = 6, we obtain a more complicated equation, namely, x 2 - 2 xVx x = 36. Before squaring, therefore, we wri te an equivalent equation which has the radical expression as one of its members; then we square both members. We solve as follows :

+

x - ✓~ = 6 x - 6 = Vx x2 -

12

X

+ 36 =

X

x 2 - 13 x + 36= 0 (x - 4) (x - 9) = 0 x =4 orx=9 PART

2.

If x = 4:

If x = 9: 9 - ✓9 =16

4- ✓4=16 ?

?

4- 2 == 6

9 - 3==6

2~6

6i6

The soh,1tion set 1s {9}. (We see that while 4 is a solution of (x - 4) (x - 9) = 0, it is not a solu tion of the origina l equation.)

I I= x -

Example 3. Solve x Solution.

PART

6.

1.

J.r i= .r - 6 6) 2 12 :r

(I .r 1)2 = (x x2 = x2 12 X = 36

+ 36

x= 3 PART

2. ?

J3J == 3 - 6

3~-3 Since 3, the on ly possible candidate for the solut ion of th e equa tion, does not satisfy the equation, the solution set is 0.

5 32

CHAPTER 11

0

EXERCISES

Solve each of the following equations.

8.~=x-2

Lv'x+S=2 2. 2 ~ = 3

9. ✓3x +4= x

3, ✓2 X + 1 + 3 = 6 4. -¼+6=4

10. ~ + x -12 = 0

Ix I= x+ 4 12. I x I = x + 12 11.

5. v'x+5 = 2

7 6. x- 2 =Vx 7. ✓;+x= 12

13. 2 XI= J

X

+ 21

14. IX- 3 I= 2 X

/2 K

15. Solve the formula v = ',J-;;; for K. 16. Solve the formula T = 2

1r✓1 for g.

17. I s the sentence x 2 + y 2 = 1 equivalent to the sentence y = ✓1 - x 2 ? 18. Show that Lhe sentence "ac = be" is equivalent to the sentence " a - b

= 0 or c = O. " More about the Pythagorean Theorem

We have referred to the Pythagorean theorem several times. Let us look at it more closely. We recall that the theorem refers to a right triangle, that is, to a triangle which contains a right angle. The two sides which fo rm the right angle are perpendicular to each other. For examB ple, in the triangle at the right, angle C is a right angle and sides AC and BC are perpendicular to each other. It is customa ary to use capital letters t o denote the vertices of a triangle and small letters to denote the lengths of the sides. We label A" ' " " - - - - - - - - - - - c the sides so that the side with length a b is opposite angle A , the side with length b is opposite angle B , and the side with length c is opposite angle C. When C is a right angle, as in the triangle above, the side with length c is called the hypotenuse. The Pythagorean theorem states: In any right triangle, the square of the length of the hypotenuse equals the sum of the squares of the lengths of the other two sides. For our drawing c2 = a2 + b2 .

533

Radicals

T he Pythagorean theorem can be used to construct a line segment whose length is the square root of any positive integer. One plan is shown in the figure at the right. Can you y describe how each of the points labeled V2, V.3, etc. on the xaxis was located? Observe that / ~ I the arcs have the origin as their I\ / V 'y / / -2' \~. \ center and V2,VJ, etc. as their /. ¼,,, I respective radii. X I 0 ~ .... l v'2 J3 The Pythagorean theorem ✓ -has an interesting and useful converse. It is: If the square of the length of one side of a triangle is equal to the sum of the squares of the lengths of the other two sides, then the angle opposite this side is a right angle.

-

'~

~ ~

Example 1.

A brace wire extends from the top of a 30-foot pole to a point on

the ground 38 feet from the foot of the pole. Allowing four feet for attaching the wire, how long is the wire? B Solution.

Let x represent the length of AB. Then x 2 = 38 2 + 30 2

30 ft.

= 1444+ 900 = 2344 X

= ✓2344

""' 48.4

The total length of the wire is approximately (48.4 + 4) feet, or 52.4 feet.

A1.---------~ c 38 ft.

Example 2. The length of the hypotenuse of a right triangle is 13 units and

the length of one leg is 7 units. Find the length of the other leg. Solution.

C

By the Pythagorean theorem, 13 2 -

(13

+ 7)( 13 -

13 2 = 72 72 = x 2

+x

7

2

7) = x 2

20 · 6 = x 2 V120= x 2-v3Q= X

A " " - - - - - - - - - -...... B X

From the table of square roots, v30 ""' 5.4772 . Therefore x ""' 2 X 5.4772 = 10.9544. If we are satisfied with expressing x to the nearest unit, x ""' 11.

534

C H APTER

11

Example 3. If the lengths of the sides of a triangle are respectively 29, 20,

and 21 inches, is the triangle a right triangle? Solution.

We ask : Is 29 2 = 20 2 + 212 a true sentence? Perfo rming th e indicated operations we have the tru e sentence 841 = 400 + 441. Consequently we know that the triangle contains a right angle and therefore is a right triangle.

0

EXERCISES

1. In each of the following , the three lengths given are the measures of the three sides of a triangle. Which of the triangles are right triangles?

a. 6 inches, 8 inches, and 10 inches

b. 5 feet , 12 feet , and 13 fee t c. 1 foot , 2 fee t , and

V3 fee t

d. 4.5 yards, 6 yards, and 7.5 yards e. 4 miles, 5 miles, and 6 miles

f. 10 inches,½ foot ,¾ foo t

2. If c represents the length of the hypotenuse of a right triangle and a a nd b represent the lengths of the other two sides, complete each of the st atements below. a. If a= 9 fee t and b = 12 fee t , then c = ?

b. If a = 2 inches and c = Vl3 inches, then b = ? c. If b = 1½ miles and c = 2½ miles, then a = ?

d. If a = VU meters and b = Vl3 meters, then c = ? e. If a =

-V:- unit and c =

V:

unit, then b = ?

3. What is the length of the diagonal of a square if each side of the sq uare is 1 inch long?

4. If the length of the diagonal of a square measures 1 inch, what is the length of a side of the square?

5. A baseball diamond is a sq uare having each of its sides 90 fee t long. What is the shortest distance from home plate to second base?

6. Mr. H olmes wants t o brace a young tree with wires attached to the trunk at a point 8 feet above the ground and reaching the ground 5 feet from the trunk. H ow long is each wire, not including the amount needed to fasten the wire to the trunk and t o the ground?

535

Radicals

() EXERCISES

7. A cube is a figure whose faces are six squares as shown at the right. a. If the length of each edge of the cube shown here is 3 inches, how long is AB, the diagonal of the square which is the base of the cube? b. What is the length of AC, a diagonal of the cube? 8. Find the hypotenuse of a right triangle if the lengths of the other two sides of the triangle are V2 + 1 feet and V2 - 1 feet.

........I

'

I',

:

....................

I

)----- / /

---

A

✓:- B"""'--- -.., /

-

9. A set of numbers a, b, and c, such that a 2 + b2 = c2 is called a set of Pythagorean numbers provided a, b, and c are integers.

+ n 2 , m 2 - n 2 , and 2 mn is a set of Pythagorean numbers when m and n are integers.

a. Show that the set of numbers represented by m2

b. Find the set of numbers represented by m 2 + n 2 , m 2 - n 2 , and 2 mn when m

= 6 and n = 4.

c. Find the set represented when m

= VS and n = V13. n2

-

10. Show that the set of numbers represented by n, - 4-

4

n2

+ 4 when

, and - 4-

n e I is a set of Pythagorean numbers.

11. A rectangular room is 30 feet long, 20 feet wide, and 10 feet high. Find the length of the diagonal from a corner on the floor to the diagonally opposite corner on the ceiling (AB in the drawing at the right).

10 ft.

,.;.- _,....::::.-:. __ _

,,,,,,,.. _ ,.,...,,,.,

,;~----

A """--- 3-0- fi-t._ _ ___..,

12. Suppose that an insect crawls from a corner of the floor of the room described in Ex. 11 to the diagonally opposite corner of the ceiling. What is the shortest possible distance that the insect can crawl?

13. In the triangle shown at the right, CD is

C

perpendicular to AB. The length of AC is V3 inches, the length of CB is 1 inch, and the length of CD is \

3inches.

Show

that the triangle ABC is a right triangle.

A ...__ _ _ _ _ _.....__ ____. B D

14. The lengths of the three sides of a certain triangle can be represented by (x + y), (x - y), and

V2 x 2 + 2 y2 .

a. Show that the triangle is a right triangle. b. Find the lengths of the sides if x = 7 inches and y = l inch .

536

CHAPTER

11

ESSENTIALS

Before you leave Chapter 11 make sure that you

1. Know what we mean by "raising to a power" and "extracting a root" and can use the proper symbols to represent each process. (Page 501 .) 2. Recognize that every positive real number has two real square roots, one positive and one negative, and that no negative real number has a real square root. Also make sure that you can explain why each situation exists. (Page 503 .)

3. Know that any real number has only one real cube root.

(Page 505 .)

4. Understand that although we cannot express an irrational number as a rational number, we can find a rational number approximately equal to the irrational number. (Page 507.)

5. Understand and can use the product property of square roots: If a 2'.: 0 and b 2'.: 0, then Va· Vb= ¼h. (Pages 508- 509 .) 6. Understand that for any real number a, W

=Ia 1-

(Page 509 .)

7. Understand and can use the quotient property of square roots: If a 2'.: 0 and b

> 0,

then

vi= f ✓-

✓-

(Pages 511 - 513 .)

8. Can simplify sums involving radicals.

(Page 515.)

9. Can simplify products and quotients involving radicals. (Pages 516518 .)

10. Can find the approximate square root of a number both with and without a table of square roots. (Pages 519- 524 .)

11. Can express a radical by means of a rational exponent and can perform operations when such exponents are involved. (Pag es 526-529.)

12. Can solve an equation involving radicals.

(Pages 530- 531 .)

13. Can use the Pythagorean theorem (a) to determine whether or not a triangle is a right triangle and (b) to find the length of one side of a right triangle given the lengths of the other two sides. (Pages 532-534 .)

14. Can spell and use each of the following words and phrases correctly. conjugates (Page 516 .) evolution (Page 50 l .) extracting a root {Page 50 l .) index (Page 501 .) involution (Page 501 .)

irrational (Page 506 .) principal square root (Page 502 .) radical (Page 50 l .) radicand (Page 50 l .) rationalize (Page 512 .)

537

Radicals

CHAPTER REVIEW

1. Which of the following expressions do not represent real numbers? a.

d. V½

V49

Y-64 £. v'o

b. \/16 c.

e.

V14

j. ~ k. \/=8i 1. ~

g. \1/w h. ~

i. v- 1000

2. Write the principal square root for each of the following numbers if such a root exists.

>0

a. 121

c. (- 7)2

e. y;y

b.¼

d. -36

f. 16 ab; a 2: 0, b 2: 0

3. Is it true that every real number has two square roots in the set of real numbers? Explain.

4. Indicate the cube root of each of the following numbers. a. -

27

d. .001

c. 11

e. x 6

f.

5. Indicate which of the following numbers are irrational. a.

v'29

b.

v'36

c.

vs

d. ~

e.

-\/16

6. Simplify each of the following expressions. a.

V3V12

d. ~ - ~ ; a 2: 0 e. VUx·Wy; x 2: 0, y 2: 0

b. (-V7) ·V3 C.

V9y2

f.

Vx·Vy2;

X

2: 0

7. Simplify:

a. \1/w

_4;-

a4

e.~

d. v'n

£. v.000021

v'j d. "'1-v'2s x

e.

c.

V

8. Simplify:

a.v'¾ b. v'¼¾

c.

2

v'2+1

£. v'l

9. Simplify:

v'6+7v'6 b. v'24+v'28 c. v'1s+v'12-v'4s d. v't-v'½ a.

4v'so- 3V8 f. ½V4S+¾V80 g. ½v';+v';; a> 0 h. ~+v'! e.

17 ab

3

538

CHAPTER

11

10. Express in simpler form: a. (2 + V7)(2 -V7)

d. VS(4+3V2)

b. (3 + VS) (4 + c. (V6- l)(V6- 6)

e. (3-VS +4)(2VS-1)

vs)

f. (6V3- 2) 2

11. Rationalize the denominator of each of the following fractions. 1 V20 • 3 VS

V3 a.vso b

V2+3 e. - r,:: v2-3

C -·--

vs

vs

d

6 · 1-V2

.V2

12. Use your square root table to find an approximation for the principal square root of each of the following numbers.

a. 71.2

b. 4.7

d. .07

c. 3700

e. .0015

f. .00061

13. Without a table find the principal square root of the following. a. 29584

c. 5490.81

b. 85849

14. Without a table find the rational number approximately equal to each of the following numbers. Express the result correct to one decimal place.

V353 b. V4732 a.

c.

e. ✓ 184,273

-V2347

f. -✓234,157

d.v'9ill2

15. Express with fractional exponents:~;

~

VJ; VS;

16. Express each of the following with a radical sign. a. (a)½ b. (½)½ c. (.7)ft

d. (- 64)½

17. Simplify: a. (- 64)½

c. (x3 )½

e. (~)-t

b. (64) -½

d. (x¾) 4

f. (49)1

g. (.027)¾ h. (81) -¾

18. Express in simpler form: i

a. 36 2 b. M½ · 64 -½

i

f. 49½ · 16½

d 25 2 • 125½

g. 25°. 25¾

e. 27ft · 31 - ½

h • a31

1

•

a"2"; a 2='.: 0

19. Solve each of the following equations.

Vx =

a. Vx = 12

C.

b. "\t'x+3 = 11

d. ✓3 x + 3 + 5 = x

X -

12

e. -Vx=2

f.

Ix I= x+ 18

539

Radicals

20. In a right triangle the two sides of the right angle have the lengths 14 inches and 12 inches. What is the length of the hypotenuse?

21. What is the length of the diagonal of a rectangle 12 yards long and 4 yards wide? 22. If c represents the length of the hypotenuse of a right triangle, and a and b represent the lengths of the other two sides, which of the following statements are true? a. a 2

= b2 -

b. b=Vc 2 - a 2

c2

CHAPTER TEST

1. If

x represents any real number, which of the following statements are true?

¾a= /x/ d. (~) 3 =x

a.W=lxl b. ¾a= X

e.-W=-lxl

c.

f.W=x

2. Which of the following statements are true? You may use a table of square roots if necessary.

-V4=-2 b. v'3 ""' 1.732 a.

c.

VS= 2.5

d. ~ = 3

3. Simplify each of the following.

a.

"\1'288

b.

-lfs4

c. V12 · V3 d

e. V32 x 2 y 4

V48

g.vi+l h.

. v'16

4. Rationalize the denominator of

a

ITT ,r~

3V3

2+

3

5. Simplify V72 + 3V18 - v'SO. 6. Solve Ix I= x + 6. 7. Solve Vx+J = x - 3.

8. Simplify

(VS+ 3)(2VS -

1).

9. Find the closest three-digit approximation to VmJ. 10. A wire is to be fastened to a telephone pole at a point 14 feet above the ground and to the ground at a point 5 feet from the foot of the pole. How long will the wire be, not including the amount required for fastening it to the pole and to the ground?

More about the Irrationality of ✓2 If an equation has a solution when the set of counting numbers C is the replacement set for each variable involved, we say that the equation has a solution in C or C X C or C X C X C • • •, depending on the number of variables involved. Some equations such as (1) 2 x

+ 5 = 9, (2) 3 x

- 7 y = 13, and (3) x2

+ y2 = z2

have solutions in C, C X C, or C X C X C, while others such as (4) 2 y - 2 x = 1 and (5) x2 = 7 do not. Observe that the ordered pairs (9, 2) and (16, 5) are solutions of(2) and that the ordered triples (3, 4, 5) and (12, 5, 13) are solutions of (3). Can you show that equations (4) a nd (5) have no solutions in C or C X C? A very interesting and easy proof of the irrationality of V2 can be based upon the fact that the equation (6) 2 x2 = y2 has no solution in C X C. If (x, y) t C X C, this equation is equivalent to

(7)

V2

= J:'._ X

Therefore, if (6) has no solution in C X C, then (7) has no solution in C X C. Moreover, if (7) has no solution in C X C, then V2 is irrational because a positive number which cannot be expressed as the quotient of two counting numbers is irrational. Thus the irrationality of V2 is established if we can prove that (6) has no solution when x a nd y are confined to the counting numbers. To show that 2 x2 = y2 has no solution in C X C, we first express the counting numbers in the base four system of numeration. Assuming that every counting number has a unique place-value representation in the base four system, we have the following table: Representation in base ten system R epresentation in base four system

1 2 3 - -

4

-

5 -

1 2 3 10 11

8

7

6 -

-

-

12 13 20

... -

23

...

-- -

41

42

...

-- -- -

... 113 ... 221 222 ...

When we consider the squares of some of the counting numbers represented in the base four system, we have l2 = l, 22 = 10, 3 2 = 21, 10 2 = 100, 12 2 = 210, 232 = 1321, and 1202 = 21000. We observe that each of these perfect squares has I for its last non-zero digit when the X O 1 2 3 digits are read from left to right. The reason for this is - - readily apparent when we study the multiplication table ~ ~ ~ ~ ~ shown. The portion of the table printed in red indicates 1 0 I 2 3 that the square of every counting number in the base four system has I for its last non-zero digit. Since every - - 3 O 3 12 21 counting number can be represented in the base four

2 0 2 10 12

540

system, we conclude that, if y f C, y2 has I for its last non-zero digit when it is expressed in this system . Similarly, if x f C, we know that x2 has I for its last non-zero digit when it is expressed in the base four system. Therefore 2 x 2 has 2 for its last non-zero digit . Why? Now a counting number having 2 for its last non-zero digit cannot be equal to a counting number that has I for its last non-zero digit. Why? Therefore 2 x 2 cannot be equal to y 2 when x and y are counting numbers. This proves that V2 is irrational since there a re no counting numbers x and y such that

V2

= 2:.' • X

The proof just given was based in part upon the assumption that every counting number has a unique place-value representation in the base four system . N otice that this assumption enabled us to assert that a counting number having 2 for its last non-zero digit cannot be equal to a counting number having I for it s last non-zero digit. Do you see why? Do you believe that every counting number has a unique representation in any numeration system whose base is a co unting number greater than I? Study the following plan for finding the unique base four representation of a number expressed in base ten . If we wish to express (473)ten in the base four sy stem , repeated division can be used as follows : 4 1473 4 1I 18 4 129 4 12 4 1_!_ 0

remainder I remainder 2 remainder I remainder 3 remainder I

473 118 29 7 I

= = = = =

4(118) + I 4(29) + 2 4(7) + I 4(1) + 3 4(0) + 1

(I) (2) (3) (4) (5)

From (I) and (2) we have : 473 = 4(4(29) + 2] + I or 4 73 = (29 . 42) + (2 . 4 1) + ( I . 40)

(6)

From (3) and (6) we have: 473 = 42(4(7) + I] + 4(2) + I or 473 = (7. 43) + (I . 42) + (2. 4 1) + (I . 40)

(7)

From (4) and (7) we have: 473 = 43(4(1) + 3] + 42(1) + 4(2) + I or 473 = (I. 44) + (3 . 43) + (I . 42) + (2. 4') + (I . 40)

(8)

From (5) and (8) we have: 473 = 44(4(0) + I]+ 43(3) + 42(1) + 4(2) + I or 473 = (0 . 45) + (1 . 44) + (3 . 43) + (I . 42) + (2 · 4 1) + (1 . 40)

(9)

Thus (473)tcn = (13 12 l )rour• Do you understand the method? Can you find the base four representation of the following perfect squares: (144)tcn, (576)ten, (1369) tcn? Can you use an argument similar to the one used to prove that V2 is an irrational number to prove that V3 is irrational? Can you use the properties of perfect squares in the base three system to show that V2 is irrational? 541

Chapur

12 Functions and Other Relations In this chapter you will study a special kind of relationship

Practically every day of our lives we pair the members of two sets of numbers. Even in the simple task of reporting the attendance at the first four football games of the season we probably pair the number of the game with the number of persons attending that game. In one school the report was: Number of game Number of persons attending

1

2

3

4

759

811

693

852

Observe that the members of {1, 2, 3, 4} are paired with the members of {759,811,693 , 852} so that 1 and 759 are in one pair, 2 and 811 are in another pair, etc. Even on the shelves of a grocery we see pairings of two sets of numbers. For example, one grocery paired the number of ounces in a can with a set of prices as follows: 8 oz. can . . . 10¢ 16 oz. can ... 19¢ 32 oz. can ... 36¢ Your teacher often pairs each test score with the number of students receiving that score to see more clearly how well the class has understood a lesson. Each hour of the day is paired with the local temperature reading by a radio station's weatherman. Relations

Each of the pairings above is a relation involving two sets of numbers. In mathematics we define a relation as follows: A relation is a set of ordered pairs. 542

543 There are several ways to indicate membership in a relation. Let us consider the two sets of numbers: B = {0, 1, 4} and D = {- 2, - 1, 0, 1, 2}. Now let us suppose that we want to pair 0 with 0, 1 with 1, 1 with - 1, 4 with 2, and 4 with - 2. Moreover, let us suppose that in each of these pairs we want the number from set B to be the first number in the pair. We may indicate membership in this relation by: (1) Listing its ordered pairs as shown below. {(0, 0), (1, 1), (1, - 1), (4, 2), (4, - 2)}

(2) Exhibiting its ordered pairs in the following table in which the top row gives the first member of each pair in the set and :Y the bottom row gives the corresponding second . member. "

_,

-

0

X 2

x when x, y e A. X B in which B = {- 2, - 1, 0, 1, 2}. In the graph of B X B use heavy black dots to graph the relation whose members are indica ted by the equation y = x 1 when x, y e B.

6. Graph the set B

+

7. Make a graph which expresses the relation whose membership is indicated by y = x + 4 if x e {- 1, - 2, - 3, - 4}. 8. List the ordered pairs of each relation indi cated by the black dots in each of the graphs below.

a.

b.

•· y

6

0

-

-

.?

4

' y

2

0

X

0

c.

y

IX

IA

-

-

0 0

'

-

-

u 2 4

'

545

Functions and Other Relations

9. Use set-builder notation to indicate membership in the relation pictured by the graph in part c of Ex. 8. y

10. Use set-builder notation to indicate the same membership for a relation as indicated by the graph at the right.

: !.{

-

-

Domain and Range of a Relation

0 :

The set of .first members of the number pairs in a relation is called the domain of the relation and the set of second members is called the ran e of the relation. Thus for the relation whose members are indicated by {(- 1, 3), (- 2, 6), (- 3, 9)}, the set of first members {- 1, - 2, - 3} is called the domain of the relation and the set of second members {3, 6, 9} is called the range of the relat ion. We have agreed to write the members of the replacement set of an open sentence in the two variables x and y in the form (x, y). The first member of the pair designates a value to replace x and the second member designates a value to replace y. Consequently, a relation expressed with the variables x and y will have the set of first elements, x, as the domain of the relation; and the set of second elements, y, as the range of the relation . When an open sentence is used to indicate membership in a relation and the domain and range are not specified, we agree to include in the domain and range all real numbers for which the sentence is true. For example, the domain of the relation whose membership is indicated by the open sentence y

=~

x-2

is all real numbers except 2. The range is the set of real

numbers except 0. Why is zero excluded?

0

EXERCISES

1. John's homework assignment required that he draw three squares, one whose sides were each one inch long, one whose sides were each two inches long, and one whose sides were each three inches long.

a. Complete the following table so that it pairs the lengths of the sides of the squares with the corresponding perimeters of the squares. Number of inches in length of side Number of inches in perimeter

H-+l+I

b. The domain of the relation whose members are indicated by the above table is {l, 2, 3}. What is the range of the relation?

546

CHAPTER

12

2. In the Community Fund drive the pledges accumulated by the end of each day were as follows: 1st day, $125,000; 2nd day, $200,000; 3rd day, $300,000; 4th day, $360,000.

a. Pair the number of the day with the number of dollars pledged by the end of that day and exhibit the ordered pairs in a table.

b. What is the domain of the relation indicated by the table? What is the range of the relation?

3. a. The three sides of an equilateral triangle are congruent. Write an equation that indicates membership in the relation whose elements are ordered pairs each having the length of the side of an equilateral triangle as its first member and the corresponding perimeter of the triangle as its second member. (Let x represent the number of inches in the length of the side and y the number of inches in the perimeter.)

b. The table at the right indicates that the domain of the relation of part a is to be {1, 1½, 2, 2½}. Write the numbers of the range of the relation in their proper places in the table and then graph the relation indicated by the table.

c. If the domain of the relation of part a is the set of all positive integers, we may indicate the do-

~!

::~n t~!erig~:~t~o:pp:; the range of the relation and graph the relation.

1~y1+.I+.I+.I+.I I .

.

.

.

.

d. Make a graph of the relation whose membership is indicated in part a for the case in which the domain of the relation is the set of all positive real numbers. e. Would it be sensible to allow the domain of the relation of part a to include negative numbers? Explain .

4. a. In the table at the right supply a range for the relation so that the table indicates membership in the same relation as that indicated by y

= X + 2;

XE {-

2, - 1, 0, 1, 2, 3}.

547

Functions and Other Relations

b. Make a graph of the relation whose membership is indicated in part a. c. Make a graph of the relation whose membership is indicated by y = x + 2 when the domain of the relation is the set of all real numbers and the range is the set of all real numbers.

5. a. If the relation whose membership is indicated by the equation

Ix -

y

=

21 has the domain {- 1, 0, 1, 2, 3, 4, S}, what is its range?

b. Graph the relation whose membership is indicated in part

a.

c. Graph the relation whose membership is indicated by the equation y = ] x - 2 I when the domain of the relation is the set of all x such that - 1 ~ x ~ 5 when x e R. What is the range of the relation? 6. a. Make a graph of the relation whose membership is indicated by the equation y = x - 1 when the domain of the relation is {x / 0 ~ x ~ 4; x e /}. What is the range of the relation?

b. Make a graph of the relation whose membership is indicated by the equation y = x - 1 when x e R. What is the range of the relation?

0

EXERCISES

7. Write the open sentence which indicates the same membership for a relation as indicated by each of the following tables. X -4 -3 -2 -1 0 1 a. - - - - - - - - - - 1 2 3 0 y -2 -1

X

-2

-1

0

1

2

3

y

-3

-1

1

3

5

7

b. - - -- - - - - -

1. 2

1

y

¼

1

X

3

2

y

2

2

X

c. - -

d. - - -

! I_:_ !14 -

1 2

-

5

2

25

-4

-1 -2 ---2 2 2

0

8. a. If U = {l, 2, 3}, graph the relation U X U using small dots. X U for which y = x. What is the domain of this relation? What is the range?

b. Use heavy black dots to indicate the subset of U

c. Make a second graph of U X U and then use heavy black dots to indicate the subset of U X U for which y > x. What are the domain and the range of this relation? d. Make a third graph of U X U and then use heavy black dots to indicate the subset of U X U for which x = 2. What are the domain and the range of this relation? 9. a. If S = {l, 2, 3, 4} and T = {4, 5, 6}, graph the set S X Tusing small dots for the points of the graph.

548

CHAPTER

b. Use heavy black dots to graph the subset of S X T for which

12

y = 6.

What are the domain and the range of the relation shown by the graph?

c. Make a second graph of S X T and then use heavy black dots to indicate the subset of S X T for which it is true that y > x. What are the domain and the range of the relation shown by the graph?

d. Make a third graph of S

X T and then use heavy dots to indicate the subset of S X T for which it is true that x > 1. What are the domain and the range of the relation shown by the graph?

10. Given that (x, y) e R X R, graph the following relations. a. {(x, y) I y < 2 x} c. {(x, y) I y = x + 2} State b. {(x, y) I y = x + 4 and 2 y = 3 x + 7} the domain and range. Functions

The graphs below indicate the ordered pairs which are members of two relations. The graph on the right shows a relation which is a function and the one on the left shows a relation which is not a function.

r

y

~

-I

~-

- t- -

I-

--

f--

>--• ,_

-

f--- f---

2

2

f---

-

-4

- 4

-

IX

C

iU

2

I

-!

-

-

B IU

2

0

4

4

_,

-IE -

-

; f (4 ;tlJ -

f---

IX

'

f--- f--- -

Graph of {(- 5, 2), (- 2, 2), (- 2, 3), (0, 2), (4, 3)}

Graph of {(- 4, - 4), (- 3, - 3), (0, 0), (I, l), (2, l), (3, I), (4, 2)}

We define a function as follows: A function is a relation in which each member of the domain is paired with one and only one member of the range. The number pairs (- 2, 2) and (- 2, 3) prevent the relation on the left from being a function because one member, - 2, of the domain is paired with two members of the range. Interpreted graphically, our definition of function means that any vertical line will intersect the graph of a function in at most one point. Point A of the graph at the right above has the x-coordinate 4, an element of the domain of the relation. If we pass a vertical line through A, it intersects the graph of the function in only the one point K (4, 2) . Point Bin the same axis does not have its x-coordinate in the domain. The vertical line through B does not intersect the graph of the function . Point C in the

549

Functions and Other Relations

x-axis of the graph on the left has the x-coordinate - 2, a member of the domain. A vertical line through C intersects the graph of the relation in the two points (- 2, 2) and (- 2, 3) . When a vertical line intersects the graph of a relation in two or more points, the relation is not a function. We have seen that a relation is a set of ordered pairs and that membership in a relation can be indicated by a list of its members, a table, a graph, a sentence, or set-builder notation. Hereafter we shall regard the phrase to describe a relation as equivalent to the phrase to indicate membership in a relation. Example l.

Draw the graph of the relation described by the sentence

2y-3x=7; -3 - 2 and x < 4

b.

x

2

2