Beginning and intermediate algebra [Fifth edition.,Pearson new international edition] 1292022876, 9781292022871, 3002902802

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Beginning and Intermediate Algebra Martin-Gay Fifth Edition

ISBN 978-1-29202-287-1

9 781292 022871

Beginning and Intermediate Algebra Elayn Martin-Gay Fifth Edition

Beginning and Intermediate Algebra Elayn Martin-Gay Fifth Edition

Pearson Education Limited Edinburgh Gate Harlow Essex CM20 2JE England and Associated Companies throughout the world Visit us on the World Wide Web at: www.pearsoned.co.uk © Pearson Education Limited 2014 All rights reserved. No part of this publication may be reproduced, stored in a retrieval system, or transmitted in any form or by any means, electronic, mechanical, photocopying, recording or otherwise, without either the prior written permission of the publisher or a licence permitting restricted copying in the United Kingdom issued by the Copyright Licensing Agency Ltd, Saffron House, 6–10 Kirby Street, London EC1N 8TS. All trademarks used herein are the property of their respective owners. The use of any trademark in this text does not vest in the author or publisher any trademark ownership rights in such trademarks, nor does the use of such trademarks imply any affiliation with or endorsement of this book by such owners.

ISBN 10: 1-292-02287-6 ISBN 13: 978-1-292-02287-1

British Library Cataloguing-in-Publication Data A catalogue record for this book is available from the British Library Printed in the United States of America

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Table of Contents 1. Review of Real Numbers Elayn Martin-Gay 1. Tips for Success in Mathematics

1 2

2. Symbols and Sets of Numbers

7

3. Fractions and Mixed Numbers

16

4. Exponents, Order of Operations, Variable Expressions, and Equations

25

5. Adding Real Numbers

35

6. Subtracting Real Numbers

43

Integrated Review—Operations on Real Numbers

50

7. Multiplying and Dividing Real Numbers

51

8. Properties of Real Numbers

61

Chapter Vocabulary Check

68

Chapter Highlights

68

Chapter Review

72

Chapter Test

75

2. Equations, Inequalities, and Problem Solving Elayn Martin-Gay 1. Simplifying Algebraic Expressions

81 83

2. The Addition and Multiplication Properties of Equality

91

3. Solving Linear Equations

101

Integrated Review—Solving Linear Equations

109

4. An Introduction to Problem Solving

110

5. Formulas and Problem Solving

121

6. Percent and Mixture Problem Solving

132

I

7. Further Problem Solving

144

8. Solving Linear Equations

151

Chapter Vocabulary Check

163

Chapter Highlights

163

Chapter Review

168

Chapter Test

171

3. Graphing Elayn Martin-Gay 1. Reading Graphs and the Rectangular Coordinate System

177 179

2. Graphing Linear Equations

194

3. Intercepts

204

4. Slope and Rate Change

212

Integrated Review—Summary on Slope and Graphing Linear Equations

226

5. Equations of Lines

227

6. Functions

236

Chapter Vocabulary Check

248

Chapter Highlights

248

Chapter Review

252

Chapter Test

255

4. Solving Systems of Linear Equations Elayn Martin-Gay 1. Solving Systems of Linear Equations by Graphing

265 266

2. Solving Systems of Linear Equations by Substitution

274

3. Solving Systems of Linear Equations by Addition

281

Integrated Review—Solving Systems of Equations

288

4. Solving Systems of Linear Equations in Three Variables

289

5. Systems of Linear Equations and Problem Solving

296

Chapter Vocabulary Check

314

Chapter Highlights

314

Chapter Review

317

Chapter Test

319

5. Exponents and Polynomials Elayn Martin-Gay 1. Exponents

II

325 327

2. Polynomial Functions and Adding and Subtracting Polynomials

338

3. Multiplying Polynomials

350

4. Special Products

357

Integrated Review—Exponents and Operations on Polynomials

364

5. Negative Exponents and Scientific Notation

364

6. Dividing Polynomials

373

7. Synthetic Division and the Remainder Theorem

380

Chapter Vocabulary Check

384

Chapter Highlights

385

Chapter Review

387

Chapter Test

390

6. Factoring Polynomials Elayn Martin-Gay 1. The Greatest Common Factor and Factoring by Grouping

397 398

2. Factoring Trinomials of the Form x2 + bx + c

406

3. Factoring Trinomials of the Form ax2 + bx + c and Perfect Square Trinomials

413

4. Factoring Trinomials of the Form ax2 + bx + c by Grouping

421

5. Factoring Binomials

426

Integrated Review-- Choosing a Factoring Strategy

433

6. Solving Quadratic Equations by Factoring

436

7. Quadratic Equations and Problem Solving

445

Chapter Vocabulary Check

454

Chapter Highlights

455

Chapter Review

458

Chapter Test

460

7. Rational Expressions Elayn Martin-Gay 1. Rational Functions and Simplifying Rational Expressions

465 466

2. Multiplying and Dividing Rational Expressions

477

3. Adding and Subtracting Rational Expressions with Common Denominators and Least Common Denominator

486

4. Adding and Subtracting Rational Expressions with Unlike Denominators

494

5. Solving Equations Containing Rational Expressions

500

Integrated Review—Summary on Rational Expressions

507

6. Proportion and Problem Solving with Rational Equations

508

III

7. Simplifying Complex Fractions

521

Chapter Vocabulary Check

527

Chapter Highlights

528

Chapter Review

531

Chapter Test

534

8. More on Functions and Graphs Elayn Martin-Gay 1. Graphing and Writing Linear Functions

539 541

2. Reviewing Function Notation and Graphing Nonlinear Functions

549

Integrated Review—Summary on Functions and Equations of Lines

557

3. Graphing Piecewise-Defined Functions and Shifting and Reflecting Graphs of Functions

558

4. Variation and Problem Solving

566

Chapter Vocabulary Check

574

Chapter Highlights

575

Chapter Review

577

Chapter Test

578

9. Inequalities and Absolute Value Elayn Martin-Gay 1. Compound Inequalities

585 586

2. Absolute Value Equations

593

3. Absolute Value Inequalities

598

Integrated Review—Solving Compound Inequalities and Absolute Value Equations and Inequalities

604

4. Graphing Linear Inequalities in Two Variables and Systems of Linear Inequalities

604

Chapter Vocabulary Check

613

Chapter Highlights

614

Chapter Review

616

Chapter Test

617

10. Rational Exponents, Radicals, and Complex Numbers Elayn Martin-Gay

IV

1. Radicals and Radical Functions

621 623

2. Rational Exponents

632

3. Simplifying Radical Expressions

639

4. Adding, Subtracting, and Multiplying Radical Expressions

647

5. Rationalizing Denominators and Numerators of Radical Expressions

653

Integrated Review—Radicals and Rational Exponents

659

6. Radical Equations and Problem Solving

660

7. Complex Numbers

670

Chapter Vocabulary Check

677

Chapter Highlights

677

Chapter Review

681

Chapter Test

683

11. Quadratic Equations and Functions Elayn Martin-Gay 1. Solving Quadratic Equations by Completing the Square

689 690

2. Solving Quadratic Equations by the Quadratic Formula

700

3. Solving Equations by Using Quadratic Methods

710

Integrated Review—Summary on Solving Quadratic Equations

719

4. Nonlinear Inequalities in One Variable

720

5. Quadratic Functions and Their Graphs

727

6. Further Graphing of Quadratic Functions

735

Chapter Vocabulary Check

743

Chapter Highlights

743

Chapter Review

746

Chapter Test

747

12. Exponential and Logarithmic Functions Elayn Martin-Gay 1. The Algebra of Function; Composite Functions

755 757

2. Inverse Functions

762

3. Exponential Functions

773

4. Exponential Growth and Decay Functions

782

5. Logarithmic Functions

786

6. Properties of Logarithms

794

Integrated Review—Functions and Properties of Logarithms

800

7. Common Logarithms, Natural Logarithms, and Change of Base

801

8. Exponential and Logarithmic Equations and Problem Solving

807

Chapter Vocabulary Check

813

Chapter Highlights

814

Chapter Review

817

V

Chapter Test

819

13. Conic Sections Elayn Martin-Gay 1. The Parabola and the Circle

827 828

2. The Ellipse and the Hyperbola

837

Integrated Review—Graphing Conic Sections

844

3. Solving Nonlinear Systems of Equations

845

4. Nonlinear Inequalities and Systems of Inequalities

850

Chapter Vocabulary Check

854

Chapter Highlights

854

Chapter Review

857

Chapter Test

858

14. Appendix: Operations on Decimals/Table of Percent, Decimal, and Fraction Equivalents Elayn Martin-Gay

863

15. Appendix: Review of Algebra Topics Elayn Martin-Gay 1. Equations (Linear and Quadratic Solved by Factoring)

869 870

2. Problem Solving

873

3. Graphing

882

4. Polynomials and Factoring

886

5. Rational Expressions

890

16. Appendix: An Introduction to Using a Graphing Utility Elayn Martin-Gay

895

17. Appendix: Solving Systems of Equations by Matrices Elayn Martin-Gay

901

18. Appendix: Solving Systems of Equations by Determinants Elayn Martin-Gay

907

19. Appendix: Mean, Median, and Mode Elayn Martin-Gay

917

20. Appendix: Review of Angles, Lines, and Special Triangles

VI

Elayn Martin-Gay

921

Index

929

Review of Real Numbers National Park Service Regions Midwest

1

Tips for Success in Mathematics

2

Symbols and Sets of Numbers

3

Fractions and Mixed Numbers

4

Exponents, Order of Operations, Variable Expressions, and Equations

5

Adding Real Numbers

6

Subtracting Real Numbers

Northeast

Pacific West National Capital Southeast

Intermountain Alaska

The National Park Service (NPS) is a federal agency that manages all national parks and many other historic monuments and properties. The NPS has 21,989 employees and had a total of 281 million recreational visitors in 2010 alone. The map above shows the different geographic regions of the NPS, and the bar graph below shows the number of recreational visits per year shown. In Section 2, Exercises 71 through 76, we shall study this bar graph further.

Number of Visitors (in millions)

Total Annual Recreational Visitors to U.S. National Park System 300 290 280

286 280

277

270

281

280

277 273

273

2005

2006

Integrated Review– Operations on Real Numbers

7

Multiplying and Dividing Real Numbers

8

Properties of Real Numbers

In this chapter, we review the basic symbols and words—the language— of arithmetic and introduce using variables in place of numbers. This is our starting place in the study of algebra.

275

266

260 250 2001

2002

2003

2004

2007

2008

2009

2010

Year

From Chapter 1 of Beginning and Intermediate Algebra, Fifth Edition. Elayn Martin-Gay. Copyright © 2013 by Pearson Education, Inc. All rights reserved.

1

Review of Real Numbers

1

Tips for Success in Mathematics

OBJECTIVES 1 Get Ready for This Course. 2 Understand Some General Tips for Success.

3 Get Help as Soon as You Need It.

4 Learn How to Prepare for and Take an Exam.

5 Develop Good Time Management.

Before reading this section, remember that your instructor is your best source of information. Please see your instructor for any additional help or information. OBJECTIVE

Getting Ready for This Course 1 Now that you have decided to take this course, remember that a positive attitude will make all the difference in the world. Your belief that you can succeed is just as important as your commitment to this course. Make sure you are ready for this course by having the time and positive attitude that it takes to succeed. Next, make sure that you have scheduled your math course at a time that will give you the best chance for success. For example, if you are also working, you may want to check with your employer to make sure that your work hours will not conflict with your course schedule. On the day of your first class period, double-check your schedule and allow yourself extra time to arrive on time in case of traffic problems or difficulty locating your classroom. Make sure that you bring at least your textbook, paper, and a writing instrument. Are you required to have a lab manual, graph paper, calculator, or some other supplies besides this text? If so, also bring this material with you. OBJECTIVE

2

Helpful Hint MyMathLab® and MathXL® If you are doing your homework online, you can work and rework those exercises that you struggle with until you master them. Try working through all the assigned exercises twice before the due date.

Helpful Hint MyMathLab® and MathXL® If you are completing your homework online, it’s important to work each exercise on paper before submitting the answer. That way, you can check your work and follow your steps to find and correct any mistakes.

2

General Tips for Success

Following are some general tips that will increase your chance for success in a mathematics class. Many of these tips will also help you in other courses you may be taking. Exchange names and phone numbers or email addresses with at least one other person in class. This contact person can be a great help if you miss an assignment or want to discuss math concepts or exercises that you find difficult. Choose to attend all class periods. If possible, sit near the front of the classroom. This way, you will see and hear the presentation better. It may also be easier for you to participate in classroom activities. Do your homework. You’ve probably heard the phrase “practice makes perfect” in relation to music and sports. It also applies to mathematics. You will find that the more time you spend solving mathematics exercises, the easier the process becomes. Be sure to schedule enough time to complete your assignments before the next due date assigned by your instructor. Check your work. Review the steps you made while working a problem. Learn to check your answers in the original problems. You may also compare your answers with the “Answers to Selected Exercises” section in the back of the book. If you have made a mistake, try to figure out what went wrong. Then correct your mistake. If you can’t find what went wrong, don’t erase your work or throw it away. Bring your work to your instructor, a tutor in a math lab, or a classmate. It is easier for someone to find where you had trouble if he or she looks at your original work. Learn from your mistakes and be patient with yourself. Everyone, even your instructor, makes mistakes. (That definitely includes me—Elayn Martin-Gay.) Use your errors to learn and to become a better math student. The key is finding and understanding your errors. Was your mistake a careless one, or did you make it because you can’t read your own math writing? If so, try to work more slowly or write more neatly and make a conscious effort to check your work carefully. Did you make a mistake because you don’t understand a concept? Take the time to review the concept or ask questions to understand it better. Did you skip too many steps? Skipping steps or trying to do too many steps mentally may lead to preventable mistakes. Know how to get help if you need it. It’s all right to ask for help. In fact, it’s a good idea to ask for help whenever there is something that you don’t understand. Make sure you know when your instructor has office hours and how to find his or her office. Find

Review of Real Numbers

Helpful Hint MyMathLab® and MathXL® When assignments are turned in online, keep a hard copy of your complete written work. You will need to refer to your written work to be able to ask questions and to study for tests later.

Helpful Hint MyMathLab® and MathXL® Be aware of assignments and due dates set by your instructor. Don’t wait until the last minute to submit work online. Allow 6–8 hours before the deadline in case you have technology trouble.

Helpful Hint MyMathLab® and MathXL® • Use the Help Me Solve This button to get step-by-step help for the exercise you are working. You will need to work an additional exercise of the same type before you can get credit for having worked it correctly. • Use the Video button to view a video clip of the author working a similar exercise.

out whether math tutoring services are available on your campus. Check on the hours, location, and requirements of the tutoring service. Organize your class materials, including homework assignments, graded quizzes and tests, and notes from your class or lab. All of these items will make valuable references throughout your course and when studying for upcoming tests and the final exam. Make sure that you can locate these materials when you need them. Read your textbook before class. Reading a mathematics textbook is unlike reading a novel or a newspaper. Your pace will be much slower. It is helpful to have paper and a pencil with you when you read. Try to work out examples on your own as you encounter them in your text. You should also write down any questions that you want to ask in class. When you read a mathematics textbook, sometimes some of the information in a section will be unclear. But after you hear a lecture or watch a lecture video on that section, you will understand it much more easily than if you had not read your text beforehand. Don’t be afraid to ask questions. You are not the only person in class with questions. Other students are normally grateful that someone has spoken up. Turn in assignments on time. This way, you can be sure that you will not lose points for being late. Show every step of a problem and be neat and organized. Also be sure that you understand which problems are assigned for homework. If allowed, you can always double-check the assignment with another student in your class.

OBJECTIVE

3 Getting Help If you have trouble completing assignments or understanding the mathematics, get help as soon as you need it! This tip is presented as an objective on its own because it is so important. In mathematics, usually the material presented in one section builds on your understanding of the previous section. This means that if you don’t understand the concepts covered during a class period, there is a good chance that you will not understand the concepts covered during the next class period. If this happens to you, get help as soon as you can. Where can you get help? Many suggestions have been made in this section on where to get help, and now it is up to you to get it. Try your instructor, a tutoring center, or a math lab, or you may want to form a study group with fellow classmates. If you do decide to see your instructor or go to a tutoring center, make sure that you have a neat notebook and are ready with your questions.

3

Review of Real Numbers OBJECTIVE

Helpful Hint MyMathLab® and MathXL® Review your written work for previous assignments. Then, go back and rework previous assignments. Open a previous assignment and click Similar Exercise to generate new exercises. Rework the exercises until you fully understand them and can work them without help features.

4 Preparing for and Taking an Exam Make sure that you allow yourself plenty of time to prepare for a test. If you think that you are a little “math anxious,” it may be that you are not preparing for a test in a way that will ensure success. The way that you prepare for a test in mathematics is important. To prepare for a test: 1. Review your previous homework assignments. 2. Review any notes from class and section-level quizzes you have taken. (If this is a final exam, also review chapter tests you have taken.) 3. Review concepts and definitions by reading the Chapter Highlights at the end of this chapter. 4. Practice working out exercises by completing the Chapter Review found at the end of this chapter. 5. It is important to place yourself in conditions similar to test conditions to find out how you will perform. In other words, as soon as you feel that you know the material, get a few blank sheets of paper and take a sample test. A Chapter Test is available at the end of this chapter, or you can work selected problems from the Chapter Review. Your instructor may also provide you with a review sheet. During this sample test, do not use your notes or your textbook. Then check your sample test. If you are not satisfied with the results, study the areas that you are weak in and try again. 6. On the day of the test, allow yourself plenty of time to arrive where you will be taking your exam. When taking your test: 1. Read the directions on the test carefully. 2. Read each problem carefully as you take the test. Make sure that you answer the question asked. 3. Watch your time and pace yourself so that you can attempt each problem on your test. 4. If you have time, check your work and answers. 5. Do not turn your test in early. If you have extra time, spend it double-checking your work.

4

Review of Real Numbers OBJECTIVE

5

Managing Your Time

As a college student, you know the demands that classes, homework, work, and family place on your time. Some days, you probably wonder how you’ll ever get everything done. One key to managing your time is developing a schedule. Here are some hints for making a schedule:

Stylephotographs/Dreamstime

1

1. Make a list of all your weekly commitments for the term. Include classes, work, regular meetings, extracurricular activities, etc. You may also find it helpful to list such things as laundry, regular workouts, grocery shopping, etc. 2. Next, estimate the time needed for each item on the list. Also make a note of how often you will need to do each item. Don’t forget to include time estimates for the reading, studying, and homework you do outside of your classes. You may want to ask your instructor for help estimating the time needed. 3. In the exercise set that follows, you are asked to block out a typical week on the schedule grid given. Start with items with fixed time slots like classes and work. 4. Next, include the items on your list with flexible time slots. Think carefully about how best to schedule items such as study time. 5. Don’t fill up every time slot on the schedule. Remember that you need to allow time for eating, sleeping, and relaxing! You should also allow a little extra time in case some items take longer than planned. 6. If you find that your weekly schedule is too full for you to handle, you may need to make some changes in your workload, classload, or other areas of your life. You may want to talk to your advisor, manager or supervisor at work, or someone in your college’s academic counseling center for help with such decisions.

Exercise Set

1. What is your instructor’s name? 2. What are your instructor’s office location and office hours? 3. What is the best way to contact your instructor? 4. Do you have the name and contact information of at least one other student in class? 5. Will your instructor allow you to use a calculator in this class? 6. Why is it important that you write step-by-step solutions to homework exercises and keep a hard copy of all work submitted?

7. Is a tutoring service available on campus? If so, what are its hours? What services are available? 8. Have you attempted this course before? If so, write down ways that you might improve your chances of success during this second attempt. 9. List some steps that you can take if you begin having trouble understanding the material or completing an assignment. If you are completing your homework in MyMathLab® and MathXL®, list the resources you can use for help.

5

Review of Real Numbers 10. How many hours of studying does your instructor advise for each hour of instruction? 11. How many times is it suggested that you work through the homework exercises in MathXL® before the submission deadline? 12. How far in advance of the assigned due date is it suggested that homework be submitted online? Why? 13. Chapter Highlights are found at the end of this chapter. Find the Chapter Highlights and explain how you might use it and how it might be helpful.

Monday

Tuesday

14. Chapter Reviews are found at the end of this chapter. Find the Chapter Review and explain how you might use it and how it might be useful. 15. Chapter Tests are at the end of this chapter. Find the Chapter Test and explain how you might use it and how it might be helpful when preparing for an exam on this chapter. If you are working in MyMathLab® and MathXL®, how can you use previous homework assignments to study? 16. Read or reread objective 5 and fill out the schedule grid below.

Wednesday

Thursday

Friday

Saturday

Sunday

1:00 a.m. 2:00 a.m. 3:00 a.m. 4:00 a.m. 5:00 a.m. 6:00 a.m. 7:00 a.m. 8:00 a.m. 9:00 a.m. 10:00 a.m. 11:00 a.m. Noon 1:00 p.m. 2:00 p.m. 3:00 p.m. 4:00 p.m. 5:00 p.m. 6:00 p.m. 7:00 p.m. 8:00 p.m. 9:00 p.m. 10:00 p.m. 11:00 p.m. Midnight

6 M01_MART5121_05_SE_C01_PCL.indd 6

28/03/12 2:34 AM

Review of Real Numbers

2

Symbols and Sets of Numbers OBJECTIVE

OBJECTIVES 1 Use a Number Line to Order Numbers.

2 Translate Sentences into

1

Using a Number Line to Order Numbers

We begin with a review of the set of natural numbers and the set of whole numbers and how we use symbols to compare these numbers. A set is a collection of objects, each of which is called a member or element of the set. A pair of brace symbols 5 6 encloses the list of elements and is translated as “the set of” or “the set containing.”

Mathematical Statements.

3 Identify Natural Numbers, Whole Numbers, Integers, Rational Numbers, Irrational Numbers, and Real Numbers.

4 Find the Absolute Value of a Real Number.

0 1

2

3 4

5

A number line

Natural Numbers The set of natural numbers is 5 1, 2, 3, 4, 5, 6, c6 . Whole Numbers The set of whole numbers is 5 0, 1, 2, 3, 4, c6 .

The three dots (an ellipsis) means that the list continues in the same manner indefinitely.

These numbers can be pictured on a number line. We will use number lines often to help us visualize distance and relationships between numbers. To draw a number line, first draw a line. Choose a point on the line and label it 0. To the right of 0, label any other point 1. Being careful to use the same distance as from 0 to 1, mark off equally spaced distances. Label these points 2, 3, 4, 5, and so on. Since e the whole numbers continue indefinitely, it is not possible to show every whole number on this number line. The arrow at the right end of the line indicates that the pattern continues indefinitely. Picturing whole numbers on a number line helps us see the order of the numbers. Symbols can be used to describe concisely in writing the order that we see. The equal symbol = means ;is equal to.< The symbol ⬆ means ;is not equal to.< These symbols may be used to form a mathematical statement. The statement might be true or it might be false. The two statements below are both true. 2 = 2 states that ;two is equal to two.< 2 ⬆ 6 states that ;two is not equal to six.
s total

T

number of days T

T

score each day T

=

3

#

( -7)

=

#

= -21 Thus, the golfer’s total score was -21, or 21 strokes under par. PRACTICE

11

A card player had a score of -13 for each of four games. Find the total score.

Lunamarina/Fotolia

Graphing Calculator Explorations Entering Negative Numbers on a Scientific Calculator To enter a negative number on a scientific calculator, find a key marked +/- . (On some calculators, this key is marked CHS for “change sign.”) To enter -8, for example, press the keys 8 +/- . The display will read -8 .

57

Review of Real Numbers Entering Negative Numbers on a Graphing Calculator To enter a negative number on a graphing calculator, find a key marked 1 - 2 . Do not confuse this key with the key - , which is used for subtraction. To enter -8, for example, press the keys 1 - 2 8 . The display will read -8 . Operations with Real Numbers To evaluate -217 - 92 - 20 on a calculator, press the keys 2 +/-

*

1-2 2

1 7

1 7 - 9 2 - 9 2

-

- 2 0 = or 2 0 ENTER -217 - 92 - 20 . -16

The display will read -16 or

Use a calculator to simplify each expression. 1. -38126 - 272 3. 134 + 25168 - 912 -5012942 5. 175 - 265

2. -591 -82 + 1726 4. 451322 - 812182 -444 - 444.8 6. -181 - 324

7. 95 - 4550

8. 58 - 6259

9. 1 -1252 2 (Be careful.)

10. -1252 (Be careful.)

Vocabulary & Readiness Check Use the choices below to fill in each blank. positive 1.

0

negative

undefined

If n is a real number, then n # 0 =

0 2. If n is a real number, but not 0, then = n

and we say

. n is 0

.

3. The product of two negative numbers is a number. number. 4. The quotient of two negative numbers is a 5. The quotient of a positive number and a negative number is a

number.

6. The product of a positive number and a negative number is a

number.

7. The reciprocal of a positive number is a 8. The opposite of a positive number is a

58

and 0 # n =

number. number.

Review of Real Numbers

7

Exercise Set Divide. See Examples 6 through 8.

Multiply. See Examples 1 through 3.

63.

18 -2

64.

20 -10

65.

-16 -4

6. -61 - 112

66.

-18 -6

67.

-48 12

68.

-2 # 8 -6 # 0

-60 5

69.

0 -4

70.

0 -9

71. -

24 8

73.

5 0

74.

3 0

76.

-45 -9

77.

30 -2

1. -6142

2. -8152

3. 21 -12

4. 71 -42

5. -51 -102 -3 # 4 -7 # 0

7. 9.

8. 10.

15 3

11. 21 -92

12. 31 -52

1 3 13. - a - b 2 5

1 1 14. - a - b 8 3

72. -

3 8 15. - a - b 4 9

5 3 16. - a - b 6 10

75.

- 12 -4

17. 51 -1.42

18. 61 -2.52

78.

14 -2

79.

80.

1 4 , a- b 5 2

81. -

5 3 , a- b 9 4

83. -

4 4 , 9 9

19. -0.21 -0.72

20. -0.51 -0.32

21. -101802

22. -201602

23. 41 -72

24. 51 -92

25. 1 -521 -52

26. 1 -721 - 72

27.

4 2 a- b 3 9

28.

29. - 111112 31. -

2 2 a- b 7 11

30. -121122

20 5 a b 25 16

32. -

33. 1 -121221 -3)1 -52

1 8 , a- b 10 11 5 5 84. , 12 12

82. -

MIXED PRACTICE

25 6 a b 36 15

Simplify. See Examples 1 through 9.

34. 1 - 221 -3)1 -421 -22

85.

35. 1-22152 - 1-112132

36. 81 -32 - 41 - 52

37. 1- 621-121-22 - 1-52

38. 20 - 1 -421321 - 22

Decide whether each statement is true or false. 39. The product of three negative integers is negative.

91.

40. The product of three positive integers is positive. 41. The product of four negative integers is negative.

93.

42. The product of four positive integers is positive. 95. Evaluate. See Example 4. 45. -1

44. -24 46. 1 -12

97. 5

47. 1 -52 2

48. -52

49. -72

50. 1 -72 2

99.

Find each reciprocal or multiplicative inverse. See Example 5. 51. 9

2 3

52. 100

53.

1 7

55. -14

56. -8

3 57. 11

6 58. 13

60. 1.5

61.

54.

1 -6.3

-62 + 4 -2 8 + 1 - 42 2 4 - 12 22 + 1321 -22 -5 - 2 -3 - 5 21 -72

59. 0.2 1 -8.9

-61 -32

-15 88. 1 - 4 90. 92. 94.

2

6 - 21 -32 4 - 31 -22 -3 - 21 -92 -15 - 31 -42

0 5 - 9 0 + 0 10 - 15 0 101. 0 21 -32 0

96. 98. 100. 102.

32 + 4 5 6 + 1 -22 2 4 - 9 - 20 + 1 -42132 1 - 5 -2 - 42 31 -62 8 - 31 -22 2 - 51 -42 -4 - 81 -22 -9 - 21 -32

0 -3 + 6 0 + 0 -2 + 7 0 0 -2 # 2 0

If x = -5 and y = -3, evaluate each expression. See Example 10. 103. 3x + 2y

104. 4x + 5y

105. 2x 2 - y 2

106. x 2 - 2y 2

3

108. y 3 + 3x 2y - 12 110. x - 4

107. x + 3y

62.

86.

-4

12 87. 9 - 12 89.

5

-91 -32 -6

Perform the indicated operations. See Example 2.

43. 1 -22 4

6 1 , a- b 7 3

2x - 5 y - 2 -3 - y 111. x - 4

109.

112.

4 - 2x y + 3

59

Review of Real Numbers 123. A deep-sea diver must move up or down in the water in short steps to keep from getting a physical condition called the “bends.” Suppose a diver moves down from the surface in five steps of 20 feet each. Represent his total movement as a product of signed numbers and find the total depth.

TRANSLATING Translate each phrase to an expression. Use x to represent “a number.” See Example 11. 113. The product of -71 and a number 114. The quotient of -8 and a number

124. A weather forecaster predicts that the temperature will drop five degrees each hour for the next six hours. Represent this drop as a product of signed numbers and find the total drop in temperature.

115. Subtract a number from - 16. 116. The sum of a number and -12 117. -29 increased by a number 118. The difference of a number and -10

Decide whether the given number is a solution of the given equation.

119. Divide a number by -33 .

125. Is 7 a solution of -5x = -35?

120. Multiply a number by -17.

126. Is -4 a solution of 2x = x - 1? x 127. Is -20 a solution of = 2? 10

Solve. See Example 11. 121. A football team lost four yards on each of three consecutive plays. Represent the total loss as a product of signed numbers and find the total loss.

128. Is -3 a solution of

122. Joe Norstrom lost $400 on each of seven consecutive days in the stock market. Represent his total loss as a product of signed numbers and find his total loss.

45 = -15? x

129. Is 5 a solution of -3x - 5 = -20? 130. Is -4 a solution of 2x + 4 = x + 8?

CONCEPT EXTENSIONS Study the bar graph below showing the average surface temperatures of planets. Use Exercises 131 and 132 to complete the planet temperatures on the graph. (Pluto is now classified as a dwarf planet.) 135. Find any real numbers that are their own reciprocal.

Average Surface Temperature of Planets* 1200

136. Explain why 0 has no reciprocal.

867

800

If q is a negative number, r is a negative number, and t is a positive number, determine whether each expression simplifies to a positive or negative number. If it is not possible to determine, state so.

600 400

330

200

59

0

137.

q r#t

138. 139. 140. 141. 142.

q2 # r # t q + t t + r t1q + r2 r1q - t2

ⴚ323 Neptune

Jupiter

Mars

Earth

600

Venus

400

Saturn

ⴚ218

Uranus

ⴚ81

200

Mercury

Degrees Fahrenheit

1000

*For some planets, the temperature given is the temperature where the atmosphere pressure equals 1 Earth atmosphere; data from The World Almanac

Write each of the following as an expression and evaluate. 131. The surface temperature of Jupiter is twice the temperature of Mars. 143. Find this temperature. 144. 132. The surface temperature of Neptune is equal to the temperature of 145. Mercury divided by -1. Find this temperature. 146. 133. Explain why the product of an even number of negative numbers is a positive number. 134. If a and b are any real numbers, is the statement a # b = b # a always true? Why or why not?

60

The sum of -2 and the quotient of -15 and 3 The sum of 1 and the product of -8 and -5 Twice the sum of -5 and -3 7 subtracted from the quotient of 0 and 5

Review of Real Numbers

8

Properties of Real Numbers OBJECTIVE

OBJECTIVES 1 Use the Commutative and Associative Properties.

2 Use the Distributive Property.

3 Use the Identity and Inverse Properties.

Using the Commutative and Associative Properties 1 In this section, we give names to properties of real numbers with which we are already familiar. Throughout this section, the variables a, b, and c represent real numbers. We know that order does not matter when adding numbers. For example, we know that 7 + 5 is the same as 5 + 7. This property is given a special name—the commutative property of addition. We also know that order does not matter when multiplying numbers. For example, we know that -5162 = 61 -52. This property means that multiplication is commutative also and is called the commutative property of multiplication. Commutative Properties a + b = b + a a#b = b#a

Addition: Multiplication:

These properties state that the order in which any two real numbers are added or multiplied does not change their sum or product. For example, if we let a = 3 and b = 5, then the commutative properties guarantee that 3 + 5 = 5 + 3

and

3#5 = 5#3

Helpful Hint Is subtraction also commutative? Try an example. Does 3 - 2 = 2 - 3? No! The left side of this statement equals 1; the right side equals -1. There is no commutative property of subtraction. Similarly, there is no commutative property for division. For example, 10 , 2 does not equal 2 , 10.

EXAMPLE 1

Use a commutative property to complete each statement. b. 3 # x =

a. x + 5 =

Solution a. x + 5 = 5 + x b. 3 # x = x # 3

By the commutative property of addition By the commutative property of multiplication

PRACTICE

1 a.

Use a commutative property to complete each statement.

x#8 =

b. x + 17 =

CONCEPT CHECK Which of the following pairs of actions are commutative? a. b. c. d.

“raking the leaves” and “bagging the leaves” “putting on your left glove” and “putting on your right glove” “putting on your coat” and “putting on your shirt” “reading a novel” and “reading a newspaper”

Answers to Concept Check: b, d

Let’s now discuss grouping numbers. We know that when we add three numbers, the way in which they are grouped or associated does not change their sum. For example, we know that 2 + 13 + 42 = 2 + 7 = 9. This result is the same if we group the numbers differently. In other words, 12 + 32 + 4 = 5 + 4 = 9 also. Thus, 2 + 13 + 42 = 12 + 32 + 4. This property is called the associative property of addition. We also know that changing the grouping of numbers when multiplying does not change their product. For example, 2 # 13 # 42 = 12 # 32 # 4 (check it). This is the associative property of multiplication.

61

Review of Real Numbers

Associative Properties Addition: Multiplication:

1a + b2 + c = a + 1b + c2 1a # b2 # c = a # 1b # c2

These properties state that the way in which three numbers are grouped does not change their sum or their product.

EXAMPLE 2

Use an associative property to complete each statement. b. 1 -1 # 22 # 5 =

a. 5 + 14 + 62 =

Solution a. 5 + 14 + 62 = 15 + 42 + 6 b. 1 -1 # 22 # 5 = -1 # 12 # 52

By the associative property of addition By the associative property of multiplication

PRACTICE

2

Use an associative property to complete each statement. b. -4 # 12 # 72 =

a. 12 + 92 + 7 =

Helpful Hint Remember the difference between the commutative properties and the associative properties. The commutative properties have to do with the order of numbers, and the associative properties have to do with the grouping of numbers.

Let’s now illustrate how these properties can help us simplify expressions.

EXAMPLE 3 a. 10 + 1x + 122

Simplify each expression. b. -317x2

Solution a. 10 + 1x + 122 = 10 + 112 + x2 = 110 + 122 + x = 22 + x b. -317x2 = 1 -3 # 72x = -21x

By the commutative property of addition By the associative property of addition Add. By the associative property of multiplication Multiply.

PRACTICE

3

Simplify each expression.

a. 15 + x2 + 9

b. 51 -6x2

OBJECTIVE

2 Using the Distributive Property The distributive property of multiplication over addition is used repeatedly throughout algebra. It is useful because it allows us to write a product as a sum or a sum as a product. We know that 712 + 42 = 7162 = 42. Compare that with 7122 + 7142 = 14 + 28 = 42. Since both original expressions equal 42, they must equal each other, or 7(2+4)=7(2)+7(4) This is an example of the distributive property. The product on the left side of the equal sign is equal to the sum on the right side. We can think of the 7 as being distributed to each number inside the parentheses.

62

Review of Real Numbers

Distributive Property of Multiplication Over Addition a(b+c)=ab+ac Since multiplication is commutative, this property can also be written as (b+c)a=ba+ca The distributive property can also be extended to more than two numbers inside the parentheses. For example, 3(x+y+z)=3(x)+3(y)+3(z) = 3x + 3y + 3z Since we define subtraction in terms of addition, the distributive property is also true for subtraction. For example 2(x-y)=2(x)-2(y) = 2x - 2y

E X A M P L E 4 Use the distributive property to write each expression without parentheses. Then simplify if possible. a. 21x + y2

b. -51 -3 + 2z2

d. -112 - y2

e. -13 + x - w2

c. 51x + 3y - z2 1 f. 16x + 142 + 10 2

Solution a. 2(x+y)=2 # x+2 # y = 2x + 2y b. –5(–3+2z)=–5(–3)+(–5)(2z) = 15 - 10z c. 5(x+3y-z)=5(x)+5(3y)-5(z) = 5x + 15y - 5z d. -112 - y2 = 1 -12122 - 1 -121y2 Helpful Hint = -2 + y Notice in part (e) that -13 + x - w2 is first rewritten as -113 + x - w2. e. -13 + x - w2 = -113 + x - w2 = 1 -12132 + 1 -121x2 - 1 -121w2 = -3 - x + w 1 1 1 16x2 + 1142 + 10 Apply the distributive property. f. 16x + 142 + 10 = 2 2 2 Multiply. = 3x + 7 + 10 = 3x + 17

Add.

63

Review of Real Numbers PRACTICE

4 Use the distributive property to write each expression without parentheses. Then simplify if possible. a. 51x - y2

b. -614 + 2t2 d. 13 - y2 # 1 -12 1 f. 12x + 42 + 9 2

c. 213x - 4y - z2 e. -1x - 7 + 2s2

We can use the distributive property in reverse to write a sum as a product.

EXAMPLE 5

Use the distributive property to write each sum as a product.

a. 8 # 2 + 8 # x

b. 7s + 7t

Solution a. 8 # 2 + 8 # x = 812 + x2

b. 7s + 7t = 71s + t2

PRACTICE

5 a.

Use the distributive property to write each sum as a product.

5#w + 5#3

b. 9w + 9z

OBJECTIVE

3 Using the Identity and Inverse Properties Next, we look at the identity properties. The number 0 is called the identity for addition because when 0 is added to any real number, the result is the same real number. In other words, the identity of the real number is not changed. The number 1 is called the identity for multiplication because when a real number is multiplied by 1, the result is the same real number. In other words, the identity of the real number is not changed.

Identities for Addition and Multiplication 0 is the identity element for addition. a + 0 = a

and

0 + a = a

1 is the identity element for multiplication. a#1 = a

and

1#a = a

Notice that 0 is the only number that can be added to any real number with the result that the sum is the same real number. Also, 1 is the only number that can be multiplied by any real number with the result that the product is the same real number. Additive inverses or opposites were introduced in Section 5. Two numbers are called additive inverses or opposites if their sum is 0. The additive inverse or opposite of 6 is -6 because 6 + 1 -62 = 0. The additive inverse or opposite of -5 is 5 because -5 + 5 = 0.

64

Review of Real Numbers Reciprocals or multiplicative inverses were introduced in Section 3. Two nonzero numbers are called reciprocals or multiplicative inverses if their product is 1. The recipro2 3 2 3 cal or multiplicative inverse of is because # = 1. Likewise, the reciprocal of -5 is 3 2 3 2 1 1 - because -5a - b = 1. 5 5

CONCEPT CHECK Which of the following, 1, a. opposite of -

3 ? 10

10 3 3 10 , , 0, , - , is the 3 10 3 10 3 b. reciprocal of - ? 10

Additive or Multiplicative Inverses The numbers a and -a are additive inverses or opposites of each other because their sum is 0; that is, a + 1 -a2 = 0 1 The numbers b and (for b ⬆ 0) are reciprocals or multiplicative inverses of each b other because their product is 1; that is, 1 b# = 1 b

EXAMPLE 6

Name the property or properties illustrated by each true statement.

Solution a. 3 # y = y # 3 b. 1x + 72 + 9 = x + 17 + 92

Commutative property of multiplication (order changed)

c. 1b + 02 + 3 = b + 3 d. 0.2 # 1z # 52 = 0.2 # 15 # z2 1 e. -2 # a - b = 1 2

Identity element for addition

f. -2 + 2 = 0 g. -6 # 1y # 22 = 1 -6 # 22 # y

Additive inverse property

Associative property of addition (grouping changed) Commutative property of multiplication (order changed) Multiplicative inverse property

Commutative and associative properties of multiplication (order and grouping changed)

PRACTICE

6

Name the property or properties illustrated by each true statement.

Commutative property of multiplication a. 17 # 3x2 # 4 = 13x # 72 # 4 b. 6 + 13 + y2 = 16 + 32 + y Associative property of addition Identity element for addition c. 8 + 1t + 02 = 8 + t 3 4 d. - # a - b = 1 Multiplicative inverse property 4 3

Answers to Concept Check: 3 10 a. b. 10 3

e. 12 + x2 + 5 = 5 + 12 + x2 f. 3 + 1 -32 = 0 g. 1 -3b2 # 7 = 1 -3 # 72 # b

Commutative property of addition Additive inverse property Commutative and associative properties of multiplication

65

Review of Real Numbers

Vocabulary & Readiness Check Use the choices below to fill in each blank. distributive property opposites or additive inverses reciprocals or multiplicative inverses

associative property of multiplication associative property of addition commutative property of multiplication

1. x + 5 = 5 + x is a true statement by the 2. 3. 4.

.

x # 5 = 5 # x is a true statement by the 31y + 62 = 3 # y + 3 # 6 is true by the 2 # 1x # y2 = 12 # x2 # y is a true statement by the

. . .

5. x + 17 + y2 = 1x + 72 + y is a true statement by the 6. The numbers -

2 3 and - are called 3 2

7. The numbers -

2 2 and are called 3 3

8

. . .

Exercise Set

Use a commutative property to complete each statement. See Example 1.

Use the commutative and associative properties to simplify each expression. See Example 3.

1. x + 16 =

2. 4 + y =

17. 8 + 19 + b2

18. 1r + 32 + 11

3. -4 # y =

4. -2 # x =

19. 4(6y)

20. 2(42x)

5. xy =

6. ab =

7. 2x + 13 =

8. 19 + 3y =

1 15y2 21. 5

22.

23. 113 + a2 + 13

24. 7 + 1x + 42

25. -918x2 3 4 a sb 27. 4 3

26. - 3112y2 2 7 28. a rb 7 2

Use an associative property to complete each statement. See Example 2. 9. 1xy2 # z = 11. 2 + 1a + b2 = 13.

4 # 1ab2

=

15. 1a + b2 + c = 16. 6 + 1r + s2 =

66

commutative property of addition

10. 3 # 1xy2 = 12. 1y + 42 + z = 14. 1 -3y2 # z =

29.

4 2 + a + xb 3 3

30.

1 18z2 8

7 2 + a + yb 9 9

Review of Real Numbers

Use the distributive property to write each expression without parentheses. Then simplify the result. See Example 4. 31. 33. 35. 37. 39. 41. 43.

41x + y2 91x - 62 213x + 52 714x - 32 316 + x2 - 21y - z2 -713y + 52

44. 45. 46. 47. 48. 49. 50. 51. 52.

-512r + 112 51x + 4m + 22 813y + z - 62 -411 - 2m + n2 -414 + 2p + 5q2 -15x + 22 -19r + 52 -1r - 3 - 7p2 -1q - 2 + 6r2

32. 34. 36. 38. 40. 42.

71a + b2 111y - 42 517 + 8y2 318x - 12 21x + 52 -31z - y2

0 + 6 = 6 1a + 92 + 6 = a + 19 + 62 -41y + 72 = -4 # y + 1 -42 # 7 111 + r2 + 8 = 1r + 112 + 8 -4 # 18 # 32 = 18 # -42 # 3 r + 0 = r

CONCEPT EXTENSIONS Fill in the table with the opposite (additive inverse), and the reciprocal (multiplicative inverse). Assume that the value of each expression is not 0. Expression

Opposite

Reciprocal

8

85. 86.

1 53. 16x + 82 2 1 54. 14x - 22 4 1 55. - 13x - 9y2 3

-

2 3

87.

x

88.

4y 1 2x

89. 90.

1 56. - 110a - 25b2 5 57. 58. 59. 60. 61. 62.

79. 80. 81. 82. 83. 84.

7x

Decide whether each statement is true or false. See the second Concept Check in this section.

312r + 52 - 7 1014s + 62 - 40 -914x + 82 + 2 -1115x + 32 + 10 -414x + 52 - 5 -612x + 12 - 1

a 2 is - . 2 a a a 92. The reciprocal of - is . 2 2

91. The opposite of -

Use the distributive property to write each sum as a product. See Example 5. 63. 4 # 1 + 4 # y

64. 14 # z + 14 # 5

65. 11x + 11y

66. 9a + 9b

67. 1 -12 # 5 + 1 -12 # x

68. 1 -32a + 1 -32b

69. 30a + 30b

70. 25x + 25y

Determine which pairs of actions are commutative. See the first Concept Check in this section. 93. “taking a test” and “studying for the test” 94. “putting on your shoes” and “putting on your socks”

Name the properties illustrated by each true statement. See Example 6. 71.

3#5

=

5#3

95. “putting on your left shoe” and “putting on your right shoe” 96. “reading the sports section” and “reading the comics section” 97. “mowing the lawn” and “trimming the hedges” 98. “baking a cake” and “eating the cake” 99. “dialing a number” and “turning on the cell phone”

72. 413 + 82 = 4 # 3 + 4 # 8

100. “feeding the dog” and “feeding the cat”

73. 2 + 1x + 52 = 12 + x2 + 5

Name the property illustrated by each step.

74. 1x + 92 + 3 = 19 + x2 + 3 75. 913 + 72 = 9 # 3 + 9 # 7 76. 1 # 9 = 9

101. a. 䉭 + (䊐 + ~ ) = (䊐 + ~ ) + 䉭 b.

= 1~ + 䊐2 + 䉭

c.

= ~ + 1䊐 + 䉭2

77. 14 # y2 # 9 = 4 # 1y # 92 78. 6 #

1 = 1 6

67

Review of Real Numbers 102. a. 1x + y2 + z = x + 1y + z2 b. = 1y + z2 + x c. = 1z + y2 + x

104. Explain why 1 is called the identity element for multiplication.

103. Explain why 0 is called the identity element for addition.

106. Write an example that shows that subtraction is not commutative.

Chapter

105. Write an example that shows that division is not commutative.

Vocabulary Check

Fill in each blank with one of the words or phrases listed below. set

1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12.

inequality symbols

opposites

absolute value

numerator

denominator

grouping symbols

exponent

base

reciprocals

variable

equation

solution

The symbols ⬆, 6 , and 7 are called . A mathematical statement that two expressions are equal is called a(n) . The of a number is the distance between that number and 0 on the number line. A symbol used to represent a number is called a(n) . Two numbers that are the same distance from 0 but lie on opposite sides of 0 are called . The number in a fraction above the fraction bar is called the . A(n) of an equation is a value for the variable that makes the equation a true statement. Two numbers whose product is 1 are called . In 23, the 2 is called the and the 3 is called the . The number in a fraction below the fraction bar is called the . Parentheses and brackets are examples of . A(n) is a collection of objects.

Chapter

Highlights

DEFINITIONS AND CONCEPTS

EXAMPLES Section 2

Symbols and Sets of Numbers

A set is a collection of objects, called elements, enclosed in braces. Natural Numbers: 5 1, 2, 3, 4, c6 Whole Numbers: 5 0, 1, 2, 3, 4, c6 Integers: 5c , -3, -2, -1, 0, 1, 2, 3, c6 Rational Numbers: {real numbers that can be expressed as a quotient of integers}

68

5 a, c, e 6 2 Given the set e -3.4, 23, 0, , 5, - 4 f , list the numbers 3 that belong to the set of Natural numbers: 5 Whole numbers: 0, 5 Integers: -4, 0, 5

Irrational Numbers: {real numbers that cannot be expressed as a quotient of integers}

2 Rational numbers: -4, -3.4, 0, , 5 3 Irrational Numbers: 13

Real Numbers: {all numbers that correspond to a point on the number line}

2 Real numbers: -4, -3.4, 0, , 13, 5 3

Review of Real Numbers

DEFINITIONS AND CONCEPTS

EXAMPLES Section 2

Symbols and Sets of Numbers

A line used to picture numbers is called a number line.

3 2 1 0 1

The absolute value of a real number a, denoted by 0 a 0 , is the distance between a and 0 on the number line.

050 = 5

Symbols: = is equal to ⬆ is not equal to

-7 = -7 3 ⬆ -3

7 is greater than

4 7 1

6 is less than

1 6 4

… is less than or equal to

6 … 6 18 Ú -

Ú is greater than or equal to Order Property for Real Numbers

000 = 0

1 3

3 2 1 0 1

For any two real numbers a and b, a is less than b if a is to the left of b on a number line. Section 3

3  0

0  3

2 3

0  2.5

2.5  0

Fractions and Mixed Numbers

A quotient of two integers is called a fraction. The numerator of a fraction is the top number. The denominator of a fraction is the bottom number. If a # b = c, then a and b are factors and c is the product. A fraction is in lowest terms or simplest form when the numerator and the denominator have no factors in common other than 1.

2 3

0 -2 0 = 2

13 d numerator 17 d denominator

# 9 = 7 T T factor factor 13 is in simplest form. 17

To write a fraction in simplest form, factor the numerator and the denominator; then apply the fundamental principle.

Write in simplest form.

Two fractions are reciprocals if their product is 1. a b The reciprocal of is . b a To multiply fractions, numerator times numerator is the numerator of the product and denominator times denominator is the denominator of the product.

The reciprocal of

63 T product

2#3 3 6 = # = 14 2 7 7 6 25 is . 25 6

Perform the indicated operations. 2#3 6 = 5 7 35

To divide fractions, multiply the first fraction by the reciprocal of the second fraction.

2 5 7 35 5 , = # = 9 7 9 2 18

To add fractions with the same denominator, add the numerators and place the sum over the common denominator.

5 3 8 + = 11 11 11

To subtract fractions with the same denominator, subtract the numerators and place the difference over the common denominator.

2 13 3 10 = = 15 15 15 3

Fractions that represent the same quantity are called equivalent fractions.

1#4 4 1 = # = 5 5 4 20 1 4 and are equivalent fractions. 5 20

69

Review of Real Numbers

DEFINITIONS AND CONCEPTS Section 4

EXAMPLES

Exponents, Order of Operations, Variable Expressions, and Equations

The expression a n is an exponential expression. The number a is called the base; it is the repeated factor. The number n is called the exponent; it is the number of times that the base is a factor.

43 = 4 # 4 # 4 = 64 72 = 7 # 7 = 49 82 + 517 - 32

Order of Operations

3#7

Simplify expressions in the following order. If grouping symbols are present, simplify expressions within those first, starting with the innermost set. Also, simplify the numerator and the denominator of a fraction separately.

= = =

1. Simplify exponential expressions. 2. Multiply or divide in order from left to right. 3. Add or subtract in order from left to right.

82 + 5142 21 64 + 5142 21 64 + 20 21

84 21 = 4 =

A symbol used to represent a number is called a variable.

Examples of variables are: q, x, z

An algebraic expression is a collection of numbers, variables, operation symbols, and grouping symbols.

Examples of algebraic expressions are: 5x, 21y - 62,

To evaluate an algebraic expression containing a variable, substitute a given number for the variable and simplify.

q 2 - 3q + 1 6

Evaluate x 2 - y 2 if x = 5 and y = 3. x 2 - y 2 = 152 2 - 32 = 25 - 9 = 16

A mathematical statement that two expressions are equal is called an equation.

Examples of equations are: 3x - 9 = 20 A = pr 2

A solution of an equation is a value for the variable that makes the equation a true statement.

Determine whether 4 is a solution of 5x + 7 = 27. 5x + 7 = 27 5142 + 7 ⱨ 27 20 + 7 ⱨ 27 27 = 27 True 4 is a solution.

Section 5 To Add Two Numbers with the Same Sign

1. Add their absolute values. 2. Use their common sign as the sign of the sum.

Adding Real Numbers Add. 10 + 7 = 17 -3 + 1 -82 = - 11

To Add Two Numbers with Different Signs

1. Subtract their absolute values. 2. Use the sign of the number whose absolute value is larger as the sign of the sum.

70

-25 + 5 = - 20 14 + 1 - 92 = 5

Review of Real Numbers

DEFINITIONS AND CONCEPTS

EXAMPLES

Section 5

Adding Real Numbers (continued )

Two numbers that are the same distance from 0 but lie on opposite sides of 0 are called opposites or additive inverses. The opposite of a number a is denoted by -a.

The opposite of - 7 is 7. The opposite of 123 is -123. -4 + 4 = 0

The sum of a number a and its opposite, -a, is 0.

12 + 1 -122 = 0

a + 1 -a2 = 0

-1 -82 = 8

If a is a number, then -1 -a2 = a.

-1 -142 = 14 Section 6

To subtract two numbers a and b, add the first number a to the opposite of the second number b.

Subtracting Real Numbers Subtract. 3 - 1 -442 = 3 + 44 = 47

a - b = a + 1 -b2

-5 - 22 = -5 + 1 -222 = - 27 - 30 - 1 -302 = - 30 + 30 = 0 Section 7

Multiplying and Dividing Real Numbers Multiply or divide.

Quotient of two real numbers

42 1 = 42 # = 21 2 2

a 1 = a# b b

7 # 8 = 56

Multiplying and Dividing Real Numbers The product or quotient of two numbers with the same sign is a positive number. The product or quotient of two numbers with different signs is a negative number.

Products and Quotients Involving Zero

-2 # 4 = -8

-7 # 1 -82 = 56 2 # 1 -42 = -8

90 = 9 10

-90 = 9 -10

42 = -7 -6

-42 = -7 6

The product of 0 and any number is 0. -4 # 0 = 0

b # 0 = 0 and 0 # b = 0

3 0# a- b = 0 4

The quotient of a nonzero number and 0 is undefined. b is undefined. 0 The quotient of 0 and any nonzero number is 0.

-85 is undefined. 0

0 = 0 b

0 = 0 18 Section 8

0 = 0 -47

Properties of Real Numbers

Commutative Properties Addition: a + b = b + a Multiplication: a # b = b # a

3 + 1 -72 = -7 + 3 -8 # 5 = 5 # 1 -82

Associative Properties Addition: 1a + b2 + c = a + 1b + c2 Multiplication: 1a # b2 # c = a # 1b # c2

15 + 102 + 20 = 5 + 110 + 202 1 -3 # 22 # 11 = -3 # 12 # 112 (continued)

71

Review of Real Numbers

DEFINITIONS AND CONCEPTS

EXAMPLES Section 8

Two numbers whose product is 1 are called multiplicative inverses or reciprocals. The reciprocal 1 1 of a nonzero number a is because a # = 1. a a Distributive Property

Properties of Real Numbers 1 The reciprocal of 3 is . 3 2 5 The reciprocal of - is - . 5 2

a1b + c2 = a # b + a # c

516 + 102 = 5 # 6 + 5 # 10 -213 + x2 = -2 # 3 + 1 -221x2

Identities

a + 0 = a

0 + a = a

a#1 = a

1#a = a

5 + 0 = 5

0 + 1 -22 = -2

-14 # 1 = -14

1 # 27 = 27

Inverses Additive or opposite: a + 1 -a2 = 0

7 + 1 -72 = 0

1 Multiplicative or reciprocal: b # = 1 b

3#

1 = 1 3

Chapter Review (2) Insert 6 , 7 , or = in the appropriate space to make the following statements true. 1. 8

10

3. -4

2. 7

5. 0 -7 0

0 -8 0

7. - 0 -1 0 9. 1.2

2

12 4. 2

-5

6. 0 -9 0 8. 0 -14 0

-1

3 10. 2

1.02

The following chart shows the gains and losses in dollars of Density Oil and Gas stock for a particular week. Day

-8 -9 -1 -142 3 4

TRANSLATING

Gain or Loss in Dollars

Monday

+1

Tuesday

-2

Wednesday

+5

Thursday

+1

Friday

-4

17. Which day showed the greatest loss?

Translate each statement into symbols. 11. Four is greater than or equal to negative three.

18. Which day showed the greatest gain?

12. Six is not equal to five.

(3) Write the number as a product of prime factors.

13. 0.03 is less than 0.3.

19. 36

14. New York City has 155 museums and 400 art galleries. Write an inequality comparing the numbers 155 and 400. (Source: Absolute Trivia.com) Given the following sets of numbers, list the numbers in each set that also belong to the set of:

a. Natural numbers c. Integers e. Irrational numbers

b. Whole numbers d. Rational numbers f. Real numbers

Perform the indicated operations. Write results in lowest terms. 21.

8 # 27 15 30

22.

21 7 , 8 32

23.

5 7 + 15 6

24.

3 3 4 20

25. 2

3 5 + 6 4 8

27. 5 , 1 15. e -6, 0, 1, 1 , 3, p, 9.62 f 2 16. e -3, -1.6, 2, 5,

72

11 , 15.1, 25, 2p f 2

20. 120

1 3

26. 7

1 2 - 2 6 3

28. 2 # 8

3 4

Review of Real Numbers Each circle represents a whole, or 1. Determine the unknown part of the circle. 29.

30. Z

~

q Q

45.

5 11

3 11

in.

in.

5 11

in.

in.

Octuplets were born in the U.S. in 2009. The following chart gives the octuplets’ birthweights. The babies are listed in order of birth. Baby

Gender

Baby A

boy

Baby B

girl

Baby C

boy

Baby D

girl

Baby E

boy

Baby F

boy

Baby G

boy

Baby H

boy

Birthweight (pounds) 1 2 1 2 8 1 3 16 3 2 16 3 1 4 9 2 16 13 1 16 7 2 16

Evaluate each expression if x = 6, y = 2, and z = 8. 49. 2x + 3y z x + 51. y 2y

50. x1y + 2z2 52. x 2 - 3y 2

53. The expression 180 - a - b represents the measure of the unknown angle of the given triangle. Replace a with 37 and b with 80 to find the measure of the unknown angle.

2

33. What was the total weight of the boy octuplets? 34. What was the total weight of the girl octuplets? 35. Find the combined weight of all eight octuplets. 36. Which baby weighed the most? 37. Which baby weighed the least? 38. How much more did the heaviest baby weigh than the lightest baby?

(4) Choose the correct answer for each statement. 39. The expression 6 # 32 + 2 # 8 simplifies to a. -52

47. The difference of twenty and twelve is equal to the product of two and four. 48. The quotient of nine and two is greater than negative five.

3 11

1 a meter

46. 5[312 + 52 - 5]

4 + 6#4

Translate each word statement to symbols.

32.

√ meter

44. 8 + 312 # 6 - 12

4 + 0 6 - 2 0 + 82

TRANSLATING

Find the area and the perimeter of each figure. 31.

3 3 42. a b 4

2 2 41. a b 7 43. 311 + 2 # 52 + 4

?

?

Simplify each expression.

?

80

37

54. The expression 360 - a - b - c represents the measure of the unknown angle of the given quadrilateral. Replace a with 93, b with 80, and c with 82 to find the measure of the unknown angle. 93

80

?

82

Decide whether the given number is a solution to the given equation. 55. Is x = 3 a solution of 7x - 3 = 18? 56. Is x = 1 a solution of 3x 2 + 4 = x - 1? (5) Find the additive inverse or the opposite. 2 58. 57. -9 3 59. 0 -2 0

60. - 0 -7 0

Find the following sums. 61. - 15 + 4

62. -6 + 1 -112

1 1 63. + a- b 16 4

64. -8 + 0 -3 0

65. -4.6 + 1 -9.32

66. - 2.8 + 6.7

b. 448 c. 70 d. 64

40. The expression 68 - 5 # 23 simplifies to a. -232

b. 28

c. 38

d. 504

73

Review of Real Numbers 97. 318 - 52 = 3 # 8 - 3 # 152

(6) Perform the indicated operations. 67. 6 - 20

68. -3.1 - 8.4

69. -6 - 1 - 112

98. 4 + 1 - 42 = 0

70. 4 - 15

99. 2 + 13 + 92 = 12 + 32 + 9

71. -21 - 16 + 3(8 - 2)

72.

11 - 1 -92 + 618 - 22 2 + 3#4

Evaluate each expression for x = 3, y = -6, and z = -9. Then choose the correct evaluation.

100. 2 # 8 = 8 # 2 101. 618 + 52 = 6 # 8 + 6 # 5 102. 13 # 82 # 4 = 3 # 18 # 42

2

73. 2x - y + z a. 15 74.

b. 3

c. 27

103. 4 #

d. -3

0 y - 4x 0 2x a. 3 b. 1 c. -1

104. 8 + 0 = 8 Use the distributive property to write each expression without parentheses.

d. -3

75. At the beginning of the week, the price of Density Oil and Gas stock from Exercises 17 and 18 is $50 per share. Find the price of a share of stock at the end of the week. 76. Find the price of a share of stock by the end of the day on Wednesday. (7) Find the multiplicative inverse or reciprocal. 77. -6

78.

85.

106. -31z + y2 107. -17 - x + 4z2 108.

1 16z - 102 2

109. -413x + 52 - 7

MIXED REVIEW Insert 6 , 7 , or = in the space between each pair of numbers.

80. 1 - 221 -142

79. 61 -82

83.

105. 51y - 22

110. -812y + 92 - 1

3 5

Simplify each expression.

81.

-18 -6

82.

42 -3

84.

2 + 1 -22

111. - 0 -11 0

31 -22 - 5

86.

0 -2

-2

1 2

114. 14 - 20

115. 41 -202

116.

4 5 a b 5 16

-20 4

118. -0.51 -0.32

1 -12 # 2

88. -52 - 12 - 202 , 1 - 32 # 3

119. 8 , 2 # 4

If x = - 5 and y = -2, evaluate each expression. 89. x - y

1 2

113. -7.2 + 1-8.12

117. -

2

112. -1

-14

-6 0

87. - 4 - 1 -3 + 52 ,

0 11.4 0

Perform the indicated operations. 2

41 -32 + 1 -82

2

1 = 1 4

4

2

90. x - y

3

TRANSLATING Translate each phrase to an expression. Use x to represent a number.

121.

120. 1 - 22 4

-3 - 21 -92 -15 - 31 -42

5 3 123. - , 8 4

122. 5 + 2[17 - 52 2 + 11 - 32]

124.

-15 + 1 -42 2 + 0 -9 0 10 - 2 # 5

91. The product of −7 and a number 92. The quotient of a number and −13 93. Subtract a number from −20 94. The sum of −1 and a number (8) Name the property illustrated. 95. -6 + 5 = 5 + 1 -62 96. 6 # 1 = 6

74

125. A trim carpenter needs a piece of quarter round molding 1 1 6 feet long for a bathroom. She finds a piece 7 feet long. 8 2 1 How long a piece does she need to cut from the 7 @foot@long 2 molding in order to use it in the bathroom?

Review of Real Numbers

Chapter Test Translate the statement into symbols. 1. The absolute value of negative seven is greater than five. 2. The sum of nine and five is greater than or equal to four. Simplify the expression.

30. -612 + 42 = -6 # 2 + 1 -62 # 4 31.

1 162 = 1 6

3. -13 + 8

4. -13 - 1 -22

32. Find the opposite of -9.

5. 12 , 4 # 3 - 6 # 2

6. 11321 -32

1 33. Find the reciprocal of - . 3

7. 1 -621 -22

8.

0 -16 0

The New Orleans Saints were 22 yards from the goal when the following series of gains and losses occurred.

-8

0 -6 0 + 2

-8 0 1 5 11. 2 6

10.

13. -0.6 + 1.875

14. 31 -42 2 - 80

15. 6[5 + 213 - 82 - 3]

-12 + 3 # 8 16. 4

9.

17.

5 - 6 3 1 12. 5 - 1 4 8

1 -221021 -32

Gains and Losses in Yards First Down Second Down

5 -10

Third Down

-2

Fourth Down

29

34. During which down did the greatest loss of yardage occur?

-6 Insert 6 , 7 , or = in the appropriate space to make each of the following statements true. 18. -3 20. 2

-7

0 -3 0

19. 4

-8

21. 0 -2 0

-1 - 1 -32

22. In the state of Massachusetts, there are 2221 licensed child care centers and 10,993 licensed home-based child care providers. Write an inequality statement comparing the numbers 2221 and 10,993. (Source: Children’s Foundation) 1 23. Given e -5, -1, 0, , 1, 7, 11.6, 17, 3p f , list the num4 bers in this set that also belong to the set of:

35. Was a touchdown scored? 36. The temperature at the Winter Olympics was a frigid 14 degrees below zero in the morning, but by noon it had risen 31 degrees. What was the temperature at noon?

? 0

31

14

a. Natural numbers b. Whole numbers c. Integers d. Rational numbers

37. A health insurance provider had net incomes of $356 million, $460 million, and - +166 million in 3 consecutive years. What was the health insurance provider’s total net income for these three years?

e. Irrational numbers f. Real numbers If x = 6, y = - 2, and z = -3, evaluate each expression. 24. x 2 + y 2

25. x + yz

26. 2 + 3x - y

27.

y + z - 1

38. A stockbroker decided to sell 280 shares of stock, which decreased in value by $1.50 per share yesterday. How much money did she lose?

x

Identify the property illustrated by each expression. 28. 8 + 19 + 32 = 18 + 92 + 3 29. 6 # 8 = 8 # 6

75

Review of Real Numbers

Answers to Selected Exercises Section 2 Practice Exercises 1. a. 6 b. 25

b. 7

c. 6

c. 25, - 15, - 99

7. a. 8

b. 9

c. 2.5

2. a. True b. False c. True d. True 3. a. 3 6 8 b. 15 Ú 9 c. 6 ⬆ 7 4. - 10 5. a. 25 7 7 3 3 d. 25, , - 15, - , - 3.7, 8.8, - 99 e. 15 f. 25, , - 15, - , 15, - 3.7, 8.8, - 99 6. a. 6 b. 7 c. = 3 4 3 4 5 d. e. 13 8. a. = b. 7 c. 6 d. 7 e. 6 11

Vocabulary & Readiness Check 2 1. whole

3. inequality

5. real

7. irrational

Exercise Set 1.2 1. 7 3. = 5. 6 7. 6 9. 32 6 212 11. 30 … 45 13. true 15. false 17. false 19. true 21. false 23. 8 6 12 25. 5 Ú 4 27. 15 ⬆ - 2 29. 14,494; - 282 31. - 28,000 33. 350; - 126 35. whole, integers, rational, real 37. integers, rational, real 39. natural, whole, integers, rational, real 41. rational, real 43. irrational, real 45. false 47. true 49. true 51. true 53. false 55. 7 57. 7 59. 6 61. 6 63. 7 65. = 67. 6 69. 6 71. 2009 73. 2009, 2010 75. 280 million < 281 million 77. 49 million > 16 million 79. 38 million pounds less, or - 38 million 81. - 0.04 7 - 26.7 83. sun 85. sun 87. 20 … 25 89. 6 7 0 91. - 12 6 - 10 93. answers may vary

Section 3 Practice Exercises 1. a. 2 # 2 # 3 # 3 6.

14 21

b. 2 # 2 # 2 # 5 # 5 46 77

7. a.

b.

1 14

7 8

2. a. 1 2

c.

8. 22

16 3

b. 11 15

c.

9. 40

7 25

3.

7 24

4. a.

27 16

b.

1 36

c.

5 2

5. a. 1

b.

6 5

c.

4 5

1 2

d.

5 6

Vocabulary & Readiness Check 3 1. fraction

3. product

5. factors, product

7. equivalent

Exercise Set 3 3 8 1 27. 2 16 53. 20 65 79. 21 1.

3.

5 7

5. 3 # 11 31. 15

33.

1 6

9. 2 # 2 # 5 35.

25 27

e.

8 125

11. 3 # 5 # 5

13. 3 # 3 # 5

15.

1 2

17.

2 3

19.

3 7

21.

3 5

30 61 21 49. 30 1 75. 7 12 23.

11 7 3 1 9 sq mi 39. sq ft 41. 43. 1 45. 47. 20 36 5 3 35 23 11 5 7 1 3 1 20 28 1 55. 57. 59. 61. 63. 65. 67. 69. 18 71. 2 73. 48 21 60 66 5 5 8 9 27 29 15 2 10 17 5 1 17 1 1 81. 83. 85. 87. 37 89. 91. 93. 5 95. 97. 55 ft 99. answers may vary 5 9 3 66 5 6 18 4 # # 2 2 3 1 9 11 21 12 2 3 101. 3 mi = = + = 103. 105. multiplexes 107. incorrect; 109. incorrect; 8 100 24 2#2#2#3 2 7 7 7 29.

6 7

7. 2 # 7 # 7

37.

25.

3 8

4 18 5 77. 7

51.

Section 4 Practice Exercises 1. a. 1

b. 25

19 8 c. 15 10 c. 2x + 9 ⬆ 25 b.

c.

1 100

d. 33

d. 9 7. No

8. a. 6x

2. a. 33 b. x - 8

d. 51112 Ú x

Graphing Calculator Explorations 4 1. 625

3. 59,049

5. 30

7. 9857

9. 2376

Vocabulary & Readiness Check 4 1. base; exponent

76

3. variable

5. equation

7. solving

b. 11

c.

5 32 or 3 9 9

c. x # 9 or 9x

d. 36

d. 2x + 3

e.

3 16

e. 7 + x

3.

31 11

4. 4

9. a. x + 7 = 13

5.

9 22

6. a. 9

b. x - 2 = 11

Review of Real Numbers Exercise Set 4 1. 243

3. 27

5. 1

27. 45

29. 0

31. 30

49. 1

51. 1

7. 5

9. 49

33. 2

53. 11

35.

55. 8

16 81

11. 7 18

27 10

37.

57. 45

1 125

13.

15. 1.44 7 5

39.

59. 27

17. 0.000064

41. 32

61. 132

63.

43.

37 18

23 27

19. 17

21. 20

45. a. 64

65. 16; 64; 144; 256

b. 43

23. 10 c. 19

67. yes

25. 21 d. 22

69. no

47. 9

71. no

73. yes

x 83. 3x + 22 85. 1 + 2 = 9 , 3 87. 3 ⬆ 4 , 2 89. 5 + x = 20 91. 7.6x = 17 4 93. 13 - 3x = 13 95. multiply 97. subtract 99. no; answers may vary 101. 14 in., 12 sq in. 103. 14 in., 9.01 sq in. 105. Rectangles with the same perimeter can have different areas. 107. 120 - 42 # 4 , 2 109. a. expression b. equation c. equation d. expression e. expression 111. answers may vary 113. answers may vary, for example, - 2152 - 1 115. 12,000 sq ft 117. 51 mph 75. no

77. x + 15

79. x - 5

81.

4

2.

Section 5 Practice Exercises 1.

2

ⴚ3

ⴚ2

3. a. - 13

b. - 32

ⴚ3

4.

8 ⴚ4 ⴚ3 ⴚ2 ⴚ1

0

1

2

3

4

5

ⴚ5 ⴚ4 ⴚ3 ⴚ2 ⴚ1

6

0

1

2

3

4

5 ⴚ5 ⴚ4 ⴚ3 ⴚ2 ⴚ1

5. a. - 3 d. 3

b. 1

c. - 0.2

10. a. - 15

6. a. - 1

3 5

b.

c. 5y

b. - 6

1 d. 70

c. 0.5

7. a. - 6

b. - 6

5 9. a. 9

8. 4°F

0

1

b. - 8

2

3

4

5

c. - 6.2

d. 8

Vocabulary & Readiness Check 5 1. opposites

3. n

Exercise Set 5 1. 9

3. - 14

27. - 2.1 51. - 18 75. 0

5. 1

29. - 8

7. - 12 31. 38

9. - 5

11. - 12 1 2 = 8 4 59. 146 ft

33. - 13.1

35.

13. - 4

15. 7 17. - 2 19. 0 21. - 19 23. 31 3 13 39. 41. - 8 43. - 59 45. - 9 16 10 61. - +6.9 million 63. - 16 65. - 6 67. 2 69. 0

53. 19 55. - 0.7 57. - 6 2 79. yes 81. no 83. July 85. October 3 101. answers may vary 103. answers may vary

77. -

99. false

25. - 47

37. -

87. 4.7°F

89. - 3

91. - 22

93. negative

47. 5

49. 11

71. - 6

73. - 2

95. positive

97. true

Section 6 Practice Exercises 1. a. - 13 b. 13

b. - 7

7. $357

c. 12

d. - 2

8. a. 28

2. a. 10.9

b. -

1 2

19 20

3. - 7

4. a. - 6

b. 6.1

15. 5

17. 37

19. - 6.4

21. - 71

c. -

5. a. - 20

b. 13

6. a. 2

b. 137

Vocabulary & Readiness Check 6 1. 7 - x

3. x - 7

5. 7 - x

7. - 10 - 1 - 142; d

Exercise Set 6 1. - 10

3. - 5

2 11

29. -

27.

51. 13

53. 6

5. 19 11 12

7.

1 6

31. 8.92 55. 9

57. - 9

9. 2

11. - 11

33. 13

35. - 5

59. - 7

75. 19,852 ft 77. 130° 79. 30° 81. no 95. answers may vary 97. 16 99. - 20

13. 11 37. - 1

39. - 23

41. - 26

43. - 24

45. 3

23. 0

25. 4.1

47. - 45

49. - 4

7 1 63. 21 65. 67. 100°F 69. 265°F 71. 35,653 ft 73. - 308 ft 5 4 83. no 85. yes 87. - 5 + x 89. - 20 - x 91. - 4.4; 2.6°; 12°; 23.5°; 15.3° 93. May 101. true; answers may vary 103. false; answers may vary 105. negative, - 30,387 61.

Integrated Review 1. negative 12. -

9 9 ; 11 11

2. negative 13. - 42

3. positive 14. 10

15. 2

4. 0

5. positive

16. - 18

17. - 7

6. 0

7. positive 18. - 39

8. positive

19. - 2

20. - 9

1 1 9. - ; 7 7 21. - 3.4

10.

12 12 ; 5 5

22. - 9.8

11. 3; 3 23. -

25 28

77

Review of Real Numbers 5 24 13 37. 20 24. -

25. - 4 38. -

26. - 24

29 40

39. 4

27. 6 40. 9

28. 20 41. - 1

29. 6

30. 61

42. - 3

31. - 6

43. 8

44. 10

32. - 16 45. 47

33. - 19

46.

34. - 13

35. - 4

36. - 1

2 3

Section 7 Practice Exercises 1. a. - 40 d. - 64

b. 12 3 8

5. a.

b. undefined

c. - 54 b.

2. a. - 30

1 15

c. -

7 2

d. -

84 9. a. 5

c. undefined

b. 24 1 5

c. 0

d. 26

6. a. - 8

b. 11

3. a. - 0.046

b. - 4

10. a. - 9

c. 5

7. a. 3 5 c. 3

b. 33

b. -

4 15

c. 14

b. - 16

4. a. 36

c. -

6 5

b. - 36

d. -

1 18

c. - 64

8. a. 0

11. - 52

Graphing Calculator Explorations 7 1. 38

3. - 441

5. 163.3

7. 54,499

9. 15,625

Vocabulary & Readiness Check 7 1. 0; 0

3. positive

5. negative

7. positive

Exercise Set 7 1. - 24

3. - 2

5. 50

7. - 12

9. 0

11. - 18

13.

3 10

15.

2 3

17. - 7

19. 0.14

8 1 29. - 121 31. 33. - 30 35. 23 37. - 7 39. true 41. false 43. 27. 27 4 3 1 1 1 11 53. 55. 57. 59. 61. - 6.3 63. - 9 65. 4 67. - 4 51. 9 2 14 3 0.2 20 18 9 75. 3 77. - 15 79. 81. 83. - 1 85. 87. - 4 89. 16 91. - 3 7 27 2 3 # 101. 103. - 21 105. 41 107. - 134 109. 3 111. 0 113. - 71 x or - 71x 115. 2 or x , 1 - 332 121. 3 # 1 - 42 = - 12; a loss of 12 yd 123. 51 - 202 = - 100; a depth of 100 ft 125. 133. answers may vary

135. 1, - 1

137. positive

139. not possible

141. negative

21. - 800

16

45. - 1

69. 0 16 7 - 16 - x 93. -

yes

23. - 28 47. 25

71. - 5

49. - 49

73. undefined

95. 2

97.

6 5

117. - 29 + x

127. no - 15 143. - 2 + ; -7 3

25. 25

129. yes

99. - 5 x 119. - 33 131. - 162F

145. 23 - 5 + 1 - 32 4 ; - 16

Section 8 Practice Exercises 1. a. 8 # x b. 17 + x 2. a. 2 + 19 + 72 b. 1 - 4 # 22 # 7 3. a. x + 14 b. - 30x 4. a. 5x - 5y b. - 24 - 12t c. 6x - 8y - 2z d. - 3 + y e. - x + 7 - 2s f. x + 11 5. a. 51w + 32 b. 91w + z2 6. a. commutative property of multiplication b. associative property of addition c. identity element for addition d. multiplicative inverse property e. commutative property of addition f. additive inverse property g. commutative and associative properties of multiplication

Vocabulary & Readiness Check 8 1. commutative property of addition

3. distributive property

5. associative property of addition

7. opposites or additive inverses

Exercise Set 8 1. 16 + x 3. y # 1 - 42 5. yx 7. 13 + 2x 9. x # 1yz2 11. 12 + a2 + b 13. 14a2 # b 15. a + 1b + c2 17. 17 + b 19. 24y 21. y 23. 26 + a 25. - 72x 27. s 29. 2 + x 31. 4x + 4y 33. 9x - 54 35. 6x + 10 37. 28x - 21 39. 18 + 3x 41. - 2y + 2z 43. - 21y - 35 45. 5x + 20m + 10 47. - 4 + 8m - 4n 49. - 5x - 2 51. - r + 3 + 7p 53. 3x + 4 55. - x + 3y 57. 6r + 8 59. - 36x - 70 61. - 16x - 25 63. 411 + y2 65. 111x + y2 67. - 115 + x2 69. 301a + b2 71. commutative property of multiplication 73. associative property of addition 75. distributive property 77. associative property of multiplication 79. identity element of addition 81. distributive property 83. commutative and associative 1 1 properties of multiplication 85. - 8; 87. - x; 89. 2x; - 2x 91. false 93. no 95. yes 97. yes 99. no 8 x 101. a. commutative property of addition b. commutative property of addition c. associative property of addition 103. answers may vary 105. answers may vary

Chapter Vocabulary Check 1. inequality symbols 2. equation 9. base; exponent 10. denominator

3. absolute value 4. variable 11. grouping symbols 12. set

5. opposites

6. numerator

7. solution

8. reciprocals

Chapter Review 1. 6

3. 7 5. 6 1 d. - 6 , 0 , 1 , 1 , 3 , 9.62 2

78

7. = e. p

9. 7

11. 4 Ú - 3 1 f. - 6, 0, 1, 1 , 3, p, 9.62 2

13. 0.03 6 0.3 17. Friday

15. a. 1 , 3 19.

2#2#3#3

b. 0 , 1 , 3 c. - 6 , 0 , 1 , 3 3 12 13 21. 23. 25. 9 25 10 8

27. 15

Review of Real Numbers

29.

7 12

31. A = 1

47. 20 - 12 = 2 # 4

1 5 sq m; P = 4 m 6 12 49. 18

51. 5

33. 14 53. 63°

1 lb 8 55. yes

35. 18

7 lb 16

57. 9

37. Baby E 59. - 2

39. c

61. - 11

41. 63. -

4 49

3 16

43. 37

45.

65. - 13.9

18 7

67. - 14

1 79. - 48 81. 3 83. undefined 85. undefined 87. - 12 89. 9 6 # 93. - 20 - x 95. commutative property of addition 97. distributive property 99. associative property of addition 91. - 7 x or - 7x 101. distributive property 103. multiplicative inverse property 105. 5y - 10 107. - 7 + x - 4z 109. - 12z - 27 111. 6 1 5 3 113. - 15.3 115. - 80 117. 119. 16 121. - 5 123. 125. 1 ft 4 6 8 69. 5

71. - 19

73. a

75. $51

77. -

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80

Equations, Inequalities, and Problem Solving

From Chapter 2 of Beginning and Intermediate Algebra, Fifth Edition. Elayn Martin-Gay. Copyright © 2013 by Pearson Education, Inc. All rights reserved.

81

Equations, Inequalities, and Problem Solving Top 10 Countries by Number of Internet-Crime Complaints

1 Simplifying Algebraic Expressions

2 The Addition and Multiplication Properties of Equality

2

3

3 Solving Linear

9

Equations Integrated Review– Solving Linear Equations

4 An Introduction to Problem Solving

5 Formulas and Problem Solving

7

1 8

5 10

Top Five States by Number of Internet-Crime Complaints 1. California

4

6

2. Florida 3. Texas 4. New York 5. New Jersey

6 Percent and Mixture Problem Solving

7 Further Problem Solving

Internet Crime

8 Solving Linear

The Internet Crime Complaint Center (IC3) is a joint operation between the FBI and the National White-Collar Crime Center. The IC3 receives and refers criminal complaints occurring on the Internet. Of course, nondelivery of merchandise or payment are the highest reported offenses. In Section 6, Exercises 15 and 16, we analyze a bar graph on the yearly number of complaints received by the IC3.

Inequalities Much of mathematics relates to deciding which statements are true and which are false. For example, the statement x + 7 = 15 is an equation stating that the sum x + 7 has the same value as 15. Is this statement true or false? It is false for some values of x and true for just one value of x, namely 8. Our purpose in this chapter is to learn ways of deciding which values make an equation or an inequality true.

Ages of Persons Filing Complaints Over 60 13.6%

Under 20 3.2%

20–29 18.8%

50–59 22.1% 40–49 22.1%

2

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30–39 20.2%

Equations, Inequalities, and Problem Solving

1

Simplifying Algebraic Expressions

OBJECTIVES 1 Identify Terms, Like Terms, and Unlike Terms.

2 Combine Like Terms.

As we explore in this section, an expression such as 3x + 2x is not as simple as possible because—even without replacing x by a value—we can perform the indicated addition. OBJECTIVE

1 Identifying Terms, Like Terms, and Unlike Terms Before we practice simplifying expressions, some new language of algebra is presented. A term is a number or the product of a number and variables raised to powers.

3 Use the Distributive Property to

Terms

Remove Parentheses.

4 Write Word Phrases as

-y,

2x3,

-5,

Algebraic Expressions.

3xz 2,

2 , y

0.8z

The numerical coefficient (sometimes also simply called the coefficient) of a term is the numerical factor. The numerical coefficient of 3x is 3. Recall that 3x means 3 # x. Term 3x y

3

5 0.7ab3c 5 z

Numerical Coefficient 3 y3 1 1 since means # y 3 5 5 5 0.7 1

-y

-1

-5

-5

Helpful Hint The term -y means -1y and thus has a numerical coefficient of -1. The term z means 1z and thus has a numerical coefficient of 1.

EXAMPLE 1 a. -3y

Identify the numerical coefficient in each term. x c. y d. -x e. b. 22z 4 7

Solution a. The numerical coefficient of -3y is -3. b. The numerical coefficient of 22z 4 is 22. c. The numerical coefficient of y is 1, since y is 1y. d. The numerical coefficient of -x is -1, since -x is -1x. x 1 x 1 e. The numerical coefficient of is , since means # x. 7 7 7 7 PRACTICE

1 a. t

Identify the numerical coefficient in each term. w d. 43x 4 b. -7x c. 5

e. -b

Terms with the same variables raised to exactly the same powers are called like terms. Terms that aren’t like terms are called unlike terms.

83

Equations, Inequalities, and Problem Solving

Like Terms

Unlike Terms

3x, 2x

5x, 5x

-6x 2y, 2x 2y, 4x 2y 2 3

2

Why? Same variable x but different powers x and x 2

7y, 3z, 8x 2

3 2

3

2ab c , ac b

Reason

6abc , 6ab

Why? Different variables 2

Why? Different variables and different powers

Helpful Hint In like terms, each variable and its exponent must match exactly, but these factors don’t need to be in the same order. 2x 2y and 3yx 2 are like terms.

EXAMPLE 2 a. 2x, 3x

2

Determine whether the terms are like or unlike. b. 4x2y, x2y, -2x2y

d. -x4, x4

c. -2yz, -3zy

Solution a. Unlike terms, since the exponents on x are not the same. b. Like terms, since each variable and its exponent match. c. Like terms, since zy = yz by the commutative property. d. Like terms. PRACTICE

2

Determine whether the terms are like or unlike. b. 5q, -3q 2 y5 d. y 5, 2

a. -4xy, 5yx c. 3ab2, -2ab2, 43ab2

OBJECTIVE

2 Combining Like Terms An algebraic expression containing the sum or difference of like terms can be simplified by applying the distributive property. For example, by the distributive property, we rewrite the sum of the like terms 3x + 2x as 3x + 2x = 13 + 22x = 5x Also, -y 2 + 5y 2 = -1y 2 + 5y 2 = 1 -1 + 52y 2 = 4y 2 Simplifying the sum or difference of like terms is called combining like terms.

EXAMPLE 3 a. 7x - 3x

Simplify each expression by combining like terms.

b. 10y 2 + y 2

c. 8x2 + 2x - 3x

Solution a. 7x - 3x = 17 - 32x = 4x b. 10y 2 + y 2 = 10y 2 + 1y 2 = 110 + 12y 2 = 11y 2 c. 8x2 + 2x - 3x = 8x2 + 12 - 32x = 8x2 - x d. 9n 2 - 5n 2 + n 2 = 19 - 5 + 12n 2 = 5n 2 PRACTICE

3

Simplify each expression by combining like terms.

a. -3y + 11y c. 5x - 3x2 + 8x2

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b. 4x2 + x2 d. 20y 2 + 2y 2 - y 2

d. 9n 2 - 5n 2 + n 2

Equations, Inequalities, and Problem Solving The previous example suggests the following: Combining Like Terms To combine like terms, add the numerical coefficients and multiply the result by the common variable factors.

EXAMPLE 4

Simplify each expression by combining like terms.

a. 2x + 3x + 5 + 2

b. -5a - 3 + a + 2 1 e. - b + b 2

d. 2.3x + 5x - 6

c. 4y - 3y 2

Solution Use the distributive property to combine like terms. a. 2x + 3x + 5 + 2 = 12 + 32x + 15 + 22 = 5x + 7 b. -5a - 3 + a + 2 = -5a + 1a + 1 -3 + 22 = 1 -5 + 12a + 1 -3 + 22 = -4a - 1 2 c. 4y - 3y These two terms cannot be combined because they are unlike terms. d. 2.3x + 5x - 6 = 12.3 + 52x - 6 = 7.3x - 6 1 1 1 1 e. - b + b = - b + 1b = a - + 1b b = b 2 2 2 2 PRACTICE

4

Use the distributive property to combine like terms.

a. 3y + 8y - 7 + 2

b. 6x - 3 - x - 3

d. 9y + 3.2y + 10 + 3

e. 5z - 3z 4

c.

3 t - t 4

OBJECTIVE

3 Using the Distributive Property Simplifying expressions makes frequent use of the distributive property to also remove parentheses. It may be helpful to study the examples below. ±(3a+2)=±1(3a+2)=±1(3a)+(±1)(2)=3a+2 means

–(3a+2)=–1(3a+2)=–1(3a)+(–1)(2)=–3a-2 means

EXAMPLE 5

Find each product by using the distributive property to remove

parentheses. a. 513x + 22

b. -21y + 0.3z - 12

c. -19x + y - 2z + 62

Solution a. 5(3x+2)=5 # 3x+5 # 2 Apply the distributive property. = 15x + 10 Multiply.

85

Equations, Inequalities, and Problem Solving Apply the distributive

b. –2(y+0.3z-1)=–2(y)+(–2)(0.3z)+(–2)(–1) property. = -2y - 0.6z + 2 Multiply.

c. -19x + y - 2z + 62 = -119x + y - 2z + 62 Distribute -1 over each term. = -119x2 - 11y2 - 11 -2z2 - 1162 = -9x - y + 2z - 6 PRACTICE

5

Find each product by using the distributive property to remove parentheses.

a. 312x - 72 c. -12x - y + z - 22

b. -51x - 0.5z - 52

Helpful Hint If a ; -< sign precedes parentheses, the sign of each term inside the parentheses is changed when the distributive property is applied to remove parentheses. Examples: -12x + 12 = -2x - 1

-1 -5x + y - z2 = 5x - y + z

-1x - 2y2 = -x + 2y

-1 -3x - 4y - 12 = 3x + 4y + 1

When simplifying an expression containing parentheses, we often use the distributive property in both directions—first to remove parentheses and then again to combine any like terms.

EXAMPLE 6 a. 312x - 52 + 1

Simplify each expression. b. -214x + 72 - 13x - 12

c. 9 + 314x - 102

Solution a. 312x - 52 + 1 = 6x - 15 + 1 Apply the distributive property. = 6x - 14 Combine like terms. b. -214x + 72 - 13x - 12 = -8x - 14 - 3x + 1 Apply the distributive property. Combine like terms. = -11x - 13 Helpful Hint Don’t forget to use the distributive property and multiply before adding or subtracting like terms.

c. 9+3(4x-10)=9+12x-30 = -21 + 12x or 12x - 21

Apply the distributive property. Combine like terms.

PRACTICE

6

Simplify each expression.

a. 419x + 12 + 6

EXAMPLE 7

b. -712x - 12 - 16 - 3x2

c. 8 - 516x + 52

Write the phrase below as an algebraic expression. Then simplify

if possible. ;Subtract 4x - 2 from 2x - 3.