Basic Algebra [Teacher's Edition] 0395278643, 9780395278642

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Teacher's Edition

THIS BOOK OrDER OF-

c4^

TUj^f^ -

A^fX. cu-~J

National "Bank of Detroit

II

Be

careful!

for

somebody if

Now why 1.

2.

It

you

2.

19

yfof.

/-^

1

?P

DOLLARS

^

your

'0

$J5^

ii:90Bi.o;se

check

No

-r-

JoinimA^

Pay TO THE

to

Write

—00, ^ when loo' 100

there are

no

cents.

ifi

might be possible else to cash the

lose

it.

not try writing

Write a check for $ Write a check for broke.

3.

Write a check for bicycle.

4.

Write a check for $ a car.

6.

Write a check for $ rock concert.

©

1974 United Feature Syndicate.

1

190

T9

Supplementary Materials This Teacher's Edition contains lesson-by-lesson commentary on the student text, with teaching suggestions, chalkboard examples, and ex-

Chapter tests and answers, an individualized assignment and a table of related references are also provided. Answers to classroom and written exercises appear on full-sized facsimiles of the tensions.

guide,

student pages. Progress Tests are a convenient

way

to

keep track of each

dent's performance.

Each

stutest Test 9

is

keyed

to the

student

text.

{Sections

appear in EdiAnnotated Teacher's the

Answers

to all tests

ions

AND NEGATIVE NUMBERS

POSITIVE

Write the answers

1

in

and

2)

the spaces provided

QUESTIONS

tion of the Progress Tests. 1

The

Solution

Key

provides

step-by-step solutions for every

Write a positive or negati\ 1.

a

liis^ i>r

S piiinls

2.

a

gam

of $12

written exercise and puzzle.

Solution

Key

r

Chapter Tests

Chapter If

AC

\.

=

1

6, find

+

X

ym/M/M/m/M/M/M/m/m/m/m/A

the value of the expression.

1

3x

2.

3.

X

3

-=-

-

5{x

4.

2)

Find the value of the expression. 5.

3

-

7

•

2

6.

(5

+

6)(7

-

7.

5)

-

(9

-

3)

-

3

2

Simplify. 8.

+

3x

7j»c

+

X

—

5a

9.

a

10. 2>x

?>a

-\-

Ay

-\-

-\-

-

X

7>y

Find the value of the expression. 11.

52

25

12.

13.

3^

power

2 to the fourth

14.

Simplify.

X X

15.

3-(5x)

16.

4-6a

4)

Find the value

if

-

26.

X X

lb)

24. 4(5

-

4m)

If not, write impossible.

possible.

8)

-

23. 6{2>a

1)

+ —

27.

X

— 28.

— X

Simplify.

+

+

4x

-

29.

3x

Tell

which of the numbers are solutions.

32.

3x

34.

X

Copyright

-

+ ©

8

5

7

=

1 1

> 20

5

30.

4, 6,

-

6b

+

6b

-

3a

or 8?

11, 12,

1977 by Houghton Mifflin

la

13

Company. Permission

to

reproduce

is

granted to users of

31.

+

33.

3

35.

3>'

2x

+

8

BASIC ALGEBRA.

\2a

+

6(5

-

=

9

1, 2,




or

'

=

X

+J

2.

4y

4.

;c

6.

2a

-2y = +

3x

-

3-??

+ +

ly 2^ 2c

=

2a

=

4x

-

3Z)

+ +

67 2c

4

•PROPERTIES OF ADDITION

10-11)

(pp.

Teaching These Pages To

reinforce the

meaning of commutative, show the students some opera-

tions that are not

commutative.

-

4

-4

2 7^ 2

4

-

2 7^ 2

-4

meaning of associative could be reinforced by showan operation that is not associative. Use numerical examples to point out that the commutative and associative prop3 8 10 erties are helpful in developing mental In a similar way, the

ing that division

is

i

1

arithmetic.

10

7

'

2

'

20

Chalkboard Examples

Some

students can check their work by substituting a value and finding the value of the expression.

for the variable

Simplify. 1.

3a

+2 +4a +6 =la +

3.

5x

+

2x

—

4v

+

X

=

^x

^

—

Ay

+6 -lb -2=

2,

4b

4.

12m

-I-

\0n

+

lOw

lb

-

+

2n

A

= 22m +

Sn

Objectives for Pages 12-21 1.

To

2.

To use exponents

3.

To use the commutative and

find the value of expressions with exponents. to simplify expressions.

associative properties to simplify expres-

sions. 4.

To use the

5.

To use

5

distributive property to simplify expressions.

the properties of

and

1

to simplify expressions.

-EXPONENTS AND FACTORS

(pp.

12-13)

Teaching These Pages is another shorthand used in matheExponents are often used to write large numbers in a concise form. For example, one bilhon is written as 10^, one trillion as 10^^^ and the distance from the earth to the sun as 150 x 10*^ km.

Point out that the use of exponents

matics.

T37

Chalkboard Examples Find the value of the expression.

=

23

1.

8

32

2.

=

9

103

3.

^

1000

4.

3^

5.

1

51

=

5

Simplify. 6.

y

'

y y = y^

1.

a

•

a

•

a

•

=

b

a^b

8.

S

X X X 'y 'y '

'

•

=

Sx^y^

=

\2k^

Suggested Extensions

Which

1.

2^; 32; 2^

2^ or 52? 32 or 23? 2^ or 42?

the larger value:

is

The students should continue the above theme by picking other numbers and recording their results.

How many squares can you find How many squares in a 3-unit

2.

in this figure?

=

42

pairs of

5

by 3-unit square? The stu-

number of squares, etc. The

dents should find a pattern in order to predict the

by

squares in 4-

6

•

by 5-unit by 2-unit square is

4-unit squares, 5-

number of squares

in a 2-

by 3-unit square

32 -f 22

is

-|-

F,

22

+

P, in a 3-

etc.

PROPERTIES OF MULTIPLICATION

(pp.

14-15)

Teaching These Pages This section extends the commutative and associative properties to multiplication.

Again, these properties can be applied to mental arithmetic. To class, illustrate how quickly one can multiply by 50.

motivate the 50

.

= (y

24

•

100)

meaning of

Point out the

•

24

=y

•

(

100

•

=y

24)

•

2400

=

1200

evaluate.

Chalkboard Examples 1.

Evaluate 2 2

•

16

•

5

16

•

= = = =

2 (2

•

5

•

•

Show how you

5.

•

16

5)

•

16

use the properties.

commutative property associative property

10-16 160

Simplify.

=

2.

3-5v

5.

(4s)i5s)

T38

I5y

=

20.v2

3.

2y3y =

6.

5

•

a

•

fl

•

6y^ 6

=

30fl2

4.

i3k)(4k)

7.

3'k-2'k =6k^

Suggested Extension Explain each step of the method in Teaching These Pages to muhiply by

Have

50.

the students use that

method

compute these products mentally.

to

1.

50

•

12

=

600

2.

24

•

50

=

1200

3.

52

•

50

=

2600

4.

50

•

76

=

3800

5.

38

•

50

=

1900

6.

50

•

98

=

4900

54)

=

Provide a sheet of questions based on the above

Have

skill.

the students try the

questions over a period of time to see the improvement in their mental tation

7

•

compu-

skills.

THE DISTRIBUTIVE PROPERTY

(pp. 16-17)

Teaching These Pages To introduce

this topic, illustrate

how

quickly certain problems can be solved

mentally by using the distributive property. 3

•

99

=

3(100

-

1)

=

-

300

^

3

297

class may be motivated by randomly picking products that involve 99 and seeing how quickly the product can be found.

The

Chalkboard Examples Evaluate each expression in two different ways. 1.

+

5(2

8)

=

50

2.

+

4(97

3)

=

400

+

3.

12(46

6.

l{a

-

2b)

9.

5(jc

-

\\y)

1200

State the expression without parentheses. 4. 1.

+ 2) = 3x + 6 8(3x - 2) = 24jc -

3(x

Use the 10.

16

distributive property.

3(2x

+

3)

-

6

=

6jc

+

-

5.

4(y

8.

6{2a

6)

+

4b)

Then combine

3

11. 4{a

+

=

b)

-

4y

=

24

+

12a

24b

=

-

la

=

5x

\4b

-

55/

like terms.

-

4a

=

4b

12.

4{m

+

n)

—

2n

3.

40

6.

99-99 =9801

= 4m +

2n

Suggested Extension Explain each step of the method in Teaching These Pages to multiply by 99.

Have

the students use that

method

to

compute these products mentally.

1.

99-

15

=

1485

2.

22-99 =2178

4.

99

65

=

6435

5.

99-75

•

=

Provide a sheet of questions based on the above try the questions over a period of time to see the

mental computation

7425

-

99

=

3960

Have the students improvement in their

skill.

skills.

T39

8

PROPERTIES OF

•

18-19)

(pp.

Teaching These Pages

A

"mind-reading trick" can be used in motivating the the students perform the following steps.

class to study zero.

Have

Results

Steps

x 2x

2.

Choose a whole number. Double the number.

3.

Add

4.

Divide by

5.

Subtract your original number.

6.

Subtract

1.

Ask the

2x

6.

x

2.

+6 +

3

3

3.

obtained

class if they all

made mechanical

an answer.

for

any students have

If

errors, retrace the steps in order to locate the errors as

well as review the

work

for the rest of the class.

Chalkboard Examples Find the value

if

possible.

6+0=6

1.

2.

(NP

indicates not possible.)

0-9=0

A^O

3.

4.

-^^^ =

8.

-?-

12

=

+

45

5.

45

6.

02

=

II 4?-

7.

NP

"

"

'

Let 9.

jc

=

4.

X

+

4

NP

Find the value of the expression.

=

8

10.

6(je

-

4)

=

^

'^^

11.

=

12.

5(3jc

-

1

1)

=

5

Simplify. 9

13.

+

16. {Ix

2fl

+

X

-

2fl

-

+

3jc)

9

=

-

18

NP

14.

8

+

17. 4fl

2x

+

9

-

2jc

-

8

=

-

4fl

+

8

=

17

15.

3(m

18.

5(3

Suggested Extension 1.

Can you

explain

always result

T40

the steps of the "mind-reading trick" given above

who can answer Question own mind-reading tricks.

Students their

why

in zero? 1

can be encouraged

to

make up

-f

+

4)

2x)

-

12 \5

= 3m =

\0x

9

PROPERTIES OF

•

1 (pp.

20-21)

Teaching These Pages

A

"mind-reading trick" can be used to introduce

this section.

Have

the

students perform the following steps. Steps 1.

Results

Choose a whole number. the next whole number

x

3.

Add Add

4.

Divide by

5.

Subtract your original number.

5

6.

Subtract

1

2.

to

2x

+ +

10

x

-\-

5

2x

it.

9. 2.

4.

All students should obtain the answer

If

1.

1

any students have made me-

chanical errors, retrace the steps.

Objectives for Pages 22-25 1.

To check whether

a given

number

is

a solution of an equation.

2.

To check whether

a given

number

is

a solution of an inequality.

10

•

INTRODUCTION TO EQUATIONS

22-23)

(pp.

Teaching These Pages To introduce this lesson, use some recent baseball standings. Ask the students which of the following statements are true, and which are false. 1.

2. 3.

Oakland

Kansas City is at the top of the standings. Texas is ahead of Minnesota in the standings. Chicago is in last place.

Chalkboard Examples

+

=

1.

Is

X

2.

Is

4 a solution of 6

3.

Is

6 a solution of 12

Which of 4.

7

-h

6

5

9 true

if

the numbers

=

9

jc

=

5?

+7 =

=

v

3 or 9?

3

6?

is

x

=

4?

no; yes

no

9?

+

shown

if

yes

a solution? 5.

a

-

.

.

.

Suggested Extension

—

Replace each * with +, need to use parentheses

=

27

1.

4

3.

16*4*2 =

*

5 * 3

X, or

,

=

(4

+

16

-4-2=2

2

to

-r-

make

a true sentence.

5)

X

3

27

3*9*3=

2.

12

4.

INTRODUCTION TO INEQUALITIES

11 •

You may

also.

=

*22

* 3

3x9 + 3=

30

12

^

3

-

22

30

=

24-25)

(pp.

Teaching These Pages Review the terms introduced in the last section and relate them to the de velopment of inequalities in this section: number sentence, true number

number

sentence, false

sentence, solution, root.

Point out that the slash

mean

Compare

no.

meanings of

=

this

and

often used to

is

example

to the

P

7^.

No

Parking

parking

Chalkboard Examples

/

1.

Is

X

2.

Is

6

3.

Is 8 7^

Tell

/:

true if

x

true if y

is

true

is

k

if

is

replaced by 3? by 6?

replaced by 4? by 9? replaced by 3?

yes;

no

yes;

no

by 8?

no

yes;

which of the numbers shown are solutions.

4.

6>k

3,4,5,6

3,4,5

5.y2

0, 1,2, 3

3

1.9

8.

jc

+

3


2f -

3

6,7,8,9

0, 10,

20

7,8,10 6,7

a true sentence.

5x2 > ? 22


Suggested Extension

Use 1.

all

the symbols to

2,3,4,

T42

14,

X,

+,
,0

vary.) 3.

=, ^, -, 24,

8, 3,

Chapter

2.

Solving Equations

In the previous chapter students learned that equations are mathematical In this chapter the students will use equations as problem-solv-

sentences.

In order to

ing tools.

word problems,

overcome the

practice

is

difficulty that students

have

in solving

regularly provided in translating words into

symbols.

Objectives for Pages 34-41 1.

To

solve equations by addition.

2.

To

solve equations by subtraction.

3.

To

solve equations

4.

To

solve equations by division.

1 •

by multiplication.

SOLVING EQUATIONS BY ADDITION

(pp.

34-35)

Teaching These Pages

Ask

the students to find a solution to each of these equations.

X

The

+

last

5

=

25

2

+m =

10

+

3Z)

=

5

2Z>

+

10

equation illustrates that the "guess and evaluate" method of find-

ing solutions will sometimes take quite a while.

Point out the need to find

organized ways of solving equations.

Use the balanced

show how an equation might be formed.

scale to

3

=A =4 -

3

=

X

X X

-

Point out that to solve the equation x

3

1

—

3

=

1

we

reverse or

"undo" the

steps shown.

Encourage students to check and 4.

their solutions mentally in this section

and

in Sections 2, 3,

Chalkboard Examples Solve. will

Encourage students

be a useful

1.

X

-

3

=

8

3.

a

—

4

=

11

to write a final statement of the solution.

The The

solution

This

problem solving.

skill in later

is

solution

is

11.

15.

=7 -

2.

8

4.

12

The

4

=y —

1

solution

The

is

solution

is

12. 19.

T43

2 •

SOLVING EQUATIONS BY SUBTRACTION

(pp.

36-37)

Teaching These Pages

The balanced

scale technique of building

and unbuilding equations can be

applied to this section also.

Chalkboard Examples Solve. I,

a

+

9

=

\5

The

solution

is 6.

3.

y +

1

=

20

The

solution

is

2.

13.

4.

19

=

y —

1

jc

+

=

10

20

The

The

solution

solution

Suggested Extension

A

path

tion

is

hidden from Start

and study the solutions

vertically.

X

Start

-

4

=

to Finish.

To

find the path, solve each equa-

to find a pattern.

Move

only horizontally or

is

is 9.

27.

Think of an input-output machine. 5.

IN]

Tell 6.

what

is

missing in the table. 7.

8.

Write each sentence as an equation. Then solve.

more than twice

a

8.

8

9.

6 less than 3 times a

10.

When

9

is

added

number

is

number

to twice a

2x

18.

3x

is 9.

+ —

number, the

8

6

= =

x

18; 9;

result

x

is

=

21.

=

5

5

2x

+ 9=21;x=6

Suggested Extension Choose a number between 10 and

sum of

For example, choose the number four.

25.

the solutions of the equations 25.

Create four equations so that the

number you chose. The equations may vary, but here is

equal to the

are

Chalkboard Examples 1.

Copy and complete. Steps

Write an equation and solve.

Twice a number plus the number

3.

What 2n

+

is

4/7

n

= —

is

36.

number?

the

36 6

Point out that the result should be checked in the original problem, not in the equation.

Suggested Extension

Use

the terms and symbols to write an equation.

may

vary.)

1.

6«, 2, 30, 2n,

2.

2x,

3.

ly, 14, 6, 2y,

3, X, 12,

-, +,

=

=

+, -,

-, -,

-

6n

2

+

2n

=

30; n

Solve.

Check. (Answers

=4

+ X - 3 = 12; x = 5 Iv - 6 - 2y = \4; y = 4

2x

=

Objectives for Pages 50-57 1.

To

2.

To use equations

7

•

translate sentences into algebraic equations.

word problems.

to solve

WRITING ALGEBRAIC EQUATIONS

(pp.

50-53)

Teaching These Pages Use a newspaper clipping

NATIONAL LEAGUE to introduce a real-life

problem. Omit some of the

entries.

Have

East

the stu-

Won

dents write variable expressions based on the standings.

Philadelphia has

The number

won

10

lost is n, the

more games than they have

number won

is

«

-|-

10.

lost.

82

?

Philadelph

? ?

68 70

New York

74

72

Chicago

70

77

68

81

St.

Louis

[Montreal

Chalkboard Examples 1.

2.

Chicago lost 5 more games than they won. Let X = number of games won. Then ? = number of games lost, x +

George

is

y

years old today.

His age in 5 years will be His age

T48

last

year was

?

V

? .

>'

—

1

+

5

5

Lost

Pittsburg

.

Georgina

3.

Let g

=

Then

is

Sam.

3 years older than

Georgina's age.

=

?

Sam's age.

g

—

3

In 3 years, Georgina's age will be In 3 years, Sam's age will be

+

"L

?

3

^

.

Suggested Extension Part of the standings shown have missing entries. Use the information that follows to complete the standings. 1.

Chicago won 6 more games than they

2.

Vancouver

3.

Chicago won

4.

Minnesota

more games than

lost 3

3

more games than times as

lost 5

lost.

St.

NHL

Louis won.

many games

Chicago

Kansas City

as

Vancouver tied. .

many games

5.

Kansas City

Minnesota

lost.

6.

Minnesota won 6 more games than Vancouver

tied.

8

•

lost as

as

Standings

Louis won.

St.

APPLIED PROBLEMS AND PUZZLES

St.

Louis

Minnesota Kansas City

(pp.

54-57)

Teaching These Pages Refer the students to the Guide for Problem Solving on student's page 54 as the following

problem

is

solved.

Chalkboard Examples

The school play ran for two nights. were 79 more people on the second

A

total

night.

of 535 people attended. There

How many

people attended

each night? 1.

2.

Read

the problem.

Let n

= number of people on the + 19 = number of people

Then n 3.

n

+

n

2n

+ 19 = +19 =

535

=

456

2« n 4.

Answer:

Check:

night.

on the second

night.

535

=228

228 people attended the (228

5.

first

228

+

+

first

night.

79) or 307 people attended the second night.

307

=

535

/ T49

Suggested Extension

On

day you are given an amount of money. Each day after that is doubled. On the sixth day you have 252 cents. much money were you given on the first day? 4 cents

the

first

your amount of money

How

Objectives for Pages 58-63 1.

To

2.

To use equations

9

solve equations having the variable

VARIABLE

•

on both

sides.

word problems.

to solve

ON BOTH

SIDES OF THE EQUATION

(pp.

58-59)

Teaching These Pages Introduce this section by posing a puzzle problem that results in an equa-

on both

tion with the variable

sides.

number decreased by number increased by 7.

Six times a

times the

6n

—

=

3

4n

+

3

is

equal to 4

1

Point out that in this section the students will learn to solve equations like this.

Chalkboard Examples Solve and check. 1.

4x 2jc

X 5.

6.

= = =

8

+

2x

25

4

Six times a

number

is

- 3m = 2m 25 = 5m 5 = m more than

15

3.

1

+

x 5 \

3 times the

= = =

6x 5x X

number.

-

4

4.

4a

+

6

2a a

6^

—

15

=

3n; n

=

= = =

2a 6 3

5

Solve the problem posed in the introduction to this lesson. Check the answer in the original problem,

10

2.

8

•

dn

—

3

—

4n

-\-

1: n

=

5

EQUATIONS WITH PARENTHESES

(pp.

60-61)

Teaching These Pages Review the various methods acquired thus far for solving equations by using examples. Encourage students to discuss how they would solve each one. .X

-f 5

=

25

+

\2

Introduce an equation with parentheses and ask the students to discuss their

first

step in solving the equation.

Chalkboard Examples Use 1.

the distributive property to simplify.

4(x

+

5)

=

+

4x

20

2.

3(x

-

5.

3(m

+

6)

=

3x

-

3.

li

-

4(2jc

1)

=

8jc

-

4

Solve and check. 4.

2(x

2x

-

5)

10

Ix

X

11 •

= = = =

8

3w +

8

18

= 2m + = 2m + m —6 1)

9

3

9

6. 4(_y ^^

+ 2) + 9 = + 8+9 = 21

=

4(27

87

"

1)

-4

ly

3=y

9

PUZZLES

(pp.

62-63)

Teaching These Pages Review the Guide

for

Problem Solving on student's page

54.

Apply those

steps to solve the following puzzles.

Chalkboard Examples 1.

Jean has $16 more than Jose.

Four times

Jose's

amount

How much money Let n 2{n

In

—

Jose's

+ 16) = + 32 ^

32= 16 =

is

the

same

as 2 times Jean's

amount.

does each have?

amount of money; n

+

16

=

Jean's

amount

An An In A2

Jose has $16; Jean has $32. 2.

One number is 6 more than another. Three times the larger number is 5 more than 4 times the smaller. What are the numbers? Let n 'in

3n

-

= 5

5

\9

The

the larger

= = =

A{n

An

number; n

—

6

=

the smaller.

- 6) - 2A

n

larger

number

is

19; the

smaller

number

is

13.

T51

Chapter

and Negative Numbers

Positive

3.

The main objective of this chapter is to teach the four operations with positive and negative numbers. The number line is used to develop the skills, as well as to help the students compare numbers. These skills are then applied to simplifying variable expressions and solving equations.

Objectives for Pages 72-75 1.

To compare

2.

To graph the solutions of an

1 •

integers.

POSITIVE

inequality.

AND NEGATIVE NUMBERS

(pp.

72-73)

Teaching These Pages

To illustrate the occurrence of positive and negative numbers in evervday life, use the weather report in the newspaper, or a portion of the stock market ings in the newspaper.

show

Discuss the need to use

a gain of 5 dollars

—4

and

to

show

CanCab

list-

+5

to

a loss of

four dollars.

Have

the students

name

other ways in which positive and negative

num-

bers might be used.

Chalkboard Examples Write a positive or negative number. 1.

a debt of S40

3.

30

m

below sea

Compare 5.

-3*^5

-40 level

—30

the numbers. Write




2?

< -2 >

2.

a deposit of SIO

4.

a profit of SIO

or

7.

0? -7

>

+10 +10

2

•

INEQUALITIES AND GRAPHS

(pp.

74-75)

Teaching These Pages

Remind tences.

the students that equations

Compare

Sentence jc

and inequalities are mathematical sen-

the following.

-

2

Graph of the Solutionis)

Solutionis)

=5

1 3

jc

'

7.

-2)= -iy y =

8

-

3(5

2)

=

-(5

9=9

5

DIVISION

•

Check:

14)

(pp.

-

14)

/

88-89)

Teaching These Pages Review the relationship between addition and 3

X

Use whole numbers

+2 =: +y =

subtraction.

5

5-2 =

3

5

5

—y =

X

to review the relationship

between multiplication and

division.

3x2=6 Have

6-2=3 show they understand how Then develop the rules for dividing

the students state further examples to

multiplication and division are related. positive

and negative numbers.

Point out the similarities of these summaries. Division

Multiplication

X positive = positive x positive = negative positive X negative = negative negative X negative = positive

= positive = negative positive -^ negative = negative negative -^ negative = positive

positive

positive H- positive

negative

negative

-=-

positive

Chalkboard Examples Divide. 1.

.

9

-(-9) = -1

-36 =

-4

T58

9

-9

-9 =

-

-18 ^ _9

3.

= -5

Suggested Extension Find the value of each expression I.

a^

5,

a

9

=4 +

b

= -5

= — 2,

a

b

=

—3, c

=6

2.

ab

6.

abc

SOLVING EQUATIONS

•

if

=

18

=

3.

a^b

7.

ab

d = —I.

3 and

= -12 +

cd

=

3

4.

-a 2 _ -4

8.

^5

= -

90-91)

(pp.

Teaching These Pages These four examples review that to solve an equation, the same number added (subtracted, multiplied, divided) to both sides of the equation. 3

=

X

+

6

= -3

Review the vocabulary related amples are completed.

is

-3v = -24

to solving equations as the

Chalkboard Ex-

Chalkboard Examples Solve and check. 1.

8;c

-

3jc

X 10

= -5 = -\

2.

3^

+

=

15

ABSOLUTE VALUE (OPTIONAL)

•

2y

3.

3x

-

{5x

-1) = -2

-4=x

j=-15 (pp.

92-93)

Teaching These Pages Introduce the symbol for absolute value. Point out that

and

\x\

= —X

if

x

|x|

=

x

if

x

>

< 0.

Chalkboard Examples Find the value of the expression. 1.

|-4|

=4

2.

-|7|

5.

|x|

= -1

-

3.

|-9|

6.

W -3=

|3|

=6

Solve. 4.

|a|

a

+2 =4 = 6 or —6

X

- 4 = 10 = 14 or —14

y =

\

or

-2 —\

T59

Chapter Skills

4.

needed

chapter.

P =

2/

-I-

will

^ =

2h' (p. 102)

(p.

/vv

D =rt

108)

students' earlier

formulas.

problem solving are developed work with the following formulas.

to apply formulas to

The students

V = Bh The

Formulas

(p. (p.

A =]-bh{^.

104)

C ^pn

112)

work with solving equations

is

(p.

related to

in this

XOA)

112)

work with

Also, throughout the chapter the students obtain practice in writ-

ing formulas.

Objectives for Pages 102-111 1.

To use

the formula

2.

To use

the formulas

f*

=

A =

2/

+ 2^

Iw and

to find perimeters.

A =

— bh

to find areas.

2 3.

1 •

To use

the formula

V = Bh

to find

PERIMETER FORMULAS

volumes.

(pp.

102-103)

Teaching These Pages Reinforce the meaning of perimeter as "distance around" by having the students find the perimeters of these figures.

6

A

cm 16

4

cm

4

cm

m/

4x

\l6m

/

\

\

/ 6

cm

2x 2x

16m

2x

Write a formula for the perimeter of each figure. ^-^

1.

3x 3x

2x 6x

P =

\lx

Suggested Extension Chain-link fencing

sells for

cost of fencing the field.

$2.10 per meter. Find the

$777

Find the area of each shaded region. 1.

.

2.

3x

3x

A =9jc2

3.

Express the volumes in Exercises

4.

A

box measures

5.

A

cube has sides that measure

12

cm by

and 2

1

cm by

15

3

24 L; 21 L

as liters.

20 cm. Find the volume.

cm. Find the volume.

3600 cm^

27 cm^

Suggested Extension

How much

NATURE

cereal

FLAKES

1.

paper is needed to cover the box shown? 1724 cm^

Find the volume of the box

2.

meters.

When

the

cereal

is

3.

in cubic centi-

25 cm

4050 cm^

8

box

is

opened, the top of the

cm from

How many

the top.

cubic centimeters of cereal are in the box?

2754 cm^

9 18

cm

cm

Objectives for Pages 112-121 1.

To

write formulas.

2.

To

solve motion problems.

3.

To

solve cost problems.

4

•

WRITING FORMULAS

(pp.

112-113)

Teaching These Pages

SOCCER STANDINGS

Sports standings from the newspaper can be

used to introduce

When

^Northampton ....

this topic.

a soccer team wins a game, they are

given 2 points. After winning n games, the

number of winning formula

w =

w

points

is

given by the

!

Won

Lost

15

4

Lincoln

Reading

Tranmere

Bournemouth

In.

Develop the formulas for distance and cost. These formulas will be dealt with in

detail in Sections 5

and

6.

Chalkboard Examples Complete. 1.

You walk

5

km/h

Your distance 2.

You rode your Your distance

3.

You buy 5 The cost C

for

D =

t

hours.

?

-

bicycle at

D =

?

.

5/

y km/h

for n hours.

vn

tennis balls at s cents each.

= _2

5s

T63

4.

In hockey, 2 points are given for a win.

Winning

6.

tie /

A

games

4

points.

1

point.

/

hockey team won

The

_A

scores

game, a iiockey team receives points. games scores _1

For a Tying

5.

2

total points

vv

games and

T = _A

tied

2w +

t

games.

/

Suggested Extension Clip the hockey standings from the newspaper.

Use

the formula developed in Exercises 4-6 above to

check the 5

total points for the various teams.

For example, games.

St.

Louis has

T = 2w +

5

•

t

=

won

+

2(10)

MOTION FORMULAS

10

5

(pp.

games and

=

tied

25

114-117)

Teaching These Pages Review the problem-solving methods of Chapter throughout

2.

They

this section.

Chalkboard Examples Jean and John

live 18

km

apart.

same time and walk toward each Jean walks at 5 km/h and John walks at 4 km/h. How long will it take them to meet?

They

leave at the

Let

= number

/

of hours until they meet.

other.

are applied

6

•COST AND MONEY PROBLEMS

118-121)

(pp.

Teaching These Pages Use several examples

to review the

formula for

cost.

Point out the mean-

ing o{ profit. 1.

A

pencil costs \5l.

2.

A

tennis ball costs $1.25.

3.

You bought

Find the cost of 10

pencils.

Find the cost of

10 pens at 35c each,

3 tennis balls.

and sold them

for

50*1;

each.

Find

the profit.

Solve the Chalkboard Example and ask the class to indicate each prob-

lem-solving step.

Chalkboard Examples George and Robin bought greeting cards at 702)

= -6a

^

\5ab^ a^b^

= -la

-aW

9ab

Sa^x^b^

9aY = 3a

5a

3y^a

-3a4/>3

—64a^b^x^

=

-\6a%^ = -4a Aab^ 36a6

= —a

=

6a

6a^

Finish

10

'EXPONENT RULES (OPTIONAL)

(pp.

156-157)

Teaching These Pages Encourage students

Discuss the exponent rules on page 156. rules,

but point out that they

when

necessary.

may

to use these

use the methods of the previous section

Chalkboard Examples Divide. 2.

x^ 11 •

— = n^ n

-3b

9a^b'^ 3.

POLYNOMIALS DIVIDED BY MONOMIALS

(pp.

158-159)

Teaching These Pages Use examples from arithmetic to illustrate ^-^ c example below illustrates a common error.

6+12 1 3

§_ 3

18 3

,

}2^ 3

-9a^b

-3

3a^b^

ab

4.

3d^b

—=

.

1

c

6+12

The second

c

+

12

Chalkboard Examples Divide.

20a

.

-

12

=

-

5a

3

=

2.

+

3a

ab .

6m'^n

4.

+ 4mn — — Imn

\2mn^

= —5m —

^

-j

1

on

—-;

3.

3ab r—.

5.

+

'iab'^

=

c,

do

3ab

Suggested Extension Find the errors in the examples below. 1.

^^

^^

=2a +6b

5—

2.

12

j"

— 4m«"^

=

Did not divide both terms by

3m^«2

Overlooked

m

2.

term in denominator.

•DIVIDING POLYNOMIALS (OPTIONAL)

(pp.

160-161)

Teaching These Pages

To introduce

this topic,

have the

(jc

Then have

+

4)(x

+9x + X

+

20

=

x

Chalkboard

1.

+

x2

+

Expl

x^ 5

to develop a

the students suggest a

rect.

Divide.

=

4

Use the example on page 160

Have

5)

9x

+

20

the class explain these equations.

x^

als.

class find the product.

+

method

to

+

9x

X

+

+

20

=x +

4

5

method

=

3ab

—

Mab"^

c

/:

,

-\-

6b

for dividing

polynomi-

check that the division

is

cor-

—

^

I

2a

—

b

Chapter

Factoring

6.

This chapter extends work with polynomials to factoring. learn three basic factoring techniques: finding a factoring trinomials with

term having coefficient of

first

The

students will

common monomial 1,

factor,

and factoring

the diff'erence of squares. In later chapters the skills developed in this chapter will be applied to

equation solving as well as other algebraic problems.

Objectives for Pages 170-175 1.

To

factor positive integers.

2.

To

find a

1 •

common monomial

factor in a polynomial.

FACTORING IN ARITHMETIC

170-171)

(pp.

Teaching These Pages Discuss the importance of factoring in adding fractions. Factoring can

common

help us find a

denominator.

9^

5

3-3-3

5-5

^25

27

60

60

2-2-3-52-2-3-5

20

12

Introduce the meanings oi prime number, prime factor,

common factor

common factor and

above example. Point out that factors can be found by division. If the remainder

greatest

the divisor

as they relate to the

is

zero,

a factor.

is

Chalkboard Examples Find the prime factors.

=

1.

24

5.

Find a

2

•

2

2

•

•

common

Find the greatest 6.

3

12,30

2.

=

42

factor of 12

common

6

2

3

•

and

7

•

=

3.

99

8.

17,27

3

•

3

•

1 1

=

4.

53

9.

18,54

53

2 or A

20.

factor.

7.

100,125

25

1

18

Suggested Extension

A

number has

example, 135

Which numbers 1.

45

yes

9 as a factor

is

divisible

if

the

sum

by 9 since

1

of

+

3

its

+

digits

5

is

is

divisible

divisible

by

by

9.

For

9.

are divisible by 9? 2.

147

no

3.

909

yes

4.

242

no

T75

A number

has 4 as a factor

ample, 2348

Which numbers 1323

5.

2 •

two

if its last

by 4 since 48

divisible

is

is

digits are divisible

by

divisible

by

For ex-

4.

4.

are divisible by 4?

no

4156

6.

324

7.

yes

COMMON MONOMIAL FACTORS

yes

(pp.

3929

8.

no

162-173)

Teaching These Pages Review the

distributive property.

+

2x(3x

Use the terms

and Sxy

6x'^

=

4y)

6x2

_,_

to introduce the

g^^^

meaning of

greatest

common

factor.

6x^

Sxy

= =

X

2

•

2

'I'l' X y

3

•

jc

•

The

'

greatest

common

factor

2x.

is

Encourage students to factor completely, that is, to find the greatest common factor, and to check their factorizations by applying the distributive property.

Chalkboard Examples Find the greatest 1.

50mn,

XOrrfin^

common

factor.

lOmn

2.

12x2y, 16X72

3.

4;cy

Uab^, 18^2^2

9^^2

Factor. 4.

3/ - 6/ =

6. 6fl2

3

•

_

24ab

3y^(y

+

\Sb^

-

=

5.

2)

6(a^

USING FACTORING

-

Aab

(pp.

+

3^2)

7^

IOjc^

^(^

+

30x2

+ y) -

_^ 50;^

x{x

=

10x(x2

+ y) =

{x

174-175)

Teaching These Pages

Ask

the class to suggest different ways to evaluate the following expres-

sion.

Some

students will suggest finding each product and then adding.

Encourage the students

to look for a

common

factor, using the

methods of

the previous section.

25

T76

X

32

+

25

X

18 -h 25

X

50

=

25(32

-F

18

+

50)

=

25(100)

=

2500

+

3x

+ y){y -

+

5)

x)

Chalkboard Examples 1.

Find the 12

= = =

X

of the sides of the box.

+ 12 X 25 + 12 X +25 + 15 +25)

15

12(15

+

15

X

12

25

12

cm

12(80)

15

960

The area 2.

total area

is

cm

25 cm

960 cm^.

Find the area of the shaded and r.

Leave your answer

part.

Area of rectangle: Ar^ Area of 2 semicircles: irr^ Area of shaded part: Ar^ — ur^

—

—

r^(4

in

terms of

77

tt)

Suggested Extension

The

surface area of a cylinder

Find an expression

shown by

is

the

model

for the surface area.

Area of 2 ends: lirr^ Area of curved surface:

iTirh

Total surface area: 277^2

+

iTrrh

=

2'nr{r

+

h)

Objectives for Pages 176-187 1.

To

factor trinomials with factors of the

form {x

+

a){x

2.

To

factor trinomials with factors of the

form (x

—

3.

To

factor trinomial squares.

4.

To

factor trinomials with factors of the

form {x

—

4

a){x

+ —

b).

a)(x

+

b).

•FACTORING TRINOMIALS— TWO SUMS

(pp.

b).

176-179)

Teaching These Pages

Have

the students find these products at sight. (x (x (x

+ + +

2)(x 3)(x

6)(x

+ + +

4) 5) 3)

=

x2

= =

x^ x2

+ + +

6jc

Sx 9jc

+ + +

8

\5

18

Discuss the method the students used to find the products. in this section

we

reverse that

method

Point out that

to find the factors that yield

each

trinomial.

T77

Discuss the examples on pages 176 and 177. Ask the class to suggest a

way

to

check that the factorizations are correct.

Chalkboard Examples

Name

the two numbers.

1.

Their

sum

is 9,

2.

Their

sum

is

12, their

product

is

36.

6,

3.

Their

sum

is

13, their

product

is

40.

5, 8

4.

Factor d^

their product

+ Ua +

30.

{a

+

3,6

18.

is

+

2>){a

6

10)

Suggested Extension

The work of

the previous section

used in

is

this alternative

approach

to

factoring. 1.

Tina factored x^

+

+

x2

2.

5

\2x

Use Tina's method

+

\2x

21 as shown. Explain the steps she used.

+

21

= = =

+ 3x + 9x + 21 xix + 3) + 9(x + 3) (x + 3)(x + 9)

x^

to factor Exercises

31-36 on page

179.

•FACTORING TRINOMIALS— TWO DIFFERENCES

(pp.

180-181)

Teaching These Pages

Have

the students find these products at sight. (x {x

(x

-

2){x

3)ix 6)(x

-

4) 5)

3)

= = =

x^ x2 jc2

-

6x 8x 9jc

+ + +

S 15

18

Discuss the examples on page 180. Be sure students understand

look for negative numbers in Step

3.

Chalkboard Examples

Name

the two numbers.

—2,-6

1.

Their

sum

is

—8,

2.

Their

sum

is

—12,

their product

is

32.

—4,-8

3.

Their

sum

is

—10,

their product

is

25.

—5,

4.

Factor m^

—

10m

T78

their product

+

24.

(m

—

is

12.

6)(m

—

4)

—5

why we

Suggested Extension

The

factoring

tion can

6

•

method used

be used for

in the

Suggested Extension for the previous sec-

this section also.

FACTORING TRINOMIAL SQUARES

(pp.

182-183)

Teaching These Pages

Have

how each

the students find the squares of these binomials and note

term of the trinomial square (jc 4-

3)2

+

5)2

{x

Develop the

test

= =

x2 jc2

+ +

6x

related to the original factors.

is

+9 +25

-

(jc

lOx

{x

3)2 5)2

= =

x^ jc2

-

+9 + 25

6x

IOjc

outlined on page 182 for recognizing trinomial squares.

Point out that the students can always use the methods of the previous sec-

how

tions for factoring if they forget

to recognize trinomial squares.

Chalkboard Examples Complete.

+

16x

+

64

1.

jc2

3.

m2 - 10m + 25

= (_X + =

(m

8)2

_J_f

=(a_J_f

+6

=(^^-4)2

y

X

2.

fl2

+

12a

-5

4.

/

-

87

6.

x2

+

16x

+

64

=

(x

8.

x2

-

\4x

+

49

=

(X

+

+

36

16

Factor. 5.

m2

7.

^2

- 18m + 81 = (m - 9)2 _ g^ ^ 15 ^ (^ _ 2,){a -

5)

+ -

8)2

7)2

Suggested Extension

We

can use what we know about trinomial squares to find squares of numFor example, we can write 25 as 20 + 5 and find 252 ^5 shown

bers.

below. (20

+

5)2

=

202

+

2(5

•

20)

Write the square of each number. 1.

7

21

441

2.

16

+

=

52

400

How many

256

3.

'FACTORING TRINOMIALS

(pp.

+

200

+

25

=

625

can you do mentally? 35

1225

4.

28

784

184-187)

Teaching These Pages

Have (;c

the students find these products.

+

8)(x

-

6)

=

x2

+

2x

-

48

(x

-

8)(jc

+

6)

=

jc2

-

2x

-

48

T79

Ask students page

How

the two examples.

compare

to

How

same?

are they the

Relate the students' observations to the examples on

are thev different? 184.

Chalkboard Examples

Name

the two numbers.

1.

Their

sum

is

—5.

their product

is

—14.

2.

Their

sum

is

—4.

their product

is

—5.

3.

Their

sum

is 6.

their product

is

-

5)

—7,2 —5.

1

—2.8

—16.

Factor.

A-

4.

+

2.x

-

35

=

+

(.V

7)(.v

5.

-

x^

3.x

-

=

40

{.x

-

8)(.v

+

5)

Objectives for Pages 188-193 1.

To multiply

2.

To

8

•

sum of two monomials by

the

factor the difference of

their difi'erence.

two squares.

A SPECIAL PRODUCT

(pp.

188-189)

Teaching These Pages

Have

the students tind these products. (.V

(X

Ask

the students

if

+ -

1)(A-

3)(.Y

+

1)

(A-

-2)(A

3)

(A

+

4)(A

+ -

2)

4)

they notice anything special about the products.

Chalivboard Examples Complete. 1.

(V

-

+

3)

= _J_ - _J_

-

9)

=

V-

-

5)(m-

-

5)

=

An"'

3)( V

)•-;

9

2.

(2m

-

4)(2m

4.

(4a

-

v)(4a

6.

(a

+

4)

=

_JL_

-

.JL_ 4m-;

Multiply.

+

81

3.

(V

5.

(m-

7.

Find the product (45)(55). Use the rule (a

9)(v

+

-

25

(45)(55)

T80

= = = =

-

b){a

+

2475

v)

=

16a2

=

x^

2v)(a

+

2v)

=

a-

-

b)

- 5)(50 + _ 52 2500 - 25

(50 502

+

5)

b'-.

-

-/ 4/2

16

Suggested Extension 1, Sections 6 and 7; Chapter 6, Section 6; and Example 6 above suggest methods for computing products mentally. Combine exercises for each method to have a Mental Multiplication Quiz. The students might work in teams to arrive at a class championship.

Extensions for Chapter

9

FACTORING THE DIFFERENCE OF SQUARES

•

(pp.

190-191)

Teaching These Pages Review a few examples from the previous section on finding products of the form (a + b){a — b). Discuss the examples on page 190. Emphasize that this method is for factoring the difference of two squares. Point out that x^ + 4y^ and 9^2

+

cannot be factored.

1

Introduce the need to recognize the difference of two squares. ple,

—16 + y'^.

ask the class to factor

ten as y^

—

For exam-

—16 + y^ can be

Point out that

writ-

16.

Chalkboard Examples Complete. I.

-

a^

b^

=

{a

_^){a ^I_)

+b;

-b

2.

4x2

4.

x^

-/

=

(^_ + /)(^_ - j)

2jc; 2jc

Factor. 3.

m'^

-

16

= (m -

5.

m^

-

16

=

(m2

4)(w

-

+

4){m^

4)

+

4)

=

{m

-

2)(m

+

- 9/ =

2)(m^

+

(x

-

3y)ix

+

3y)

4)

Suggested Extension 1.

— y)(x + y). Students may show that x^ — y^ = (x — y)(x + y). Then since x"^ — y"^ y£ x^ + y^ (unless y = 0), x^ + y^ Show why

The x^

x^

+ y^

:^

(x

students might also assign values to x and

+ y^

and (x

— y)(x + y)

to

show

that the

y

:p^

(x

— y){x + y).

then evaluate

two expressions are not

equal.

10*

MANY TYPES OF FACTORING

(pp.

192-193)

Teaching These Pages Review the various factoring techniques developed the class work a few examples using each method.

in this chapter.

Have

T81

Point out that in this section

more than one technique may be needed

to

factor each example.

Chalkboard Examples Factor. 1.

3.

^

2x2

i2;c

_

3x2

48

+

= =

= =

18

3(^2

3(x

2(jc2

2(x

+ 6x + + 3)2

- 16) + 4){x -

2.

9)

x3

6jc2

-x2 +

4.

+

6jc

Sx

+

= =

16

4)

x(x^

x(x

- 6x 4- 8) - 4)(x - 2)

= - 1(^2 -6x - 16) = -l(x -S)(x + 2)

Suggested Extension Factor. 1.

2.

3.

-

+ 9)(x2 - 4) (x2 + 9)(x - 2){x + 2) (^2 - 9)(x2 + 4) x^ - 5x2 _ 36 (x - 3)(x + 3)(x2 + 4) x* - 13x2 + 36 = (x2 - 4)(x2 - 9) .= (X - 2)(x + 2)(x - 3)(x + x^

11 •

+

5jc2

MORE

36

= = = =

(jc2

3)

DIFFICULT FACTORING (OPTIONAL)

(pp.

194-195)

Teaching These Pages Introduce this section by having the students find products. (3x (2x

+ -

l)(x

l)(x

+ +

2) 3)

Discuss the examples on page 194. the

first

term of the trinomial

is

= =

3x2 2x2

+ +

+ -

7x 5x

Point out that

not

1,

it

may

2 3

when

the coefficient of

be necessary to

try

many

factors before finding the correct ones.

Chalkboard Examples Factor. 1.

3x2

T82

_

i2x

+

4

=

(3x

-

l)(x

-

4)

2.

3/ +

5/

-

2

=

(3/

-

1)(/

+

2)

Chapter

7.

Graphs

Every day we are confronted with graphs occurring in newspapers, magaand so on. Thus it is important to be able to interpret information displayed by means of a graph. This chapter develops skills for working with pictographs and bar graphs and then extends the development to include broken line graphs as well as graphs of linear equations. zines, reports,

In preparation for the next chapter, the graphs of equations in two un-

knowns

are studied

and related vocabulary

is

introduced.

Objectives for Pages 204-215 1.

To read

2.

To read and draw

a bar graph.

3.

To read and draw

a broken line graph.

4.

To

a pictograph.

identify

and graph ordered

pairs of

GRAPHS YOU OFTEN SEE

1 •

numbers on

(pp.

a coordinate plane.

204-207)

Teaching These Pages Throughout

this chapter,

emphasize the usefulness of a graph

to display

information in a concise manner.

Have zines. sion.

the class collect examples of graphs from newspapers and magaTextbooks for other subjects might also provide examples for discus-

Use the example on page 205

to introduce the nature

of misleading

graphs.

Point out that the scale on a graph should always start at zero.

When

would cause the size of the graph to be too large, we delete a portion of the scale and indicate the deletion by a jagged rule. (See page 206.)

this

Chalkboard Examples

NUMBER OF OLYMPIC MEDALS WON

1896-1972

Sweden Great Britain

USSR United States = 100 medals 1.

What

information does the graph show?

The number of medals won

in

the Olympics from 1896 to 1972.

T83

2.

How many

medals does each symbol represent?

3.

What does

a portion of a symbol represent?

4.

Draw

2

•

100

Fewer than 100 medals.

shows the same information.

a bar graph that

GRAPHS WITH LINES AND CURVES

(pp.

208-211)

Teaching These Pages Record the temperature

for a certain city for 7 consecutive days.

Make

a

table to illustrate the data.

Then show

the class the use of a bar graph to

show

the data, the construc-

graph from the bar graph, and the drawing of a "smooth" line graph. Discuss how the three graphs are related and why we can draw a smooth curve in the third graph. Have the class create questions based on the information shown by the tion of a

broken

line

graphs.

Chalkboard Example Each year a group of students hold

a car

WASH-ATHON EARNINGS

wash-a-thon to raise money for charity. The

amount of money collected shown on the bar graph.

for the last 6 years

is

Refer to the graph to answer the questions. 1.

Estimate the amount of 1972 and in 1975.

2.

How much more money 1976 than in 1971?

3.

money was

collected in

$60

Estimate the average amount of lected each year since 1971.

4.

collected in

$482; $525

Predict the

might

amount of money

collect in 1977.

money

col-

about $500 the group

probably more than

1971 1972 1973 1974 1975 1976

$500

Suggested Extension Each day the newspaper publishes high and low temperatures for various Have the class, or groups of students, select a city and collect data

cities.

about the daily temperatures. 1.

Draw

2.

Use the graph

T84

a graph to

show

the data.

to predict the next day's temperature.

3

POINTS ON GRAPH

•

212-215)

(pp.

Teaching These Pages

The temperature data used

in the previous section

can be used here to

introduce graphing points on a coordinate plane.

Introduce the vocabulary related to the graph on page 212: ordered pair, origin, coordinate, horizontal axis, vertical axis.

Chalkboard Examples Give the coordinates of each point.

A (-4,2)

I.

Name

C

the point by

(1,3)

4.

2.

E

5.

(2,-2)

3.

Z)

6.

(-1,-3)

(-2,0)

its letter.

(-1,3)

G

/

Give the coordinates of the point closest to each point.

F E

1.

Name

10.

S.

G E

4 points that are

H

9.

J

at the vertices

of a

rectangle.

Objectives for Pages 216-227 1.

To

2.

To draw

3.

To

4

•

identify solution pairs for a linear equation in

two

variables.

the graph of a linear equation in two variables.

find the slope of a line.

SOLUTION PAIRS

(pp.

218-219)

Teaching These Pages

Show

the class an equation in one variable

and an equation

in

two

variables.

3x+6 = List

some

X

12

solutions for the equation in

X

=

\,y

=

5

X

=

2,y

+y =

6

two variables.

=

4

x

= -3,y =

9

Point out that the equation in one variable has only one solution, the equation in two variables has

many

2,

but

solution pairs.

Introduce the use of an ordered pair {x,y) to record a solution for an

equation in two variables.

Chalkboard Examples

=

2x

+7 =

1.

Find X

3.

Which ordered pair is not a (1,-4), (-2,-5), (6,1)

if V

3 in

x

+

4

1 1.

solution pair for

x

—

2y

ii

=

x

=

2 in

x

S'!

(6,1)

=

4.

Find a solution pair

5.

Write an equation to go with the table.

for

Find y

2.

3/

9.

x X

y

y = 3x

+y =

Suggested Extension 1.

Draw

the graph of

x

2.

Draw

the graph of

x

3.

Name

6

•

+y = —y =

the point that Hes

SLOPE OF A LINE

5.

1

on the same

on both hnes.

(pp.

set

of axes.

(3, 2)

222-223)

Teaching These Pages Introduce the meaning of slope by comparing the slopes of the roofs of various houses.

The roof of an A-frame cottage has a steeper ing rain and snow to run off more quickly. Introduce the meanings of rise and run.

slope than

Chalkboard Examples Find the slope.

l./ 3.

=

f+

How

is

2

i

same

y = -2x +

3

the slope of each graph of the equa-

tions in Exercises efficient

2.

1

and 2 related

to the co-

of the X term in each equation?

the

many

roofs, allow-

Suggested Extension

Draw

y = 3x +

I.

Do

3.

2?

Draw

the graphs in Exercises

They

Do

7.

6?

•

and 2

set of axes.

y = 3x -

\

cross each other?

No.

notice about the slopes of the graphs in Exercises

1

and

notice about the slopes of the graphs in Exercises 5

and

3

6.

the graphs in Exercises 5

They

same

are the same.

What do you

8.

1

the graph of each equation on the

y = 2x +

5.

2.

\

What do you

4.

7

the graph of each equation on the

and 6

same

set of axes.

y = X - 4

cross each other?

Yes.

are different.

FUNCTIONS

(pp.

224-227)

Teaching These Pages Discuss the examples of functions on pages 224 and 225.

Although

this section takes a

very intuitive approach to functions, some

students will learn in later courses that a function can be thought of as a set

of ordered pairs in which no two different pairs have the same

first

element.

Chalkboard Examples 1.

A machine adds 2 to the Complete the table.

input.

X

=

INPUT

^

—

vj»j

r

1

vj 1

10 2.

3.

An

equation relating x and

y

in Exercise

1

is:

Study the function in Example 2 on page 224.

y

Is

lating the age of the car to the value of the car?

Suggested Extension

=

?

.

8

x

+

-10

2

there an equation re-

No.

Chapter

8.

Equations with Two Variables

In this chapter students will learn to solve a pair of hnear equations in

two

The graphic method is first used, but for greater accuracy the methods of substitution and add-or-subtract are developed. These skills are

variables.

applied to problem solving in Sections

5, 7,

and

8.

Objectives for Pages 236-243 1.

2.

To

find the solutions of pairs of linear equations

To recognize whether a no solution pairs, or all

1 •

by graphing.

pair of linear equations shares one solution pair, solution pairs.

THE GRAPHING METHOD

(pp.

236-239)

——r~

Teaching Tliese Pages Use a

map

street

\

— CO to

review locating points on a

coordinate system and to introduce the meaning

of intersection.

Rogers Ave.

"CO \

"(5

ro

O

\

Davis Ave.

c QQ

O Ebos Ave.

Park Ave. CO

Hillcrest Ave.

Chalkboard Examples Complete the 1.

.V

+7 = X

table.

12

y 10

-2 6 5

CO

NO SOLUTION, MANY SOLUTIONS

2 •

(pp.

240-243)

Teaching These Pages Use Ask

the following graph to introduce this lesson. the students to name a pair of

equations that have intersecting graphs. Introduce the meaning o( parallel

You may want

lines.

parallel lines

Ask

lie

to point out that

same

in the

the students to

name

plane. a pair of

equations that have graphs that are Point out that the slopes

parallel lines.

are the same.

Point out that the graphs of

X

-\-

y = A and 2x

same, and so

2x

4-

2y

=

a:

-\-

2y = % are the 4 and

+/ =

8 share all solutions.

Chalkboard Examples Find the slope of the line defined by each equation. 1.

^ = 6x +

3

6

2.

jc

+

V

=

4

-

1

3.

X

+

2y

=

4

Chalkboard Examples Solve.

X

1.

+J =

8

2.

x-y=4 P =

4.

21

Find

4

+

P

2w-

I

=

y = 4x X

(6,2) 4k,

in terms of k.

3.

-y = -9

+

2b

=4 -26

(8,

-2)

w = 2k, P = \2k

THE ADD-OR-SUBTRACT METHOD

•

a

b-3a=

(3, 12)

(pp.

1A6-1A9)

Teaching These Pages Use the substitution method

3x-y = 2x+/ =

to find a solution pair for these equations.

l

(1)

8

(2)

y = 2x

+

3x-l

-7) =

(3x 5jc

-

= =

7

X Introduce the add-or-subtract method. Point out that (1)

and

(2)

we

will obtain the starred equation.

pairs of equations the add-or-subtract

method

if

8* 8 2>,y

=

2

we add equations

Suggest that for certain

more convenient than

is

the

substitution method.

Chalkboard Examples Solve.

3^

1.

2y

Encourage students to check the solution

- 2jc = + 2x=4

2.

1

(-1,3)

3y 2y

+ +

in the original equations.

2x 2x

=4 =

2

(-1,2)

Objectives for Pages 250-257 1.

To use a

2.

To use

pair of linear equations to solve problems.

multiplication

and the add-or-subtract method

to find the solu-

tions of pairs of linear equations.

To use

3.

5

•

a pair of linear equations to solve cost problems.

WORD PROBLEMS, TWO

VARIABLES

(pp.

250-253)

Teaching These Pages Review the Guide

for

Problem Solving on page

54.

Point out that

we

fol-

low similar steps when solving problems with two variables. Solve a problem with one variable and compare its solution to the solutions of the

Chalkboard Examples.

T91

Chalkboard Examples Solve. 1.

Use two

variables and two equations.

The sum of two numbers is 26. One number is 6 more than the other. Find

2.

y = X X

=

How many

score?

+ y = 26 — 6 =y

=

Let X

the smaller.

=

16;/

Write a problem to

=

X X

10

"fit" these

Lesley's score;

+y = —y =

•

2S

—y =

2

The sum of Alec's and Georgina's ages The difference in their ages is 2. Find

6

+y =

is

1

X

=

76,

28 years.

their ages.

USING MULTIPLICATION

(pp.

254-255)

Teaching These Pages

Show an example

of a pair of equations that cannot be solved by the sub-

method or the add-or-subtract method. Review that we can multiply members of an equation by the same number to obtain an equivalent equation. Thus we can transform the second

stitution

equation and solve the pair as shown.

Ax 2x

-\-

5y

+

2>y

= =

1 3>

Ax Ax 4x

Point out that tional

number

-\-

5y

+

6y

-

5

= = =

1

6

7;

X

=

3

we can multiply members of an equation by if it is more convenient.

the

same

Chalkboard Examples Solve. \.

Zx 2x

T92

-2y = + y =

10

9

2.

(4, 1)

3x

2x

-2y = -1 + 2>y = 4

y =

Mike's score

\45

Solve your problem.

equations.

X X Sample:

more

points did each

number;

the larger

Suggested Extension 1.

points alto-

Lesley scored 7 points

than Mike.

the numbers.

Let X

Mike and Lesley scored 145 gether.

(-1,2)

frac-

/ =

69

7

•

COST PROBLEMS

256-257)

(pp.

Teaching These Pages Review the cost formula from Chapter 4. Guide for Problem Solving on page 54.

C =p X

n.

Also review the

Chalkboard Examples 1.

2.

A

A

hot dog costs 50c.

a.

Write the total cost of x hot dogs and

b.

Write the total cost of

hamburger

costs

60(1;.

y hamburgers.

hot dogs and 2y hamburgers.

3jx:

50x

+

6O7 cents

150x

+

120^^ cents

Movie tickets cost $3 for adults and $2 for children. On Saturday, $66 was paid for 28 tickets. How many adult tickets were sold?

— number

Let a

a

3a

+ +

of tickets for adults; c

= number

of tickets for children.

z=2S

c

=

2c

66

There were 10 adult

tickets sold.

Suggested Extension Clip advertisements from the newspaper to obtain prices for various items.

Have

8

•

the students create

and exchange

cost

problems based on the

BOAT AND AIRCRAFT PROBLEMS (OPTIONAL)

ads.

(pp.

258-259)

Teaching These Pages Review the

skills

developed in Chapter 4 for solving motion problems with

one variable.

Show

the students the difficulty in attempting to solve the

example on

page 258 with only one variable.

Chalkboard Example 1.

Janie and Jacques traveled 12 return trip took only 3 hours. the rate of the boat (in

Let X

=

rate in

6{x

-y) = +y) =

upstream in a boat in 6 hours. The Find the rate of the flowing water and

water).

km/h of the boat km/h of the current.

rate in

y = 3(x

still

km

in

still

water;

12 12

X

= 3;y =

\

T93

Chapter

9.

Working with Fractions

In earlier chapters the students have developed skills for working with pol-

ynomials, and applied these

skills to

solving problems.

In this chapter stu-

working with fractions in algebra, essential to solving problems. Throughout the chapter skills with fractions in algebra are related to similar skills the students have learned in working with arithmetic. Once the students have learned to add, subtract, multiply and divide dents develop

skills for

algebraic expressions involving fractions, these

skills

become

useful for

This work with equations

solving equations that involve fractions.

is

then

applied to solving problems that deal with rates of work.

Objective for Pages 268-273

To use

1.

1 •

factoring to simplify fractions.

FRACTIONS IN ALGEBRA

(pp.

268-269)

Teaching These Pages Review

that a

Remind

number has

different

9

12-3

students that a fractional

4

names.

5+4

32

number

also has different names.

32

8

Point out that to simplify a fraction

means

to find a simpler

name

for the

fraction.

Use the examples on page 268

to

compare simplifying

fractions in arith-

metic to simplifying fractions in algebra. it would be helpful to review the rules for numbers (page 88), and the exponent rules

Before assigning the exercises dividing positive and negative

(page 156).

Chalkboard Examples Simplify.

-8fl

. '

\6a

_ _\_ ~ 2

2 '

30m _ _ -25m ~

6^

5

3 '

-Xla^b -24ab^

^ ~

_a_

-36ab^c^

^ '

2b

^

3b^c

-Ud^bc

a

State the value of x for which the fraction has no meaning. 5.

^

— X -2

T94

2

6.

-^^— \

-

X

1

7.

^r^—r 2x - 4

2

8.

-r-^ 6-3x

2

Suggested Extension Simplify each fraction then add or subtract.

-36a^b J

— lAm^n — Arn^

= -la

Uab

j6rV

=

— 3>r'^s

4

—Ars

SIMPLIFYING FRACTIONS

2 •

(pp.

—Arn^n^

-.Sn

2„2 Irrrn

6^W

-16A:5m4

3km

Ak^m^

—Ik'^m

270-271)

Teaching These Pages Review the various factoring techniques students learned a^

+ 6m +

m^

9

=

+ 3f

(m

in

Chapter

6.

+ 2a -S =(a + A)(a - 2) - / = {2x - y){2x + y)

4x2

Chalkboard Examples Simplify.

X

+

1.

2x

2ab 3.

3^2

3

•

2

+A

jc

2(x

-6a _ _ 9^ -

4-

3y-6 ^ 3{y-

2

+

2

2)

3

2a(b

-

2b(b

-3)~Jb

_

3)

THE -1 FACTOR

+6 +4k +A

2a

(pp.

2)

3k 4.

k^

= y-2

3

3(k (k

+

+

2){k

2)

+

2)

k

+

2

272-273)

Teaching These Pages

Ask

the students to

compare these X

-

pairs of binomials.

1

a

—

-X

b

—

I 1

Use the

b

x^

— j2

a

yi-

_

I

show

distributive property to

-

-\{x

1

that

\)

1

—

jc

= -X ^ = -JC

x^

= — l(jc —

1).

\

1

Point out the usefulness of the ;c \

-

1

-X

-

factor in simplifying fractions.

1

X

-\ -

-l(jc

1

1)

-1 T95

Chalkboard Examples Simplify.

1.

—

.V

X

= —

= —X +

2.

1

X

-\-

V

3.

y

—

— jc2

4-

2xy

— y'^

= ^

—y

Objectives for Pages 274-283 1.

To

solve problems involving ratios.

2.

To

solve problems involving proportions.

4

•

RATIO

274-275)

(pp.

Teaching These Pages Introduce the meaning of ratio with familiar examples. 1.

2.

number of students over 165 cm number of students under 165 cm tall.

Ratio of the to the

tall

Ratio of the number of pages in the math book to the

number of pages

in the dictionary.

Point out that the following two statements are equivalent.

The

ratio of the

teachers

is

20 to

number of

students to the

number of

1.

There are 20 students

for every

1

teacher.

Chalkboard Examples Give the ratio

in simplest form.

1.

20 meters to 100 meters.

3.

A

team won 24 games and

What

is

—20— = — 1

lost 18.

the ratio of losses to wins?

2.

What

•

the ratio of wins to losses?

— — 4

3

;

3

5

is

2 years to 2 months.

4

PROBLEMS INVOLVING RATIOS

(pp.

276-277)

Teaching These Pages Introduce this topic by posing a problem.

The

ratio of the

number of

tended the school play

tended the play.

T96

How

adults to the

number of

children

who

at-

Three hundred seventy-five persons many adults were there? is

2 to

3.

at-

—24- = —12

Point out that the ratio as

Ix

if

the ratio of two

numbers

2 to

is

3,

then we can write

to 3x.

2x 3x

Review the Guide

to

_ 2_ ~y

Problem Solving on page

54.

Chalkboard Examples ratio of the number of adults to the number of children who attended the school play was 2 to 3. Three hundred seventy-five persons attended the play. How many adults were there?

The

1.

2jc

There were 150 adults 2.

in the

size cars

How many

3 to 5.

3x X

=

375

=:

75

there.

There are 480 cars is

+

parking

3x

+

compact cars were there?

ratio of

cars

to full-

= 480 =60

5x jc

There were 180 compact

The

lot.

compact

cars.

Suggested Extension

A

box

is filled

with 500 beans, brown beans and white beans. Have the

brown beans number of samples, have

students take a sample from the box and record the ratio of to white beans.

them

6

•

predict the

After the students have taken a

number of brown beans

PROPORTION

(pp.

in the box.

278-279)

Teaching These Pages Review

that fractional

numbers have

In a similar way, ratios

may be

2_^

4.

3

6

~

equal.

2 to 3

equals

3

4 to 6 t

I

1

names.

different

=

1 6

Introduce the meaning o{ proportion and the property of proportions.

T97

Chalkboard

Objectives for Pages 284-289 1.

To multiply

2.

To divide

3.

To add and subtract

8

•

fractions.

fractions.

fractions with the

MULTIPLYING FRACTIONS

same denominators.

(pp.

284-285)

Teaching These Pages

Compare how we

3-2 5-7

1.1 5

Compare an

multiply in arithmetic to

7

_6^ 35

3

X

y

.

multiply in algebra.

xk

k

m

y

'

m

arithmetic example to an algebra example to

often simplify the product of two fractions.

4

how we

4

3

ym show how we

Suggested Extensions 1.

Have

2.

To review

a

la

— 2b l(a - b) -l(b - a) 2a

work as well as emphasize the need to watch out have students discuss the "proof" that 2 = — 2.

= = = = =

for

b

lb lb

Multiply both sides by

— la l(b - a) lib - a)

-1=1 •

on page 287.

earlier

restrictions,

10

on the denominators of the

the students discuss the restrictions

fractions in the exercises

2.

Subtract the same values from both sides, (la Factor.

a

—

b

=

—{b

—

a)

Divide both sides hy b

—

a.

ADD, SUBTRACT— SAME DENOMINATORS

(pp.

288-289)

Teaching These Pages Point out that in algebra as well as in arithmetic the

same denominators by adding 1

5

2^ 1+2

x_

Ix

5

5

5

5

Chalkboard Examples Simplify.

1.

^+ a

^

5a

Ix

we add

fractions having

the numerators.

X

+

Ix

3£ 5

=

lb)

And

in algebra, to

add or subtract

fractions that

have different denomina-

we need to rename the fractions. This section provides practice in renaming fractions and the next section appHes this skill to adding and subtors

tracting fractions.

Chalkboard Examples

_ ^•3-6 -7

X

12

•

12 2)X

2a

?

4a

3.

8fl

ADD, SUBTRACT— DIFFERENT DENOMINATORS

Teaching These Pages

Compare

the steps of each example.

11_3 2

3

3-2

2_32 2-3

6

6

_6_

IOjc

16x2

8jc

3+2

5

6

6

(pp.

292-293)

\6ab

Mb

Chalkboard Examples Simplify.

X

-2

X

2{x

\

-2) -{x -

-2

JC

-

\)

-

jc

5(x

1

-

2)

-

3

4

4

4

2

X

-

2{x

-

3x

1)

2.

10

10

a

-

3.

+

-2 =

a

1

1

3fl

14 •

a

\

2(a

-

1)

+

3(a

1)

-

(a

-

2)

EQUATIONS WITH FRACTIONS

(pp.

=

4a

+

3

6a

6a

6a

2a

+

296-297)

Teaching These Pages Review the basic skills developed earlier following examples reviews a skill. X

+

6

=

12

-

for solving equations.

Each of the

Suggested Extension

Which ,

greater?

is

1

1

2

2

+ +

a

+

Let X X2 4.

•

b and

+/

+ + >>

2^2 (a

A:_y

15

1

2

1

+

-,4 —

1

2.

or

= +

4+1 5

1

a

+

_

5

+

3.4or3 +

1

2

1

Evaluate.

A.

2Z?2

^)(a

4+

1

(-y

+7)

2a

5.

b)

(a

WORK PROBLEMS

(pp.

-

b){a

+

b)

298-301)

Teaching These Pages Discuss different types of jobs where the job.

dents in class

may

deliver newspapers.

time to deliver the newspapers that the time

gether.

many

persons contribute to complete

(Building houses, making cars, delivering newspapers.)

is

Discuss

not reduced by

why

or

why

Ask whether

when another person

— when

2 students

it

takes

helps.

—

Some as

stu-

much

Chances are

do the paper route

to-

not.

Chalkboard Examples 1.

Steve delivers newspapers and it takes him 70 minutes. His sister Sandra can do the job in half the time. How long would it take Steve and Sandra to deliver the newspapers together?

Let n

= number

of minutes needed to do the job together.

l

2+1

3

+

1

2+1

Suggested Extension 1.

A

rectangular tank has dimensions 20

Which of

the following will

empty

A: 2 holes in the bottom

mx

m

10

and

is

6

m

deep.

the tank faster?

or

B:

hole in the bottom of

1

of the tank each with

the tank with radius

diameter 4 cm?

3

The

rate at

cm?

which the tank empties depends on the area of the

hole.

B Area of Holes

A — l"n

= Thus B

16

•

is

Stt

'1?

cm^

cm^

Area of Hole

A =

=

TT

•

977

32 cm^ cm^

the fastest.

BINOMIAL DENOMINATORS (OPTIONAL)

Teaching These Pages This section extends the work of Sections

Compare

these examples.

4

,

10, 12

and

13.

(pp.

302-303)

Chapter

Decimals and Percents

10.

As consumers, students need to use their skills with decimals, fractions, and percents. The first three sections of this chapter review the skills needed to work with decimals and fractions, and Section 7 deals with the topic of percents. Once the skills have been established they are applied to equations, which are then applied to real-life problems that involve interest and investment. In the

last section,

mixture problems are solved using the

skills

developed in the chapter.

Objectives for Pages 312-321 1.

To add,

2.

To express

3.

To

solve equations with decimal coefficients.

4.

To

solve problems using decimal equations.

1 •

subtract, multiply,

and divide decimals.

fractions as decimals.

DECIMALS

(pp.

312-313)

Teaching These Pages To motivate

the class,

show examples of

different

ways

that

we

use deci-

mals every day.

(Cottage

Cheese

QQ 9 -WW tt^I

Review the meaning of place value. This read decimals but also to understand points

when we add

to line

only to

up the decimal

or subtract decimals.

26.396 means 2-10-f-6-l

Compare place

will help the students not

why we need

+3--^-h9- -i- +

1

6

1000

value and face value as they relate to the example.

Chalkboard Examples

Add 1.

or subtract.

4.63

+

389.2

=

393.83

2.

0.57

-

0.3

=

0.27

3.

492.19

-

273.284

=

218.906

T105

Multiply. 4.

69.8

X

=

0.04

2.792

x

0.69

5.

46.21

=

31.8849

6.

7.32

x

0.12

Suggested Extension

Newspaper advertisements can be used

to

provide additional practice.

TOOTY-FROOTY shampoo, $1.42

1.

A

case of

cost? 2.

How much cost?

2

shampoo holds 24

bottles.

does a case of shampoo

would 6 cans of shaving cream and

3 boxes of facial tissues

$8.13

DIVISION WITH DECIMALS

•

How much

$34.08

(pp.

314-315)

Teaching These Pages Review that to rename 19.55 -=- 2.3 we can think of the fraction form. We rename by multiplying numerator and denominator by the same number, namely 10.

X 10 X 10

19.55

19.55

195.5

2.3

2.3

23

Before beginning the exercises, review the meaning of rounding to one dec-

imal place, to two decimal places. Introduce the symbol

=

for "is approximately equal to."

Chalkboard Examples Divide.

Round

to

one decimal place. 25.82

8^ 1.

2.3)19.55

-^

23)195.5

2.

0.17)4.39

-^

=

25.8

I7J439

Suggested Extension 1.

a nickel is about 2.15 cm. How many nickels placed edge to edge are needed to stretch 1 km? about 46,512

The diameter of

T106

=

0.8784

How many

Measure the diameter of a quarter.

2.

km?

stretch

1

Which

is

3.

quarters are needed to

about 40,816

worth more: a kilometer of nickels or a kilometer of quarters?

quarters

3

•

DECIMALS FOR FRACTIONS

316-317)

(pp.

Teaching These Pages Use the

sports pages to

rational

number. The

the best in the

show that it is useful to find a decimal form for a do not help us quickly to decide which team is standings. To compare the standings we need to find the ratios

decimal forms for the rational numbers.

Chalkboard Examples Express the fraction as a decimal. 1.

-=

0.75

43 = 0.666

2.

4

4

•

3.

.

^=

0.151515

33

EQUATIONS WITH DECIMALS

4.

4 = 0.375

318-319)

(pp.

Teaching These Pages

Have

the class

same?

How

compare the following two equations.

How

are they the

are they different?

8j

=

24

0.8/

Point out that to solve equations,

it is

easier to

=

2.4

work with whole numbers

than with decimals.

Review multiplying by

and 1000.

10, 100,

Chalkboard Examples

Would you

multiply by 10, 100, or 1000 to obtain an expression with the

smallest whole 1.

0.4x

4.

0.26x

number

10

-4.1x

100

coefficient? 2.

40.23/

5.

0.25/

+

0.5/

8.

0.127

+

0.037

100

1000

3. 4.036A:

100

-

6.

0.025m

9.

0.15(7-2) =0.3

0.25«

1000

Solve. 7.

0.9jc

9x X

= =

3.6

36

=4

12_y

+

= 3_y = y =

0-3

30 2

15(7

-

= = 7 =

2)

7-2

30 2

4

T107

Suggested Extension

—

1.

- as a decimal. Express ^ 33

3.

Use your answers 0.090909

.

for

1

0.030303

and 2

.

2.

.

to predict the

Express

——

decimal form of

as a decimal.

0.060606

^.

.

Express each fraction as a decimal. 6 ^^^0.1818...

4.

5.

5

•

^=

0.2424

6.

33

33

USING DECIMAL EQUATIONS

(pp.

4133

=

0.4545

1.

33

320-321)

Teaching These Pages Review the Guide

Problem Solving on page 54

for

as

you discuss the ex-

amples on page 320.

Chalkboard Examples 1.

On Monday

Jenny jogged 2.6 times as far as she did on Tuesday. The total distance she jogged on the two days was 14.4 km. How far did she jog each day? Let X

=

distance jogged on Tuesday; 2.6x

2.6x 26jc

+

+

X

IOjc

36jc

X She jogged 4

km

= = =

=

distance jogged

on Monday.

14.4

144 144

=4 on Tuesday and

10.4

km

on Monday.

Suggested Extension

The data below was taken from a

cereal box.

lems based on the data they find on nutrition

VITAMINS AND MINERALS

Have labels.

^=

students create prob-

0.9696

Objectives for Pages 322-331 1.

To express decimals

as percents.

2.

To express percents

as decimals or fractions.

3.

To

6

solve problems involving percents.

PERCENTS

•

(pp.

322-325)

Teaching These Pages Use examples from the newspaper used in everyday

to illustrate the

many ways

percents are

life.

^

100/0

^

Down Payment

Chalkboard Examples Express as a percent. 0.48

1.

= 48%

2.

0.375

6.

10%

=

37.5%

3.

44 = 0.25 =

25%

4.

^ = 0.2 =

8.

4.5%

20%

Express as a decimal. 5.

38% =

The

circle

9.

What

0.38

What

0.1

7.

=

46.5%

0.465

=

0.045

graph shows how Mike spends his time. percent of the day does

50%

he sleep and eat? 10.

=

percent of the day does

he spend in school?

25%

Suggested Extension 1.

2.

3.

What

percent of

about

61%

What

percent of

about

68%

What

percent of

about

67%

its

games has Rochester won?

its

games has

its

games has Nova

Springfield lost?

Scotia

won?

1

AMERICAN LEAGUE

7 •

USING PERCEN TS

(pp.

326-327)

Teaching These Pages Introduce this topic with an advertisement of a sale from the newspaper.

Save 250/0! was $16. ..now only $12!

Introduce the meanings of original price, discount, and sale price as they relate to the advertisement.

Chalkboard Examples Compute. 1.

4.

40%

A A

400

of 1200

pen

2.

35%

of 1800

set is regularly $8.

15% discount

is

5.

offered at a sale.

X

8

=

3.

75%

A

badminton set regularly sells on sale at a 20% discount. Find the sale price.

0.2

1.20

X

7.5

=

1.5;

$7.50

-

Suggested Extension

Have

the students collect sale advertisements

and create problems based on

the ads.

8

•

PERCENTS IN EQUATIONS

(pp.

328-331)

Teaching These Pages In this section, as well as the sections on interest, investment, and mixtures, the students will be solving equations involving percents.

of the equations that occur in the 1 1

to

show students

real-life

problems

Chalkboard Examples

The stadium holds 4500

80% of the How many 0.80

TllO

X

seats.

seats are filled.

4500

seats are occupied?

=

Point out

some and

in Sections 9, 10,

the need to learn skills for solving equations involving

percents.

1.

3600

of $23

$17.25

for $7.50.

It is

Find the discount. 0.15

450

3600 seats are occupied.

1.50

=

$6.00

A

2.

total

What

of 360 bottles were returned and 18 of them were broken.

percent were broken?

J^ = 0.05

= 5% 5%

of the bottles were broken.

360 After the game, the spectators were asked

3.

if

they enjoyed the game.

63 persons said they enjoyed the game. If

6%

0.06x

6x

X

of the persons interviewed enjoyed the game,

= = =

how many

persons were interviewed?

63

6300 1050 persons were interviewed.

1050

Objectives for Pages 332-339 1.

To

solve simple interest problems.

2.

To

solve investment problems.

3.

To

solve mixture problems.

9

INTEREST

•

332-333)

(pp.

Teaching These Pages Introduce this topic with a bank brochure showing the current rate of interest

paid on deposits. Collect different brochures to show that different rates

of interest /

may be

paid by different banks. Discuss the interest formula

= prt.

Chalkboard Examples 1.

How much money does $200 earn in one year at 6% interest? = 200 X 0.06 X = 12 The interest is $12 for one year.

/ 2.

1

George deposited $250 Find the 13.75

0.055 3.

= =

in the bank.

amount of

After one year, the

interest

added

to his account

250r

The

r

interest rate

5.5%

is

In January Jan put $150 into an account that earns

Four months

How much Interest

The

was $13.75.

interest rate.

later she

added $100

interest did the

on $150:

total interest

money earn

at the

7=150x0.06=9 earned

is

$9

+

6%

interest per year.

to the account.

$4

=

end of the year? Interest

on $100:

7=100x0.06x^ =

4

$13.

Till

10

•

INVESTMENT

(pp.

334-335)

Teaching These Pages Point out

some of

investment

ways money can be invested: stocks, bonds, Obtain advertisements that give the interest rates

the different

certificates.

for various types of investments.

that occur in the exercises

Relate these rates of interest to the rates

on page 335.

Chalkboard Examples 1.

The school band saved $2400

One

part of the

The

interest

How much Let X

money was

earned is

in

for their trip.

invested at 6%, and the rest at 8%.

one year

is

$180.

invested at each rate?

= amount

invested at 8%; 2400

—

x

= amount

invested at 6%.

Chapter

1 1.

Squares and Square Roots

In problems that apply the Pythagorean Theorem, the need arises to find square roots. This chapter provides skills for working with radicals as well as finding their decimal values from a square root table. Work on square roots leads to the introduction to irrational numbers.

The

skills

of finding square roots are also used in solving equations. The

Pythagorean Theorem provides an opportunity to solve a new type of problem, as well as apply the skill of solving equations of the type x^ = a^. The chapter concludes with developing skills to multiply and divide, as well as add and subtract radicals. An important skill for simplifying computation is to rationalize the denominator of a radical expression.

Objectives for Pages 348-355 1.

To recognize

2.

To use

3.

To recognize

4.

To simplify

1 •

perfect squares

and

find their square roots.

a square root table to find square roots.

and

rational

irrational

numbers.

radical expressions.

SQUARE ROOTS

(pp.

348-349)

Teaching These Pages Review the meaning of the square of a number and grams below.

yA

- 3^ or 9 units

relate

/4

it

to the dia-

= 4^ or 16 units

Point out the following. 1.

The square of

3

is 9.

3.

A square root of 9 is 3. — 3)2 is also equal to 9,

4.

The

2.

(

so

—3

is

also a square root of 9.

positive or principal square root of 9, represented

by

\/9,

is 3.

For example, we are given a measure of each side? We need to know how to find square roots to answer the question. Relate the symbol V^ to the example, pointing out that the side of the square would measure ^/\i cm. Point out the need to find square roots.

square region.

T114

What

is

the

A

=

12cm2

Summarize

that the next few sections will

Introduce the symbol

show how

to find a value for

r!z.

Chalkboard Examples

Which numbers 36

1.

are perfect squares?

\es

37

2.

no

3.

65

no

4.

64

Find the value.

-

Vm =

5.

\/36

=

8.

Fmd

the perimeter of a square

\/62

X VsT =

4

=

4

6

X

6.

9

=

36

if

-

\/l02

the area

The perimeter

is

= - 10 is

7.

V52

-

42

= V26 -

16

= V9 =

3

81 cm^.

36 cm.

Suggested Extension

The numbers 3, 4, and 5 have a special property, namely 3^+42 = 5^. Such numbers are called Pythagorean triples. Find other Pythagorean

Some

triples.

2

•

are

10; 9, 12, 15; 5, 12, 13.

6, 8,

USING A SQUARE ROOT TABLE

Teaching These Pages Introduce the use of tables to show values concisely.

For example,

in su-

permarkets the cash registers often have a table summarizing the sales tax posted on them. Thus the operator does not need to calculate the tax each time but refers to the handy chart.

To provide some background

V6

lies

between

3

and

for the square root table, point out that

2.

V9=3>\/6>\/4 = An

approximate value for \/6

table

is

is

—^—

or 2.5.

2

Note that the value

2.449.

Point out that a computer has calculated the square roots for

numbers

in the

^

all

the

in the table.

Chalkboard Examples

Use the

table to find the value.

=

1.

\/39

5.

-V21=

6.245

-4.583

=

2.

\/74

6.

-V53=

8.602

-7.280

3.

V98 =

9.899

4.

11

7.

\/80

=

8.944

8.

V28 =

=

-3.317 5.292

T115

Round

one decimal place.

to

=

\/35

9.

5.9

10.

\/86

=

9.3

Find the length of a side of the square correct

13.

\/T3

= =

4v'l3

9.8

- V67 =

12.

8.2

one decimal place.

to

3.6

Find the perimeter of the square correct

14.

V97 =

11.

to

one decimal

place.

14.4

13cm2

/^

-

/4

= 59 cm2

Suggested Extension the students find the length of a side of the square to one decimal

Have

place and then find the perimeter using that value.

Axl.l =

Now

\^ =

have the students find the length of a side correct to two decimal

places and then find the perimeter using that value.

4

X

7.68

Have

1.1\

30.8

=

\/59

=

7.68;

30.72

the student round 30.72 to one decimal place

and compare the an-

perimeter they found, 30.8. The difference will provide a useful source for discussion about adding numbers that have been rounded.

swer to the

3

•

first

IRRATIONAL NUMBERS

(pp.

352-353)

Teaching These Pages numbers are numbers that can be expressed as the two integers. Point out that the prefix "ir" in irrational means "not" and thus irrational means not rational. Have the students give other examples of words with the prefix "ir." (For example, irresponsible, irreleReview

that rational

ratio of

vant.)

Review that rational numbers have decimal forms that do not terminate, but repeat. Discuss the examples of bers on page 352.

or

either terminate irrational

num-

Chalkboard Examples Is the

number

rational or

is it

irrational? irrational

irrational

rational

1.

5.

1—

9.

Find the area

A

T116

6.

irrational

to

='7rr'^= 3.14

•

\/9

rational

one decimal place. 3^

=

28.26;

4.

y\2

8.

3.1515

irrational

-V5

V5

A =

28.3

7.

^/l5

irrational

rational

4

•

SIMPLIFYING SQUARE ROOTS

(pp.

354-355)

Teaching These Pages This section uses square roots of small perfect squares, such as to find the square roots of larger numbers.

Provide examples to illustrate the rule \/a^

=

\fa

•

4, 9,

16, 25,

y/b.

Chalkboard Examples Find the value. \/3600

1.

= V36 ^A00 =

60

•

V32 =

2.

VT6^ =4V2

3.

V^ =

\/9

Suggested Extension

Have the students investigate the prime numbers that are not perfect squares. 100

=

2

•

2

•

5

•

factors of perfect square as well as

125

5

=

5 -5 -5

Point out that the prime factors of squares occur in pairs and that write the square root by taking one

100

=

The students can use

member

Vm=2'5

2 -2 -5 -5

this

method

to

we can

of each pair.

=

lO

do Exercises 1-20 on page

355.

Objectives for Pages 356-361 1.

To

2.

To use

5

•

solve quadratic equations by taking the square root of each side.

the Pythagorean

Theorem.

SOLVING EQUATIONS

(pp.

356-357)

Teaching These Pages Review x^

=

tions

two square

roots,

25 and have the students

name

that 25 has

—5 and

5.

Introduce the equation

the solutions.

Point out that the solu-

can be found by taking the square root of both sides of the equation.

Chalkboard Examples

•

3

•/ ^ 3yV3

6

•

LAW OF PYTHAGORAS

Teaching These Pages

may have dealt with this topic in earlier courses. To review, draw diagram on page 358 on the chalkboard and have the students count the number of squares in each square region. Have the students studv the relationship among the numbers 9. 16. and 25. Thev will discover 9 + 16 = 25. Write this equation as 3- + 4- = 5- and relate it to the sides of the triangle and the squares drawn on each side of the triangle. Students

the

Chalkboard Examples Find the length of the third side of the triangle. 3.

?

4vl3

?

4.

A

wire

is

attached to a pole as shown.

Fmd

6\5

the length of the

wire correct to one decimal place. f2

c

= 16 + 64 = = v'^ = 8.9

80

8

4

Suggested Extension Discuss taking short cuts across corner

lots. Use the shown to see how much walk along AB and BC than to walk

dimensions of the comer further

it

is

to

lot

along AC.

Objectives for pp. 362-369 1.

To

2.

To multiply and divide

3.

To

4.

To simplify

T118

find the square root of a fraction.

rationalize the

radical expressions.

denominator of a radical expression.

radical expressions

by addition or subtraction.

m

m

7

•

QUOTIENTS OF SQUARE ROOTS

362-363)

(pp.

Teaching These Pages

^

\/a

Illustrate the rule

,

b

:^ 0,

with simple examples.

V^ . u

= V/74 =

,

2 but also

9

100

=

=

\/4

— = —V36 V9

/36

/

V

=

100

100

25

25

2 but also

25

2

9

_ ~

10

_ ~

5

Chalkboard Examples Find the square root.

—=

9

v/81

10

2.

9

^/9

1

36

3 1

1

4.

6

25

49

49

3^

'25/

\/25f2

5_^

3^2

100

VIOO

10

64

_ ~

5

7

aV3

5.

/225

225

V

15

Find the square root of

7.

/35

Express your result correct to one decimal

5.916

35

/25

25 •

35

25

place.

8

36

25

5

=

1.183

=

1.2

5

MULTIPLICATION AND DIVISION

364-365)

(pp.

Teaching These Pages

Show how

can be written in a different way to be useful in different + c) = ab + ac can be written ab + ac = a(b + c)

rules

ways. For example, a{b to

show how

to factor out a

In a similar

show

how

us

way we can

to multiply

y/ab

=

\fa

•

common monomial

factor.

rewrite the rules of the previous sections to

and divide with \/b

radicals.

can be written

^Ja

•

\[b

=

\fc^

can be written

V5

^Tb

Chalkboard Examples Simplify.

L V3

•

V27 =

\/f^ =

VsT =

9

2.

2\f5'A^/2

= %^/5~^ =

8\/l0

T119

3.

^96

3Vx'2lVx = S\Vx^ =S\x

4.

=

Vn

= 2V3

V8 '125

/T25

[5

/72x^

V72]

5.

50

9

•

= V2F = xV2

36

RATIONALIZING THE DENOMINATOR

(pp.

366-367)

Teaching These Pages

Have

the students find the value of

\A5 =

3.87

and

Vl =

one decimal place, using

to

V2

1.41,

15

.

3.87

V2 The

—VT5— =

2.1 A

=

2.7

1.41

calculation will take about 5 minutes of tedious work.

time that ate the

is

actually taken.)

method of

With

(Record the

this experience, the students will appreci-

rationalizing the denominator.

——

Introduce the method

\/l5

and have students

find the value of

again, recording the

v2

time they take. 15- y/2

Vl

Vl- V2

30

5.48

2

This calculation will take students only about

Chalkboard Examples Express

in simplest form.

=

2.14

2 1

minute.

=

2.7

amount of

A common error that should be pointed out y/6 = \/lOand3V6+6V3==9\/9.

if it

occurs

is

to write

V4+

Chalkboard Examples Express

in simplest form.

1.

3V6+2V6 = 5V6

4.

^/S

5.

\fl2y

11 •

2.

V4^

+ VYS =

3.

3V5+V5-3V3=4V5-3\/3

V9^

= 2V2 + 3\/2 = 5v^ - ^/3y - V25 3^ = 2 V3y - \/3y-5\/3y = -4^3^ = V4

+

V^

- V3y-

8V7-10V7=-2V7 •

•

3_>;

RADICALS AND BINOMIALS (OPTIONAL)

(pp.

370-371)

Teaching These Pages

—

Introduce this topic by having students consider the value of

we could

find the value

evaluate as shown. 1

may

Students

indicate there

we would

is

2

+

1

3.414

1.414

a simpler

is,

way

Point out

to find the value.

an expression that does not have a radical we would like to rationalize the denominator.

l-(2-

1

+ V2

(2

+

\/2)(2

To

+ v2

like to find

denominator, that

2

1

+ V2

2

that

2

\/2)

- v^)

2

- \^

2

in the

- V2

4-2

2

Chalkboard Examples Express

L 3.

in simplest form.

-

\/5(2 (2

-

=

V^)

3 \/3)(3

2

V5 - VSO

V2 -2)=6\/2-4-9\/6

2.

4V3(6 + V^) = 24V3

4.

(1-

\/3)(V3

-

1)

+ = -4

12

V2

Rationalize the denominator. 3

1

3

- V3 ^/3

1

_ V4

^3

+ V3

'

6

V3(l

+

V4)

-3

\/3+2\/3 -3

2

3\/3

-3

4 -

3(2

+ V5

VS)

= -6 -

3\/5

5

= -V3

Simplify. 8.

(1

+

V5)2

=l+2V^

+ 5=6+\/5

9.

(2V3-

3 \/2)2

=30 - 12V6 T121

Chapter

12.

Quadratic Equations

In earlier chapters the students have dealt with the graphs of linear equa-

The intersection of lines was related to the solutions of equations having two variables, and the meaning of function was introduced. In this tions.

chapter, the students erty: if fl^

=

solve quadratic equations using the zero prop-

first

^

then a

Q ox b

=

0.

Quadratic equations are then solved

using factoring, taking square roots, and finally using the quadratic for-

mula. These

skills

are applied to solving problems that require quadratic

The chapter concludes with a study of

equations.

the graph of this nonlin-

ear function, the quadratic function.

Objective for Pages 380-385

To use

1.

1 •

the zero property

ZERO PRODUCTS

and factoring

(pp.

to solve quadratic equations.

380-381)

Teaching These Pages Introduce this section by having the students complete each of the following examples.

4

3

Summarize that the product of two numbers, one of which is zero, must be Then ask the students what they know about the factors in the fol-

zero.

lowing examples.

5x

=

14^

y^ =

=

ky

=

if two numbers have a zero product, one of Extend this to equations with a binomial factor.

Students should conclude that the

numbers must be

zero.

Chalkboard Examples Complete. x{x

1.

If

2.

If (X

4)

-h

-

=

then x

l)(x -H 5)

=0

= _J_

then

or

jc

-|-

_2_ =

4

= _?

0;

or

^^ =

-

5)

jc-1;x +

0.

5

Solve. 3.

y(y

+

6)

=

^ = 0orv +

4.

6=0 y = -6

T122

3k(k

=

5.

k=0ork-5=0 k

=

5

(2x

+

8)(4x

-

16)

=

2x-H8=0or4x-16=0 X

= -4

x=4

Suggested Extensions

-

1.

Find _JL-

in (x

2.

Find

in (2x

3.

The

=

.)

the solution

if

.)(x

—

4) if the solution

solution to an equation

is

x

the form {ax

2

+

3)(x

-

+

-

b)(cx

=0.

d)

=— {4x

or

-

=—

jc

SOLVING QUADRATIC EQUATIONS

•

=

3 or

4 or

jc

jc

=

—7.

7

= —

Write the equation in

.

-

l)(3x

x

is

=

x

is

=

5)

382-383)

(pp.

Teaching These Pages

The

developed in

skills

this section

and

subsequent sections

in

will

be ap-

plied to solving problems that involve the quadratic equation (in

Section

7).

Have

the students find the product in each example. (x

-

4)(x

-

5)

=

x2

-

9x

(x

+

2)(x

-

5)

=

x2

-

3x

Point out that in this section

we

+ -

will solve quadratic

20 10

=0 =0

equations like

—

9x + 20 = 0, but the first step will be to factor them. Review the types of factoring before assigning the exercises, and remind students to always check for a common monomial factor first.

x^

Chalkboard Examples Solve.

+ 8x + + 6)(x +

x2

1.

(x

12 2)

= =

(a

a-S=0ora

x-f-6=0orx+2=0 X 3.

= —6

X

-5a -24 =0 - S)(a + 3) =

a^

2.

= —2

a

/

+ 12;; + 36 = + 6)(7 + 6) = y + 6=0 or y + 6=0 y = —6 y = —6

4.

=

+

3=0 a

S

= -3

m2 - 16 = - 4){m + 4) = m — 4 = or m + 4 = m=4 m = —4 (m

(y

Suggested Extension Solve. 1.

2/ +

3/

+

1

y

or

1

=

y = —

2.

1

2A:2

k

=

+

^ 3

_

3

=

or k

=

3.

1

6x2

_

13;^

+

6

^

2 3 x=jOrx=-

T123

3

SOLVING QUADRATIC EQUATIONS

•

(pp.

382-385)

Teaching These Pages Use the examples on page 384 side of the equation.

x^

X

—X =6 - =6.

into

jc(jc

to point out the

need

to write zero

A common error students make is — 1) = 6 and then concluding that

on one

factoring to change

x

=

6 or

\

Chalkboard Examples Solve.

-5x = (x - 8)(jc + 3) = X = ^ or X = —3 x^

1.

+ 2)iy - 13) = -50 y = 3 or y = ^

24

X

iy

-4 ^ -3 X

X

=

X 2 or X

+ = —2 1

Objectives for Pages 386-389

To

1.

4

solve a quadratic equation by taking square roots.

USING SQUARE ROOTS

•

(pp.

386-387)

Teaching These Pages 4, Chapter 11, in preparation which students will solve equations of the form (x + of- — c. Throughout the exercises, students should be encouraged to solve the equations by taking square roots.

This section

is

primarily a review of Section

for the next section in

Chalkboard Examples Solve. 1.

x2

X

5

•

= 36 = ±6

2.

X

= 12 = ±^/n

X

= ±2\/3

jc2

USING SQUARE ROOTS

(pp.

3.

a

388-389)

Teaching These Pages

Have

the students

compare the following equations. x2

/

= =

Use the above examples and

T124

36

^

(x

49

^^

(y

fl2

+ +

find the roots

1)2

3)2

= =

36

49

by inspection.

= 64 = ±i

4.

= 50 = 25 w = ±5

2m2 m2

Some jc

+

+ 3)^ = 16 as x + 3 = 16 and problem occurs, encourage those students

students will solve (x

= — 16.

3

If this

as 4^ before taking the square roots of

both

to write 16

sides.

Chalkboard Examples Complete. 1.

+

{X

2)2

+

X

2

= =

9

2.

-

(x

X

4)2

=

16

3.

+

(2x

-4 =^L

2x

1)2

+

1

= 25 = _1

Solve. 4.

{X

+ X

3)2

+

3 jc

X

= 16 = ±4 = -3 = —1

5.

-

(x

X

3)2

-

±4 or

3

X

= 12 = ±2\/3 = 3 ±2V3

6.

-

3(x

X

2)2

-

2

X X

1

= 12 = ±2

=2±2 =

or 4

Suggested Extension Students can solve any quadratic equation by the process of completing the square, illustrated below.

Solve x2 1.

2.

—

2x

—

2

=

0.

Write the equation with the constant

term

in the right

Add

to each

x2

—

2x

-

2x

=

2

=

3

=

3

member.

member

the square of

+

1

one-half the coefficient of x. 3.

Write the

left

member

as a perfect

(x

-

1)2

X

-

1

square. 4.

Take the square root of both members.

X If the coefficient

of x2 in a quadratic equation

lent equation in

which the

coefficient of x2

is

is

not

1,

first

= ± \/3 = ± \/3

find

1

an equiva-

1.

Students can use this technique to solve the quadratic equations in Section 6.

Objectives for Pages 390-399 1.

To use

2.

To use quadratic equations

3.

To draw the graphs of quadratic

the quadratic formula to solve quadratic equations. to solve problems.

functions.

T125

6

•

THE QUADRATIC FORMULA

(pp.

390-393)

Teaching These Pages So that students

them attempt

will appreciate the

6x2

The formula

Some

"power" of the quadratic formula, have by factoring.

to solve the following equation

is

+

i3;c

+

6

=

first Chalkboard Example. work with the formula if they record

applied to this equation as the

students

may

the values of a, b,

find

and

Chalkboard Examples

it

c for

easier to

each exercise.

Let

y =

the width;

-4) =

y(y

llXj

—

15)

The length

is

15

+

(y

7 — 4 =

the length.

165

=

(Is

cm and

possible to have a negative length?)

it

the width

11

is

cm.

The perimeter of a rectangle is 32 cm. The area of the rectangle is 55 cm^.

2.

Find the length and width. 21

+ 2w = / + w = /=

32 16 16

— w)w = w =

(16

The length

w =

use

is

-w 55 or

11 1 1

w =

5

cm and

the width

5

is

cm. (Point out

why we do

not

11.)

Suggested Extension

Two numbers

1.

The sum of

by

differ

4.

their squares

is

208.

Find the numbers. (x

2.

+

4)2

+x^ =

208

The numbers

are 8

A

m

garden

is

4

The diagonal of

and

12 or

longer than the garden

—8 and wide.

it is

is

—12.

20 m.

Find the length and width. (x

+

4)2

+

The length

8

•

x2

=

400

is

16

m

and the width

QUADRATIC GRAPHS

(pp.

12 m.

is

396-399)

Teaching These Pages

Have

the students

draw the graphs of the following equations.

y =X Discuss the graphs.

How

y

=

y = —x^

x^

are they the

same?

How

are they different?

In-

troduce the terms parabola and vertex. Introduce the meaning of a function as

it

relates to the

graphs of 7

=

j>c2

and /

=

—x^.

Till

X

Chalkboard Examples 1.

2.

+

+

Draw

the graph of v

What

are the coordinates of the vertex of

this 3.

=

parabola?

What

is

x'^

4x

4.

(-2,0)

What

the smallest value of y?

the largest value

of/?

There

is

no

value. 4.

How many

values of x

5.

How many

values

make 7 =

1?

two

=

0?

one

of/ make

a:

is

largest

"^

Basic

Algebra

Basic

Algebra Richard G. Brown Geraldine D. Smith

Mary

P.

Dolciani

Editorial Advisers:

Andrew M. Gleason Robert H. Sorgenfrey

Houghton

Mifflin

Atlanta

Hopewell,

New

Company,

Dallas

Jersey

Geneva,

Palo Alto

Boston Illinois

Toronto

The Authors Richard G. Brown

Mathematics teacher at The PhilUps Exeter Academy, Hampshire. Mr. Brown has taught a wide range of mathematics courses for both students and teachers at several schools and universities, including Newton, Massachusetts High School and the University of New Hampshire. He is an active participant in professional

New

Exeter,

organizations and the author of mathematics texts and journal articles.

Geraldine D. Smith

known

in both publishing and in mathematics Smith has done extensive work in developing and directing a wide variety of major elementary and secondary mathematics programs. She also serves as a special consultant for many

education for

Well

many

years. Miss

other educational materials.

Mary

P.

Professor of Mathematics, Hunter College, City Uni-

Dolciani

New

Dr. Dolciani is the author of many mathematics both elementary and secondary, and has developed programs of basic mathematics review for beginning college students. versity of

York.

texts,

The

Editorial Advisers

Andrew M. Gleason

Hollis Professor of

Philosophy, Harvard University. search mathematician.

National

Academy of

His

many

Mathematics and Natural is a well-known

Professor Gleason

affiliations include

membership

re-

in the

Sciences.

Robert H. Sorgenfrey

Professor of Mathematics, University of CaliforLos Angeles. Dr. Sorgenfrey has won the Distinguished Teaching Award at U.C.L.A. and has been Chairman of the Committee on Teaching nia,

there.

He

has been a team

member

of the

NCTM

summer

writing

projects.

©

by Houghton Mifflin Company part of this work may be reproduced or transmitted in any form or by any means, electronic or mechanical, including photocopying and recording, or by any information storage or retrieval system, without permission

Copyright

1980, 1977

All rights reserved.

in writing

No

from the publisher.

Printed in the United States of America.

ISBN: 0-395-27863-5

IV

Contents UNIT A CHAPTER

1

Working With Variables

Using Letters for Numbers 2 2 Parentheses, Order of Operations 3 Coefficients and Terms 8 4 Properties of Addition 10 5 Exponents and Factors 12 6 Properties of Multiplication 14 1

•

6

•

•

•

•

•

7

•

The

8

•

Properties of Zero

Distributive Property

Properties of

16

18

One

20 10 Introduction to Equations 9

•

22

•

•

1

Introduction to Inequalities

Reviewing Arithmetic Skills Career Notebook 27 Consumer Corner 28 Reviewing the Chapter 30

CHAPTER

2

24

26

Solving Equations

1

•

Solving Equations by Addition

2

•

Solving Equations by Subtraction

34 36

More Equations 38 4 Equations with More Steps 42 3

•

Solving

•

5

•

6

•

7

•

A

Mind-Reading Trick Combining Terms 48

46

Writing Algebraic Expressions

50

Applied Problems and Puzzles 54 9 Variable on Both Sides of the Equation 10 'Equations with Parentheses 60 1 1 Puzzles 62 Consumer Corner 64 Reviewing the Chapter 66 8

•

•

•

Cumulative Review

68

58

UNIT B CHAPTER

Positive and Negative Numbers and Negative Numbers 72 2 Inequalities and Graphs 74 1

•

3

Positive

•

3

•

Addition

4

•

Subtraction

5

•

Simplifying Expressions

6

•

Multiplication

7

•

Simplifying Expressions

8

•

Division

•

Solving Equations

9

76 78 82

84 86

88

90

10 Absolute Value (Optional) 92 Reviewing Arithmetic Skills 94 Career Notebook 95 Consumer Corner 96 Reviewing the Chapter 98 •

CHAPTER 1

•

4

Formulas

Perimeter Formulas

102

Area Formulas 104 Volume Formulas 108 4 Writing Formulas 112 5 Motion Problems 1 14 6 Cost and Money Problems 1 18 7 New Formulas from Old 122 8 Formulas from Mechanics (Optional) Reviewing the Chapter 130 Consumer Corner 128 2

•

3

•

•

•

•

•

•

Cumulative Review

VI

132

126

UNIT C CHAPTER

Working with Polynomials

5

1- Addition

136

2

•

Subtraction

3

•

Multiplying Monomials

138

140

142 4 Powers of Monomials 5 Polynomials Times Monomials 146 6 Multiplying Polynomials •

144

•

•

7

•

Multiplying at Sight

8

•

Square of a Binomial

9

•

Division with Monomials

10

•

11

148

152

154

156 Exponent Rules (Optional) 'Polynomials Divided by Monomials

12 'Dividing Polynomials (Optional)

Reviewing Arithmetic Skills 163 Career Notebook Consumer Corner 164 Reviewing the Chapter 166

CHAPTER

6

162

Factoring

1

•

Factoring in Arithmetic

2

•

Common Monomial

3

•

Using Factoring

170

Factors

172

174

4 Factoring Trinomials •

— Two Sums 176 — Two Differences

5

'Factoring Trinomials

6

•

Factoring Trinomial Squares

7

•

Factoring Trinomials

8

-A

9

•

Factoring the Difference of Squares

10

•

Many

Special Product

1

188

Types of Factoring

•

Reviewing the Chapter

Cumulative Review

180

182

84

More Difficult Factoring Consumer Corner 196 1

158

160

190

192

(Optional)

194

198

200

Vll

UNIT D CHAPTER

Graphs 7 Graphs You Often See 204 2 Graphs with Lines and Curves 212 3 Points on a Graph 1

•

•

208

•

216

4 'Solution Pairs 5

•

6

•

7

•

Graphs of Equations Slope of a Line 224 Functions

220

222

Reviewing Arithmetic Skills 228 Career Notebook 229 Consumer Corner 230 Reviewing the Chapter 232

CHAPTER

8

Equations with

1

•

The Graphing Method

2

•

No

3

•

The

Solution,

Many

Substitution

Two

Variables

236

Solutions

Method

240

244

246 4 • The Add-or-Subtract Method 250 5 Word Problems, Two Variables 254 6 Using Multiplication •

•

Cost Problems

256 Boat and Aircraft Problems (Optional) Consumer Corner 260 Reviewing the Chapter 262 7

•

8

•

Cumulative Review

viu

264

258

UNIT

E

CHAPTER

Working with Fractions

9

1

•

Fractions in Algebra

2

•

Simplifying Fractions

3

•

The -

4

•

Ratio

268

270

Factor 272 274 Problems Involving Ratio 1

5

•

6

•

7

•

8

•

Multiplying Fractions

9

•

Dividing Fractions

278 Proportions Applying

276

Proportion

10 'Add, Subtract

Renaming

280

— Same

Fractions

1 1

•

12

'Add, Subtract

13

"More

284

286

Denominators 290

— Different Denominators

Difficult Fractions (Optional)

Equations with Fractions 296 -Work Problems 298 16 Binomial Denominators (Optional) Reviewing Arithmetic Skills 304 14

288

292

294

•

15

•

Career Notebook 305 Consumer Corner 306 Reviewing the Chapter

CHAPTER

302

308

Decimals and Percents 1 Decimals 312 2 'Division with Decimals 314 3 'Decimals for Fractions 316 4 Equations with Decimals 318 320 5 Using Decimal Equations 322 6 Percents 7 Using Percents 326 8 Percents in Equations 328 9 Interest 332 10 Investment 334 1 1 Mixture Problems 336 Consumer Corner 340 Reviewing the Chapter 342 10

•

•

•

•

•

•

•

'

•

Cumulative Review

344 IX

UNIT

F

CHAPTER 1

•

2

•

3

•

4

•

5

•

6

•

7

•

8

•

9

•

Squares and Square Roots 11 Square Roots 348 Using a Square Root Table 350 352 Irrational Numbers 354 Simplifying Square Roots 356 Solving Equations Law of Pythagoras 358 362 Quotients of Square Roots 364 Multiplication and Division 366 Rationalizing the Denominator

368 Radicals and Binomials (Optional) 1 Reviewing Arithmetic Skills 372 10

•

Addition, Subtraction

•

373 Career Notebook Consumer Corner 374 Reviewing the Chapter

CHAPTER

376

Quadratic Equations 1 Zero Products 380 382 2 Solving Quadratic Equations 384 3 Solving Quadratic Equations 12

•

•

•

Using Square Roots 386 388 5 Using Square Roots 390 6 The Quadratic Formula 394 7 Problem Solving 396 8 Quadratic Graphs Consumer Corner 400 Reviewing the Chapter 402

4

•

•

•

•

•

Cumulative Review

404

370

EXTRA PRACTICE EXERCISES 406 TABLES 436 CUMULATIVE REVIEWS 438 ANSWERS TO SELF-TESTS 449 GLOSSARY 452 INDEX 457

XI

Symbols

Diagnostic Tests in Aritiimetic 1.

2.

What

is

the

denominator

in

—? 3

The area shaded

3.

in figure

D

is

repre-

sented by which fraction in row E?

e15' 15' 15' A

D

1

4.

Which of

the figures below repre-

sents the fraction

F

—'^^

Write the fraction represented by the

5. .

set

diagram shown below. ^

-

6.

711-^ 12-11755 88 54

Fractions

— Addition 3

1.

7.

Fractions

3- + 4- = ^L^

2.

3— 12

3.

— Subtraction

\4

1.

9

1

- + - = ^L5

i±

3^

2.

7

3.

.1

,1

Fractions— Multiplication

e:

4.

9.

Fractions

'

-

5.

6

.7

1

^4

11

3

^ 9

^

15

3— V4— ?2 .2 3.^X4-

^4

,

= 5.1^X2^ 5 2

'"'^To

Tpia

21

2—gX^-^4 V — — "^4

21

14^

6.

3

Z.

29

-J

= -^To 44x^ 6 5

3

9

6

1— V — — "^27" 1.3X^-^27

9 4A

5.

2

5I

i5 8.

- + - = _IAO

7I

4.

8

1

4.

— Division 1

5

1 '

5.

——= '9

45

2

-4=^^ 41

5-

_ 32°"^

8

10.

?

-

6— — = '"4 6.

?

^ J_ *6"2

3 J_

8

5-- 5-=

9

?

3^

=:

4

?

A^A— *9'7 5

?

9

28

9

6

8

^32

Decimals 1.

Write the decimal which represents "two and

2.

—5— = _J_ 1000

5.

Round

(Decimal)

3.

0.206

=

206 1000

0.005

thousandths." 2.0O6

six

(Fraction) ^_ 103

4.

3- = _i_

(Decimal)

3.80

5

500

40.5656 to the nearest tenth.

6.

1.033

+

0.1

+

10.066

=

_^11-199

40.6 7.

11.

856.175

-

20.05

310.1

^_ 836.125 =

71.60

8.

-

5QQ42

+

Percents

735.02

9.

^3-96

10.

6639

7.709

X.0031

727.311

^K^^9

625.98 2.

1.

0.69

= _J_7c^^

3.

- = -J-.7c 33^3 3

% 240.5

6.

289%

=

?

»

38-% = _?_ 2

9.

89

(Mixed numeral

4.

6- =

_^% 650

2 in simplest

form)

Too

1

7.

.07)21.707

11.

(Decimal)

8.

25% of 23 = ^^5.75

0.385

of 20 = 8 _^% 40

10.

20% of

= 340 _^ 1700 XV

Why

Study Algebra?

You've probably asked

this

question at least once. Here are a few

reasons to think about.

Through algebra

you'll learn to organize

and

ex-

press your thoughts concisely. That's important in

today's world.

You know

worked with a

calculator, for

this is true if

you've ever

you have

to

have a

plan for solving the problem before the machine

can begin

to help

you

at all.

You'll get a chance to view from a different angle all

those concepts you've already learned in math.

This will help you firm in your mind what math is all

about and

Algebra

and

will

how

it

works.

prepare you for further work in math

the sciences, if this

is

for you.

Also, hundreds

of careers rely upon a good, solid math back-

ground. Algebra can help give

this to

you.

Once you've learned basic algebra skills, you'll probably find yourself using some of them to solve many problems in your everyday life.

XVI

Here's what you'll learn in this chapter: 1.

2.

3.

4.

To find the values of expressions with variables. To simplify expressions. To find the value of expressions with exponents. To check whether a given number is a solution of an equation or inequahty.

Chapter 1

Working With Variables

1 •

Using Letters for Numbers

Imagine a calculating machine that adds it. It might look like this.

INPUT

5 to

any number you put into

When we

We

call

;>r

use a letter to represent a

+

5

and y

number we

—

3 variable expressions.

X

+

call the letter

a variable.

Here's a hint to help you

remember

,/-3

5

A

find the value of

EXAMPLE

1

If «

an expression

=

10, find

EXAMPLE

2

If

jc

=

and

14

in the following way.

+

the value of «

«

;;

=

+

3,

2

= =

10

+

2.

2

find the value of

x

- /.

11

Suppose you had a machine which multiplied by

is

.

12

=

There

another way to show multiplication, too.

7.

It is

with a dot. You'll

see this used very often.

7x4 < variable

is

means

the

same

as

and the dot are usually

In algebra, the times sign

>

left

7

•

out

use^.

EXAMPLE

3

If «

=

2nk

If

the value of the expression.

-

^

+

6 10

31. /7^ 96

1

2 • Parentheses, Order of Operations When you

have several numbers

to add, subtract, multiply, or divide,

helps to show which operation to do

Here we add

+

(3

Multiply

first.

5)-2

7

-

16

When

there are

Step

1:

Step

2:

Subtract

first.

I (2

•

and

divisions in order

EXAMPLES

from

multiplications

all

additions and subtractions in order from

how If

6

+ 5-2 +

+

5

V 10

=

-

+

5

left to right.

left to right.

2-3

V

8-6

13-6

V

when

to use these ideas

X

2)

steps to follow.

13

see

+ 11

all

3

let's

-

1

3

Now

(8

5

no parentheses, there are only two

Do Do

first.

I

+

5

3)

7-6.

8-2

it

Parentheses are used.

first.

4,

there are variables.

find the value of 2(x

+

+

and 2x

5)

5.

This means that you multiply here.

7 2{x

+

5)

Vi Classroom Practice

~1^

= 2(4 + = 2-9 = 18

2x

5)

+

5

= 2-4 + = 8 + 5 = 13

5

f/M/M/M/M/M/M/M/M/M/M/M/M/M/M/M/A

Find the value of the expression. 1.

If

5.

(7

x

-

=

2(jc

3)-2

3, find

+

8

2.

2

+ 3-4

14

+ ^ _

3.

3

7.

3(x

1)6'

-

4.

7

8.

3x

(3-2)

the value of the expression.

5) 16

6. 2jc

+

5

1

-

1) 6

-

1

8

1

)

,

^/i

Written Exercises

W/M/M/M/M/M/M/M/M/M/M/M/M/M/M/M/A

Find the value of the expression. 1.

(16

-

4) -2 24

4.

(16

-

4)

7.

3-4 + 517

10. 2

^

•

-

2

1

-

(3

1)

3

3

13. 8

+

(2-3)14

16.

4

+

8

19.

If

X

=

4, find

20. If

X

=

21. If

b

-

+

2.

16

-

5.

(8

+

8.

6

+

4) 8

•

-

+

(4

2) 2

4 38

+ 7-431

11. 3

+

14. (2

2 8

(4-2) 8

-

(4

+

2-2 +

+

17. 6

3)

25

1)

29

the value of 2(jc

+

6)

and 2x

+

6. 20,

7,

find the value of 5(x

+

1)

and 5x

+

1.

=

9,

find the value of S(b

-

4)

and Sb

-

4.

22. If 7

=

8,

find the value of

3{y

+

2)

and 3y

+

2. 30, 26-

a

=

5,

find the value of 4{a

—

2)

and

-

24. If 5

=

3,

find the value of 6(5

23. If

Tell

what

is

+

4)

4xy

xy

-

9.

-9

lab

— xy; = 5a

Ixy

-\-

f/M/M/M/M/M/M/M/M/M/M/M/M/M/M/M/A

Complete the statement. 1.

In the expression 5x the coefficient of

2.

In the expression 6x

3.

In the expression Aa

understood

to

x

?^

is

+ ly the coefficient of x + b the coefficient of ^ is

.

?

is

^

not written.

It is

be -JJL.

Simplify. 4.

X

+

X

S.

y +y

+y

r/A

6.

+

lab

3ab

Written Exercises

1.

3a

+

a

+

+

3

2

4a + 5

Sab

3y

2x

W/M/M/M/M/M/M/M/M/M/M/M/M/M/JF/M/M

Simplify.

A

1.

5.

9.

+

4x

-

6y

2x 9y 3«

2.

6.

^

5a

4-

6xy

+

4a

\1.

10.

8^-25-2

21.

3a

+

6x7 -^

14.

25. 7 y

a-4

18.

4a - 4

-

xy

+

5x/ +

-

2y

29.

3«

if

+

3 not possible

33. ar

+

4ar 5ar

3a

2jcy

1.

2a

+ +

6b

+

3b

-

3« 2a

+ +

b 86

a - 2 4a - 2

5a

+

11.

4

+

7a

15. 3

+

19.

6

-

22. 9c

1

-

2c

+

+ 6 7a + 6

+

5y

+

+

4y

4c

23.

5d

6y

1 1

9x

26. 2jc

+

6d

x

+

4y

If

27. 7aZ)

-

4s

+

3y

+

12.

2x

+

4x

+

5y

-

-

2ab

31.

16.

3a

+

+

3a

+

20.

7 + 2y -

+

3b

+

35. 8« 2 not possible

3

7

3/ - 7

3d

24. 8 y -f 3 y

+

5/?

-

^

28.

cd

-}-

4c(^ 5ct/

32.

2a

-

2a - 2a

+

5a 7a

36.

4v Ty

possible.

3a^

1

6a + 3

not possible

34. r 4s not possible

4y 12/

6x + 7

5a6 + 56

you cannot simplify, write not 30. 4^

5y

8e

om tvb uuuau.

THE eBLATlVS SPFEO-A VAUUe THE" FIFTH *^

POSA/Bfi K>\THF1?

THE WIZARD OF ID

TTf

TFUM CUSED.

iA4F«fiSlPtgI^F THAT VVEPB- TlJUe, VV& WOULD

BE

IN/

•V:fTAU

Cy\KKNE55

AT

by permission of John Hart and Field Enterprises,

THIS

Inc.

I 13

6 • Properties of Multiplication Multiplication has

some

properties which are like the properties of

addition.

THE COMMUTATIVE PROPERTY In multiplication, the order of the factors

(AT

•

3

=

3

•

difference.

Algebra Example

Arithmetic Example 2

makes no

ab

2

=

ba

THE ASSOCIATIVE PROPERTY In multiphcation, the grouping of the factors

•

3)

•

4

=

2

•

(3

(ab)c

4)

•

=

a(bc)

the addition properties, you do not need to memorize the names. Just be sure you can use the properties. They can make your

As with work

easier.

EXAMPLE

EXAMPLE

1

2

Evaluate 5

SimpUfy 2

•

•

82

•

2.

5

•

3

Simplify

•

{5x)

4

•

2 •

(5

82

•

2)

•

82

10-82 820

= =

(2

•

5)

•

X

\0x

= =

(2

•

4)(r

•

r)

8r2

Simplify 6 -a- a- 5 -a. 6

14

5

(2r)(4r).

(2r)(4r)

EXAMPLE

= = = =

2

•

{5x).

2

EXAMPLE

82

'

a' a' 5

'

a

= =

difference.

Algebra Example

Arithmetic Example (2

makes no

(6* 5)(a

30^3

'

a- a)

Vi Classroom Practice

f/m/M/M/M/M/M/M/M/M/M/M/M/M/M/M/k

Find the value of the expression. 1.

17 -25 -4

1700

Show how you use

the properties.

7 • The Distributive Property Jan at

is

buying two

$3 each.

Her

2($1

+

felt-tip

$3)

at $1 each and two books can be figured in two ways.

pens

total bill

f/M/M/M/M/M/M/M/M/M/M/M/M/M/M/M/A

Vi Classroom Practice

+

=

1.

Check

2.

Is

3.

Once you've checked to see you should be able to tell

——

2(4

to see if 3(2

•

=

3)

(2

•

4)

•

(2

1)

•

(3

•

2)

+

(3

•

true

1) is true,

3) true? no

the statement in Exercise 2

if

2{ab)

if

=

(2a)

•

(lb)

is

true.

is

true, Is it?

no

True or false? 4.

3(a

-\-

=

b)

3a

+

3b

5.

3(cd)

=

(3c)

•

true

(3^?) false

State the expression without parentheses. 6.

2(x

+

7.

5)

2x + 10 10.

^/i

3(x - 4) 3x - 1 2

1.

f

11. a(a a2

Written Exercises Check

to see

3(4

+

5)

3(x + 2) 3x + 6

if

(3

3)

- 3a

4(a + 1) 4a + 4

12. bib b^

-

3)

- 3b

9.

5(7

+

a)

35 + 5a 13. y(2

-

y)

1y - y^

W/M/M/M/M/M/M/M/M/M/M/M/M/M/M/M/M

the statement

=

-

8.

is true.

n

8 • Properties of O

r

• Any number times

is 0.

7-0 = •

No number

fl-0

can be divided by

=

0.

• Zero divided by any number (except 0)

0=0

0=0.

6

to a

number

+

=

gives that

6

n

Adding and subtracting A2

+

5

—

5

the

=

same number

+

«

1

Let X

=

4.

Find the value of 9(x

%x -

EXAMPLE

2

Simplify 2

+ 2

3x

-

4)

= 9(4 = 9-0 =

=

= =

King Features Syndicate 1974

18

-

4).

4)

2.

+ 3;c-2 = 3x + 3jc

3x

+

number.

^^ -

Let's put these ideas to work.

EXAMPLE

Divisions are often written with a bar.

6

Adding

is 0.

2-2

«

is

like

adding

0.

T/i

Classroom Practice Find the value

if

possible.

If not,

say impossible.

8-0

3.^

2.0

5-0

1.

f/M/M/M/M/M/M/M/M/M/M/M/M/M/M/M/A

9

7 impossible

Let

jc

=

5.

Find the value of the expression.

5.0

6.

jc

X

f/A

Find the value

if

8.

35(x

2.

0-31

3.

.

7.

impossible

9.

Let X

10.

Let X

IL Let 12.

JC

Let X

Simplify.

= = = =

1.

2. 3.

0.

Find Find Find Find

5)

If not, write impossible.

18

+

35

-

18

4.

16

the value the value

5

•

— 1). of (jc 2) ^ 4. of {2x - 6) ^ 4. of x ^ 3.0

the value of 5(x the value

-5

8.

impossible

+ 16

35

16

-

11

W/M/M/M/M/M/M/M/M/M/M/M/M/M/M/M/A

possible.

12-0

5

7.

5

Written Exercises

1.

-

+

16

•8-0

.

9 • Properties of

1

Whether you add to a number or multiply the same result the original number.

number by

the

you get

1,

—

original original

The second sentence above

+ x

number number

= =

1

states a property

original

original

of

number number

1.

• Multiplying a number by

1

number.

gives that

6*1=6

ft'

• Any number (except 0) divided by

=

\

n

itself is

1.

^=\

1=1 4

1

1.

Remember, you can

Any number

divided by

write a division with a bar.

1

is

that

6 Q -=8

number. n -=«

V. Here

are a couple of examples to

EXAMPLE

1

Simplify 8

show

these properties.

-f

=

EXAMPLE

2

Simplify 4x

-

X

4.

This gives you 4;c

^4 =

4x

4

= 20

X

4 4

—X,

or

jc.

Vi Written Exercises Let X

A

1.

^

= 1

4.

W/M/M/M/M/M/M/M/M/M/M/M/M/M/M/M/A

Find the value of the expression. 2. 4a;

16

3.

(4

-

x)

•

1

1

6 Simplify.

10 • Introduction to Equations STRIKE THREE!

THE GAME 15

NEVER OVER

UNTIL THE LAST ^

MAN

(« OUT.'

ICAN5TILL BE

A

HERO.,

1959 United Feature Syndicate, Inc.

Consider whether the following statements are true or

Hank Aaron Charlie

He

is

is

Brown

a baseball hero is

true

a baseball hero

false

maybe

a baseball hero

(It

The sentences below 6

true,

maybe

+

4

=

10

true

8

false

=

1

maybe

-\-

1

(It

true,

maybe

false

depends on what x

is.)

These number sentences are called equations. Equations always have an = sign, but are not always true. Is

X

+

2

= X

7 true

+

2

=

if

7

x

=

6?

false

depends on who he

are like the ones above.

5+1 = X

false.

is.)

8

Vi Classroom Practice Tell

,

r/A

in color is a solution.

7 -

=

jc

+

3

=

9 6

7 or 6?

2.

3.

4

+

X

=

10 6

9 or 6?

4.

3n

=

\2 4

5.

3x

8 3

2 or 3?

6. 4jc

-

12

7.

m-

16 or 34?

8.

5a

=

35 7

-

=

1

=

9

25 34

Written Exercises

+

=123

9

3.

3n

=

36 12

5.

3«

+

7

7.

5(r

9.

;c

=

+

2)

9

=

+

5a

-

13. n^

=

13 2

=

30 4

4x 3

=

1

3fl

+

7

1 1

18 or 4?

18

4 or 9?

=

7 or 8?

20 8

30 or 7?

W/M/M/M/M/M/M/M/M/M/M/M/M/M/M/M/M

which of the numbers shown

.1.x

11.

B

which of the numbers shown

1.

Tell

A

f/M/M/M/M/M/M/M/M/M/M/M/M/M/M/M/A

9 5

25 5

in color is a solution.

-

=

3 or 21?

2. jF

12 or 33?

4.

4x

=

24 6

2 or 4?

6.

2a

-

6

3 or 4?

8.

7(5

7

-

or 4?

10.

2b

5, 6,

or 7?

12.

4(a-

0, 5,

10?

14. a^

15.

Write an equation which has 9 as a solution.

16.

Write an equation which has

17.

Write an equation which has

18.

Write an equation which has no

=

4 or 5?

b

5 or 6?

21 6 -\-

1 2

= a+

5)

=

6 or 20? 45

=

5

-\-

\7

\^

Answers

may

Find at least one solution. You

=

3)

2, 3,

8 or 22?

15 22

to exercises

1, 2,

or 3?

6, 7,

or 8?

0, 1,

or 2?

15-1

vary.

as a solution.

1 1

as a solution.

number

may need

to try

as a solution.

many numbers

for

some

exercises. 19.

JC

+

5

=

23.

Zj

-

4

=

27.

C

31.

7 +

1

3a 7 •

=

13 8

2 8

41 40

20.

7 -

24. b

28.

X

^ +

=

17

5

5

20 37

= 525 =

25 20

21.

=

5«

30 6

-0 = Answers may

25. «

29.

x

+

=

3

=

42 2

32.

2m +

9

=

21 6

33. 2/?

+

4

=

6x

26.

x

+

5

=

116

30.

m -

9

=

27 36

34.

x^

36 6

vary.

+

x

Answers may

=

22.

3

vary.

4

=

x

0,

1

23

II Not

all

•

Introduction to Inequalities

number

sentences are equations.

Some number sentences are We call a number sentence

statements that two numbers are not equal. like this

an inequality. Here are some examples of

4

is less

than

7.

inequalities.

f/i

W/M/M/M/M/M/M/M/M/M/M/M/M/M/M/M/M

Written Exercises


.

or

Al.

6?7


Tell

which of the numbers shown

in color are solutions.

9.

jc

13.

;c

>

14 20

15.

«




17.

X

< 9 4,

6

4, 6, 9,

10

18.




7

+

27. 83

5

•

-

29. (24 For

4) ? 7

more

-

•

8)

•

Tell

10. c

0, 1, 3,

7

12.

7

> 9 10,

14.

y

>a+

4«

7

4, 6, 10,

1

22. 4jk

0, 3, 5,

14

-

9

7

3, 5,

3^^

2, 5, 7,

2

13 7

6 5

4, 3, 6,

2

2, 4, 5,

7

1, 0, 2,

5

or =.




150 calories in a can of cola

200

->

200 calories in a piece of pizza

number of number of

Check:

r/A

Written Exercises

A

1.

An

= =

X

Answer:

calories in 3 pieces of pizza calories in 4 cans of cola

=

=

3

4

150

•

•

200

=

= 600 600 /

W/M/M/JF/M/M/M/M/M/M/M/M/M/M/M/M/A

apple has 30 more calories than a peach.

Five peaches have as

How many

many

calories as 3 apples.

calories are in each?

peach: 45 calories; apple: 75 calories 2.

A

donut has 50 fewer calories than a glass of milk. Four donuts have as many calories as 3 glasses of milk.

How many milk: 3.

200

calories are in each? doughnut: 150 calories

calories;

One number

is

7

more than another.

Twice the larger is 22 less than 4 times the smaller. Find the numbers. 18, 25 4.

One number Five times

is

t+ie

5 less

Find the numbers. 62

than another.

smaller 7,

number

12

is

1

less

than 3 times the larger.

!

5.

Maria has twice as much money as Paul. Paul has $8 more than Rocky. Together they have $104.

How much 6.

Rich

is

Ruth

is

B

7.

8.

Paul: $28, Maria: $56; Rocky:

$20

3 years older than Carla.

twice as old as Rich.

Their ages

How

does each have?

old

is

total 33 years.

each person?

Caria: 6 yrs. old; Rich: 9 yrs. old; Ruth:

Warren is 14 years older than Chuck. Next year Warren will be 3 times as old How old is Warren? 20 years old Francine

is

as

18

yrs. old

Chuck.

6 years older than Carol.

Last year Francine was three times as old as Carol.

How C

9.

See

old

if

is

Carol? 4 years

you can solve

msBsn

this

old

problem.

'A

MAN HAS A PAU6HTERANP

A

IS

THREE VEARS ." OLPER THAN THE DAUGHTER. SON.. THE SON

1972 United Feature Syndicate, Inc.

For

more

practice, see

The man

is

THE MAN WILL &E OLP A$THE PAU6HTEK IS N0W,ANP in TEN i/EAfiS HE WILL 5E fOmiEH 4EAK5 OLPEKTHAN THE COMBlNEP A6E5 OF HIS CHILPREN... WHAT IS THE MAN'S PRESENT A6Er" IN ONE VEAR SIX TIMES AS

41 years

I'M

SORRV,

kJE

ARE LWA5LE TO

COMPLETE Y0i;RCALL..PLEA5E OIECK

THE NUMBER ANP PIAL A6AIN

old.

page 411

iSELF-TESTi— Solve. 1.

4.

3x

=

48

Lou has

—

X 12

twice as

2.

6(a

much money

-

3)

=

12 5

3.

4(m

-

3)

=m +

15 9

as Jo.

Jo has $11 more than Sherry. Together they have $89.

How much 5.

does each have?

Jo: $25; Lou: $50; Sherry:

Al has twice as much money as Vic. Vic has $5 less than Connie. Together they have $125. How much does each have? Vic: $30;

Al:

$14

$60; Connie: $35

63

S ER CCRIVIER f ^^T

^^j

Metric units such as the milligram (mg), gram used to measure quantity.

The

prefix milli-

means

1

-^

->1000 milligrams

1000 grams

Here are some objects usually measured

Here are some objects measured

V 64

in

=

1

=

1

(g),

and kilogram

gram

kilogram

-4

f/m/m/m/M/m/M/M/M/M/m/m/m/m/M/m/k

—

—

2 • Inequalities and Graphs The number X

-

line

3

X

= =

is

useful for picturing a solution of an equation.

-3-2-1012345

2 5

Think about inequalities. Sometimes there are an unlimited number of solutions. Take the inequality x < 8.

Some

More

In order to

solutions: 7, 6, 5, 4, 3, 2,

solutions: all

show

0,

1,

numbers between

all the solutions,

we make

—1, —2, —3,

and

8

7,

.

.

.

means "and

so

on"

between 1 and

6,

\

a graph like

this.

x

x

H-4-3-2-1 ^ EXAMPLE

=

5

2

the solutions of

\

3

-2.

\

\

1

x

2

1

< -

1

\

h

2

3

4

2

3

4

1.

^

-4-3-2-1 74

1

3

4

5

r/i

W/M/M/M/M/M/M/M/M/M/M/M/M/M/M/M/M

Written Exercises

Graph the given numbers. Write an

Sample

A

-2,3 > -2

1.

-2 -7

8. 0,

-4

-4
-4

number

the solution of the equation on the

Graph

-6

-6 < -1; -1 > -6

7.-1,-7 -7 < -1; -1

M

I

-1,

3.

> -5

-5
-7

_i; _1

4 4

'

-1

-7,

6.

.7

-3,1

compare them.

inequality to


^

12x« 17.

11.

(2x)(2x2) 4x^

23. (-;c4)(-3jcyz2)

24. (-6a^b^)(abc^)

3x^yz'^

25.

(a^b)i-5a^b^)

26.

(-x3)(-5x2y)

27. (;c73)(_2x3y2)

28. (-a^b^){a^bc^)

-2x4/5

141

4 • Powers of Monomials Study the examples below and see rule of exponents.

(y^)^

To

find the

can discover another important

if you

= y^

= y^^

power of a power of a number, multiply the exponents. (x^f

=

-y^ 'y^

=

EXAMPLE

1

(x^f

EXAMPLE

2

(ySy -^8-2 _^16

x4-3

Suppose you have an expression

=

x^^

jci2

like (xy)^.

You can

rewrite

it

following way.

(xyY

Make

=

ixy){xy)

a note of this rule of exponents.

(xyT

EXAMPLE

3

EXAMPLE

4

(ab)^

=

=

x"/"

a^h^

WARNING! 2^2 la'

2^2 and {laf are not the same.

^

(2^)2


xY equal 3^2^

2;>c^,

?

times. 2

^Jq^^

j^

equal

9a: 2?

or does

it

equal

Sjc^? Sx^

qj-

Qx^

Simplify. 5.

(^4)2 38

9.

(X2)3 x«

6.

(C2)5 cl«

10. (2a)2 4a2

7.

11.

(«6)4 a^b*

(4«2)2i6„4

8.

(5^)2 25a2

U2. (-3x7)2 9xV^

r/i

Written Exercises

W/M/M/M/M/M/M/M/M/M/M/M/M/M/M/M/A

Simplify.

A

1.

(;c2)3

5.

(C^)

3\5

2.

3\4 (^3)

6.

4M0 (n4) ,40

9.

(

5 • Polynomials Times Monomials Now

that

you can multiply monomials, you can put the

distributive

property to work.

You know

3{a

1

+

3b)

= =

(3

2

-\{2x -3y)

EXAMPLE

3

a(a^

EXAMPLE

^/i

4

y

+

2qb

-2x(9x3

Classroom Practice

+

-

3a

EXAMPLE

+

5(3

same way

In the

EXAMPLE

that

+

a)

+

5(3x

+

{3

-

4)

+

=

4y)

(5-3)

+

= =

3x)

+

+

207

+

(a

(5

•

\5x

(5-4)

(5

•

4y)

3b)

9b

= {-\'2x)-{-\' 3y) = -2x + 3y

b^)

3x^

= =

+

(a

a^

x)

•

a^)

+

+

(a

•

+

2a^b

2ab)

•

b^)

ab^

= {-2x'9x^) + (-2x'3x^) + {-2x'x) = -18x4 - 6^3 - 2x2

f/M/M/M/M/M/M/M/M/M/M/JW/M/M/M/M/A

Multiply.

L

2{a + 4j 2a + 8

2.

3(x + v) 3x + 3y

4.

x(x-2)

5.

«(«

x^

— 2x

a2 8.

+

3.

2b)

6. 2c(dz

_

2c)

9.

x(x2

-5a2 + 10c 10.

-x(x2

-

2x

+

4)

IL x2(x2 x^

144

-

+

b)

2ac + 2bc

+ 2a6

-5(«2

-l(2x + 3y) -2x - 3k

3x 3x3

X^

+

+

1)

x^

12.

+ +

2x

+

4)

2X2

+

4;f

a2^2(_^ _ fy^ -3^62 - 32^3 1

^/i

W/M/M/M/M/M/M/M/M/M/M/M/M/M/M/M/M

Written Exercises Multiply.

A

1.

5.

2{x + 4} 2x + 8

-6(«

+

- 6n -

9.

2m)

+

4a(a

2b

19.

22.

+

23.

-

2jc

4.

5(^2 _^ ^j Sa^ + 56

-

Z?)

8.

x{x

- 1 5x2 -

+

5jc

+

2v)

12. 2;c(3x

^

Qx/

Bx^

^2)

-3c(2c2 _6c3 -

+

2x

+

ab{a^

18.

a^b

-

4c

I2c2

1)

+

+

z

/

laZ?

2a^b^

-

1)

- ab

-y\y^ -

2y2 + 4y) -/5 + iy4 _ 4-pi

21.

5) 15c.

-

- 2x

- \(2x + y + z) _ -

15.

4x2

_ 2v + 1) -4/4 + 8/2 _ 4j^^

3y)

+ 3xk

+

-4v(y3

20.

+

x^

-5;c(3jc

11.

-4 - 20x -

17.

4)

- 8x

y)

- ab

a^

4Z))

-4(1

14.

3)

-Jc2(x + 2jc2) -x3 - 2x4

A

b^

- 2a^b + 4ab^

2x3 - 4x2

is lOn centimeters long by (n + 6) centimeters wide. area as a polynomial. (lOn^ + eon) cm2

rectangle

Write

B

-

2x(x2

a(a

7.

b"^}

-

-«Z)(2a

10.

Z))

+

4x + 4y

-5a -

4a2+8a6+12a 16.

3. 4(jc

- l(5a +

6.

2m

- ca - cb

13.

b)

3a - 36

1

+

-c{a

\a -

2.

its

You have

collected (3«

+

1)

dimes.

What

their value in cents?

is

(30/1

+

1Q) cents

Solve.

24. 4(2« 26. 5;c

+

3)

-

2

-

2(2x

-

6)

-

+

28. 2i5x 30. 3(;c

-

4)

32. 6(1

-

3x)

'SELF-TES

+

3(n

-

=

+

6)

=

3(2;c

-

4)

2(2x

-

1)

2(2x

+ +

1)

-3,

25.

10

=

=

5)

,

4 2

=

^r

_^

40 -2

'

+

-(«

3)

-

27. (2^

-

3)

29. 3(1

-

2a)

^3i. 2(«

-

6)

33. 4(2a

-

+

3)

2(n

(7

+

(6

+

6)

-

5(2«

-

+

2(«

7)

=

2^)

=

-lU

63 72

= -7

+

4)

=

32 2

-

8)

=

22 3

1

H6

•

Multiplying Polynomials

To multiply by

a binomial

you

also use the distributive property.

use a vertical form. Your work

is

«

+ +

n^

-\-

n

+ +

3«

n

«2

1

3

Here are a few more examples

EXAMPLE

1

X X x^

x^

2

2y

+ + +

-

n(n

4«

Be sure

+ +

+

3(«

5x

5

14 14

1)

that like terms are lined up.

to study.

x(x

-

+

3

+

7)

Ix

2x

1)

3

7 2

Let's

a lot like multiplying in arithmetic.

-2{x

+

7)

5

2

f/M/M/M/M/M/M/M/M/M/M/M/M/M/M/M/A

Vi Classroom Practice Multiply. 1.

+

{x

4.

7.

{x

f/A

-

3)(;c

x2

- 2x -

+

{2a

5.

5)

3/)2

+

5){x 2x2 - X

8.

1

-

1

-

+

(a

6.

5) 5

+

4)(5jc

0x2

4)(a 1) + 3a - 4

a2

6){2a 4) + 8a - 24

2a2

1)

8x - 4

-

+

6){n 3) 3n2 - 3n - 1 8

11. (3«

2)

- 10

Written Exercises

-

{2x

+

1)(a:

-

+

{a

3.

7)

+ 11x +

2x2 3Z))

+

2)(n

+ 9n + 14

+

(2jc

1

+

Z>)(«

+

(«

/)2

+ 7a6 +

-

{2x

2.

4)

+ 6x + 8

+ 2a2

10.

+

2){x

x2

12.

-

(4;c

+

\){2x

-

8x2

3)

0x - 3

1

W/M/M/M/M/M/M/M/M/M/M/M/M/M/M/M/A

Multiply 1

+

X

4.

7.

+

3)(«

+ 8a + 15

-

;c

-

X

B

25.

6

+

^)(«

14.

+

n

b)

-

/,2

+

2)(x2

m—

(x

+

l)(jc

(«

+

+

a/)

(y

-

17.

2)(m^

6.

+

c)

9.

+ ac +

/)c

b)(a

-

3)(y

_ 7/ +

1

20. (a

-

+

6x2

-

2b)ia

23. (6jc

1

-

33.

+

28.

4)

2a-

30. {n

4)

3a/)2

+

a3

ti^

-

ab + 4b^)(5a^ -\- ab -\- b^) - 3a^b + 2^a^b^ + 3ab^ + 4b*

lOa" 37.

n

7^

For

1)(«4

-

4n'*

_

3«3

+ 4n3 -

moi''e practice,

24. (3>^

a

-

+ n^

+

-

8«

4)(w2

-

„2

+

+ n -

see page 417.

1) 1

-

-

38. {a

_

Z7)(a2

-

a2/7

-

7jc

+

-

+

-

b){a'^

+

1

1)

- 2

y

16)

Z72)

+

b^

12)(jc2

_

3jc

-

1)

4)(2x4 1

b) 6

2)(2>^

-

+

+

\){a

- 10x3 + 32x2 _ 29x - 12

3x4

1)

-

+ 2a6 -

X*

1

-

6x2

+ 48„ - 64

12/72

a/)2

34. (x2 36. (;c 2x5 -

-

4)

2x + 1) + 5x + 2 6y + 9) 27/ + 27

+

+

l)(4;c

6/2

+ 3X72 + 9/2 +

{y

n^

35.

+

-

2a2

2){x^ + 4x2

/3

2« + l)(n2 + 2« + 1) + 4/7=' + 6n2 + 4a7 + 1

+

n2

21. {2a

3

_ y _ 20 1

x3

32. (a

+

18. (4jc

5)

+

(;c

5){y

1)

+

4/7

1

2b)

-

5)(6jc

26.

2x - 8

— 2m +

3a2;b

(7

36x2 _ 60x + 25

31.

+

-

-

/2

- 462

4)

-

3)(«

/72

/73

a^

-

1

15.

3)

+ x/ +4x + 4/

12. (n

4)

+

9)(>^

+ 12/ + 27

+ y){x +

{x

x2

6)

+ 24

0/7

1

+

(J^

+

4)(«

+

k2

+ 8x + a2

4jc

+

7) 7

iy + 3)iy - 4) _ K- 12 (4x + l)(4x + 1)

2« + 1) + 3n + 1

-

6x2

+

+

{n n2

3.

+ 8x +

/2

2y^

-

1)(«2 + 3n2

-

X

3)(7 + 4) + 7/ + 1 2

,2

— x){6 — x) 36 - 12x + x2

x3

29.

11.

2y)

+

5jfj^

n3

27.

a2

2) 2

+ X -

a2

22.

8.

2) 2

+

1)(X

+

a

/2

- 3x +

+ y){x +

2x

+

(;;

x2

/2

-

\){x

2x2 19.

5.

5)

+ 2x/ +

x2

16.

+

+ y){x + 7)

X

x2

13.

2.

2)

+ 3x + 2

a2

a

x2 10.

+

l)(x

x2

-

3jf3

+

5x^ - 7x^ + 10) + 28x2 + 1 0x - 40

a^b

+

^2^2

_^

^^3

+ a^

^4)

-

b^

147

7 • Multiplying at Sight When

you multiplied binomials by using the vertical form, you may have noticed

a

pointed out

pattern.

The

at the right.

pattern

is

X

Be sure

to

watch the

EXAMPLE

EXAMPLE

2

3

signs!

Multiply (x

-

4)(x

-

2).

Step

1:

{x

-

4)(jc

-

2)

=

x^

Step

2:

(x

-

4){x

-

2)

=

x^

-

6x

Step

3:

(x

-

4){x

-

2)

=

x^

-

6x

Multiply

(;c

-

4){x

+

2).

(;c

-

4)(jc

+

2)

=

/

x2

-

2;c

T

-

\

.

.

+

.

4^ S

- 4^2 A =

29.

a

2) 2

6.

5)

9.

-

2)(«

-

+

5)(a

+ 2a - 15

+

b){a

b)

-

/)2

-

2^

-

+

1

+

3b) 3b'^

6x

- y)(4x + _

+

1

its

its

area. 6/^

Find the area. Use

+

l)(x

+ + +

27. (4x 4x2

—

2)

+

4)

+ 9x + 4

5)(2x

6x2

-

5)

_ 25

4)(3;c

-

4)

9x2

-

-16

2y)(2x + 3j) ^ 1 2xy + 6/2 3y)(x

—

5y)

_ 23x/ + 15/2

+ 2p){7n - /?) 21n2 + llnp - 2p2

30. (3«

2y) 2/2

3

+ 6x +

9

2a

x+3 x2

- 68 -

+

1

a+4 2a2

x) centimeters by {x

—

4) centimeters.

by (3/

+

1)

+ 5a -

x^

A rectangle measures (2^ + Find

+

X

0x + 24

area. 2ix

-

5)(;c

+ 3x - 10

33.

4

A rectangle measures (17 — Find

4)(/ + 1) - 3/ - 4

Iw.

x + 6 x2

3) 1

+

24. {3x

5)

-

Qjfy

—

2)

x2

21. {3x

32. x +

-

iy

3x - 5

- ^Oab +

b){a

6)(a:

- 9x +

4x2

3a2

3a

— x2

18. {2x

+

\){2x

6)(x

+ 8x + 12

2x2

3) 3)(27 4/2 - 9

-

3x

+

+ x2

15. (2jc

3)

-

(x

12. {x

1)

a2

(x

/2

+ n - 2

24x2 +

31.

35.

+

6x2

3a

Find the area. Use

34.

a

-

3)(a

a-

l)(2n 3) 10n2 + 13/7 - 3

3jc

n

-

- 8a + 15

- 4

7)(6a - 7) 36a2 - 49

+

-

3.

5)

- 3x +

n^

6)(v - 2) + 4y ^ 12

5n

«

+

l)(;c

a2

Y^

6fl

x2

1

+

x2

19.

3)(x

+ 8x + 15

X

+ 2a - 3

^+ X

5.

1)

+ x2

X

+ 3x + 2

x2

a-

13.

+

2.

3) centimeters

centimeters.

1/ + 3

A =—bh. 2

B

37.

36.

h = 4n

b = 2n -^ h=n + 3 (2n2

b 150

6n2

- 2n

+ 5n -

3)

1

2

Find the area of the side of the house

38.

shown

at the right.

1(2x2 - 3x -

9)

+

2x2

+ X -

3

x-1 2x

A

39.

shown at the found by using

trapezoid

may

be

A =

]-h{a

+ h

a b

Find the

^

.

right.

the

Its

3

area

formula

b).

= = =

2x 3x 3x

—

\

-\-

\

area. Sx^

The box shown

40.

^

is

+

hold a cup

at the right

and

saucer.

is

designed to

The

top,

the

bottom, and two sides are rectangles. Two sides are trapezoids. Find the area of the

whole surface of the box. 318x2 - iox

lOx

There are 3 red hats and 2 black hats in a drawer. Melba, Fran, and Brent line up in a single file, and a hat is placed on each one's head. They are asked to figure out what color hat they are wearing.

who could look ahead and see Fran and Melba, says, "I don't know." Fran, who could see only Melba, says, "I don't know." But Melba, who could see nobody, says, "I know." Brent,

What color is Melba's Melba's hat

is

hat and

how does

she

know?

red.

151

X

8 • Square of a Binomial You know

The same two

that

is

factors,

when you square

a

number you multiply

6^

=

6 '6

x"^

=

by

it

itself.

X'

when you square a binomial. All you do is to write the then multiply as you did in the last section.

true

{a

+ bf = =

{a

a^

+

b)(a

-\-

lab

+

The square of b)

-\-

b^

-

4)2

binomial

1970 1975

1980 (estimated)

It is

graph above

AUTOMOBILE REGISTRATION

is

The bar graph below

is

in the advertisement

misleading.

Do you

see

why?

At is

first

glance

it

looks as

if

Tru-Vision

twice as popular as Brand A.

This

TV

is

because the scale on the side of the graph does not start with 0. Instead, it starts with 18.

Perhaps the graph

at the right is

more

honest.

^/A

Written Exercises The graph

at the right

W/M/M/M/M/M/M/M/M/M/M/M/M/M/M/M/A

shows the

increase in sales for a newspaper in 1.

one year.

NEWSPAPER SALES FOR ONE YEAR

During which months did the sales decrease?

Aug

July. 2.

The bar

for

December

is

twice as long as the bar for

January.

Does

this

mean

doubled during the year? No that sales

3.

What was sales

the increase in

from

January

s^^ x

3jc

-9 +

8.

^'-Bx^B

"° 1

;;

-

2a;

=

9.

°"^

4

- 2x = -3 + j = y

2,

2x

^"

W/M/M/M/M/M/M/M/M/M/M/M/M/M/M/M/M

Find the slope of the lines whose equations are given.

A

\.

y = Ax ^

5.

y -

9.

2x

Tell

2.

j =

y + 2x =

=

6 2

6.

-y =

3 2

10.

2>x

y

-j-

4jc

5x

+ 34

=

4 -2

-5

2

3.

/ = jc+1i

1.

4x

U. y

whether the equations share one solution

-y =

=

-

S

4

7x -7

no solution

pair,

1

1

4.

y = ^x +

S.

4x

12.

2

-^

+y =

y = x -

5

-4

^

pair, or all

solution pairs. '3

13.

y = 2x

14.

y =

15.

)-x

fio

y = 2x

11.

21.

X X

-y= +y =

y = -X

one

^

18. 2jc

y

6

=

1

+y = 2x = + y

9

j^

25. 2jc

3x

-^

+

y 4x A

-\-

2jc

26. no-

7

=

8

16.

-h

= =

y

2

9

19.

^ one -^ 6i

-\-

2x

=

+^^y = — 3x = y 3x

S

4 ^

20.

one

2

2x = 4)^^'-'^^'-4x = 9^„ y - 9 = 4a: / +7 = 8

= -3x + y — X = 5 y

-^

3

one

x

-/ =

x

—y

4

one

3x+y = 7^„22.^ + y

242

S

jc

rit)

3

-\-

7 +

27.

24.

-y = / — 4x = y 4x

_ no

J-

x+y=-3

-^

,

-\-

6

y - 2x = -6

-y = 4 28. y + one . / = 4 —3x y — y 3x

=.6

2x 2x

= =

4

-

^^^

B

29. a. Will the

graphs of these equations intersect? no

y = 2x y = 2x -\-

b.

Will the graph of /

c.

Can

30. a.

= —x +

1

3 \

intersect either graph? yes

the three equations share a solution pair? no

Are the

lines in color at the right parallel?

(Check

their slopes.) no b.

Will they intersect? yes

c.

Do

the equations of the lines share a solution

pair? yes

3

•

The Substitution Metiiod

There are more exact ways of solving a pair of equations than by graphing. Here is one method.

EXAMPLE

Solve

1

Step

1

Step

2:

y +

\

=

+y =

3x and 2x

Solve one equation for y.

:

9.

->

;;

+

1

=

3x

+y =

2x

Substitute the value of ^

2x

in the other equation.

+

-

(3Ar

Solve for x.

5jc

-

9

=

9

1

5x= Step

y = 3x y = 3-2y = 5

Substitute your x value in

3:

the equation in Step

The

Check:

Is (2,5)

y

-\-

\

5

+

1

6

solution pair

a solution of both equations? Yes!

2x

+y =

2-2 +

3-2

4

/

+

The

Solve x

2

first

=

2y and x

says that x

=

y = +7 = 3y = y =

X

equation

^

2y.

+y =

-\-

2>'

Substitute 2y for x.

9

5 5

9

EXAMPLE

1

is (2,5).

=3x 6

10

\

1.

Find the value of y.

Answer:

9

1) z=

/

6.

6 6

6 2

Substitute.

Substitute 2 for in

X X X

j'

one of your

original equations.

Answer: 244

The

= 27 — 2y = 2-2 =4

—J x

15)

(5,

+y =

3.

7 X

20

- y = 2 (5, 3y + X = 14

X

3)

7.

=X +y =

X +7 = 5x = 3y

2

4.

(7, 5)

\2 (1,2)

3

-

^.

1

= 2>y 5y - x =

X

x - 2y 3x + y

(12, 4)

%

= 5 = -6

(-1, -3)

Ua Written Exercises

W/M/M/M/M/M/M/M/M/M/M/M/M/M/M/M/M

Solve by the substitution method.

A

1.

y = 2x

(3, 6)

x+7 = 5.

7 = X + 4

x+7 9.

=

2.

9 (9,

13)

6.

22

+7 = 2 3x + 7 = 8

X

10.

y = 5x 7 — x =

X = 57 3x = 7y

10)

= 2 - 7 (-5, 27 + x = 9

2x 4x

3.

+7 = -7 =

x

=7+

x+7 =

X

(3,-1) 13.

(2,

8

7) 7.

4x+7 =

5

11.

1

3x 2x

14.

16 (10,2)

4.

(5, 2)

7

7 = 3x +

1

(I- 4)

8.

8

- 7 = 13 + 37 = 16

(1,3)

+

3

12.

7 = x +

1

x+7 =

5

7 = 2x -

3

4x+7 =

9

2x 5x

15.

(5,1)

^ + 2Z> = 7 2a = 3b

(3,-1)

(2, I)

-7 = 9 + 27 = 27

(5,2)

X - 57 = 8 4x + 2y = 10

(2, 3)

16.

p - 5q = 3p ^

-

2q

6

=

5

(1,-1)

(3,2)

Solve.

Sample

D= r

=

Find

B

17.

D

bh 2h Find A in terms of h. A = 2/»2

20.

/.

18.

=

Solve for x, y, and

C

in terms of

^ = b

D= D= D=

Answer:

rt

5t

V=

Bh

B =

2h^

Find

V in

\9.

rt

(50/

^

Substitute.

5/^

V=

Iwh

= 3/2 w = 2h

/

terms of

V =

h.

2/)3

Find

V

in terms of h.

V =

6/|3

z.

+ 7 + z = 180 = 3xx = 20 7 z = 5xK = 60 X

z=100

21.

X x

+ 7 + z = 62 = 2z — 5x=i9

^^3^_5y, =

3i

z=12

22.

x + 27 2x + 7 3^ +

+ =

3z

=

6 x = 3 = o

^^8y

Z5=-1 245

4 The Add -or- Subtract Method •

When

add or subtract the one variable.

solving a pair of equations, you can often

equations to get a

EXAMPLE

new equation with

Solve:

1

just

+y = -y =

(2x

[2x

l S

5jc

+7 = -7 = + =

5x

=

2jc

Step

1

Add. This makes the y

:

term drop

^

3jc

out.

new

equation.

Step

2:

Solve the

Step

3:

Substitute 3 for x in one

X

7 8 15

15

=3

+y = 2-3+7 = 6+7 = 2x

equation. Find the value

of J.

Answer:

EXAMPLE

2

Solve:

1:

The

solution pair

+ +

[5^

67 27

= =

make

Subtract to

is (3,1).

You may check

the

5>:

x

+

new

equation.

*

4y

47

3:

Substitute

—2

for

7

3

5x

in

value of X.

+

67

5;c

246

solution pair

is

(3,-2).

=

+ 6(-2)= 5x - 12 = X

The

j

= -8

y=-2

one equation. Find the

Answer:

11

= -8

5x Step

=

67

^. 5x + 2y =

+ Solve the

it.

1

term drop out.

2:

7

3

-^

Step

7

7=1

l5x

Step

7^

= =

3 3 3 15 2>

You may check

it.

f/A

f/M/M/M/M/M/M/M/M/M/M/M/M/M/M/M/A

Classroom Practice

Would you add or would you does the y term drop out? 1.

5.

+ 7 = 6 add; jc— 7 = 2/ X

= + 7 =

2x 2x

6 sub; 2 Jf,

3y

-\-

2x Ix

2.

Does the x term

subtract the two equations?

+ -

X

6.

+y = —y =

3x

2y 2^

5 add;

3.

2>y

= =

7 add;

7.

5/'

5x 3x

+ +

2j 2y

= =

X 2x

+ -

3^

= =

3/

9 sub;

13.

f/i

X =

0;

K = 2

14.

X =

3;

/ = 2

15.

9K

3;

-\-

2y = - 7 = -\-

2.

x =

1

;

/ =

1

16.

x =

1;

/ =

1

1

/ = -1

= 5 sub;y = 2x

Ay

-\-

4x 4x

S.

add;

;

x =

X x

4.

1y

9-16. Solve the pairs of equations in Exercises 1-8. 1 1 x = 1 / = 2 1 0. X = 2; / = 1 9. X = 4; K = 2 .

or

6 sub; 3

W/M/M/M/M/M/M/M/M/M/M/M/M/M/M/M/M

Written Exercises

Solve by the addition method.

A

I.

5.

X X

+y = —y =

2a4a +

b b

2

2.

10(6, -4)

= =

3

+ —

5;c

3x

+7 = —7 =

6. 2;c

9

{2, A)

= =

47 47

3x

4x -1) 2x 3.

1

7(1,

10 5

7.

(3, 4)

-7 = +7 =

3;c

-

x

-\-

8

—2(1, -4) 2p

—

3q 3q

4r

-

7^

4r

-\-

= =

4.

27

2y

8

8.

'

8(4, 2)

5p

-\-

= =

\0

4

(2, 0)

= 13 = -29

Is (-2, -3)

Solve by the subtraction method. 9.

X X

+ +

67 27

= =

10

(-2, 13.

3x

X

+ +

27 27

10.

2

X X

- 7= - 37 =

3x

11.

+7 =

18

14.

14

-

2x 2x

= =

57 37

(2,

(2, 6)

7

12.

-2x + 7 = -8

10

(25, 5)

2)

= =

20

-2)

(3,

14

15.

2a

2a

10

+ -

=7 = -5

3b b

-2)

-

3x 2x

(-1,

47 47

= =

(3,

-

16. 2s

6s

18

-3)

= =

5r

5r

21

17 \

(-4, -5)

3)

Solve by either the addition or subtraction method. 17.

+7 = —y =

X X

9

18.

+ —

X 3x

5

27 27

3x 4x

+

27 27

= =

(3,

25.

-X 4x

27 27

13

22.

8

4x 2x

-

37

37

-2)

= =

3a

-

\2b

19.

2x

+

2x—

5

37

^9

= =

9

23.

3

-4x + 7 = 4x

+

30.

2x

-

1

=

=

8

7=—

20.

8

a

-{-

2a

+

(-2,4)

3x 6x

+7= -7 = (2,

(3, 1)

10 26.

(2,-1) 29.

7

(3,2)

(7,2)

21.

= =

7 27. 8x

37 (-1-3)

5

X

6x-ll7=10

(-1, -1)

(-2, -2)

37

24.

18 5

5

(3,5)

6 9

4a 2a

- 7b = -lb =

5x 5x

+

13 3

(5, 1)

28.

4 37 (1,-1) 8

= =

(3,1)

-6)

= =

- 27 = 5x-27 =

17 '= 18 31. 6x

a-\2b=\\

+ -

3b 3b

32.

4x

^-4x

27

= =

7

2 37 (1,-1)

- 7 = 15 + 37=-5 (5,5)

247

Jf

Can you work

with fractions?

The

following exercises have fractions in the

solutions.

2x Sx

Sample

-\-

—

4y 4y

Sx

X

= = =

=

2

^2-i +4^ =

2

2x

2 2

Substitute.

4

+

4^

2

=

1

+

4;;

4y

= =

2 1

1

/ = 7 Answer:

B

33. 6jc 6jc

+ +

9^ 3y

= =

(4.

(hi)

4

5«

34.

10a

+

^ Z)

= =

3

35. Sa

3

8a

+ 6b= - 4Z> = 4

(I--)

1)

Pass the center of one string under the string circling the other person's wrist, over that person's

A

String

Duet

hand, then back under the string again.

Here is a stunt you might of your friends.

like to try

with one

Tie a piece of string to your wrists.

Tie

another piece of string to your friend's wrists so that the two pieces of string interlock as

shown. Now try to separate yourself from your friend without cutting the string, untying the knots, or taking the string off your wrists. It can be done! is the branch of mathematics that can explain why this stunt can be done. Topology deals with figures and how they can be bent and

Topology

stretched.

248

a:4)

r/i

Mixed

f/M/M/M/M/M/M/M/M/M/M/M/M/M/M/M/M/M

Practice

Solve by the graphing method. 1.

j=-jc +

3

2.

= 2x-A^

y

_.

;c+7 = 3x + y = -S

7 + 3a; =-7 y - 4x = 1

3.

x +y = A 2x -\- y = 4

(-2,-1)

(-4,4)

(1^, if)

4.

(0,4)

Solve by the substitution method. 5.

= 3y + y=

X x

6.

y = X + X

12(9,3)

+7 =

x X

1.

I

5(2, 3)

2y 3/

-{-

+

= =

2

H.

13(-20, 11)

y - 2x = 4 / + 4x = 16 3,8)

Solve by either the addition or subtraction method. 9.

X 2x

-\-y

-

y

= =

7 5

10. (4. 3)

= =

2a-\-3b 2a + b

3(i,

-

3x

11.

5

2^ 4y

-3x +

I)

= =

12.

1

7

(3, 4)

-

4x 6^

= =

ly 7y

9

3 (-3, -3)

Solve by the method which seems easiest.

=X +2 + y = 11

n. y

2jc

-^

14. (3, 5)

X

+/ =

4(7,

+7= - yy =

2x 3x

15.

8

x-3y^ =

1)

9

6

16. (3, 3)

= =

x + 2^ 3x + y

5

10

(3, 1)

17.

-

2x

X

= 8 18. = lib.-\

3/

-\-3y

^

21.

jc

x

= +

2/ 3y

19.

=

5(2,1)

3/

3a; - >' = 8 X + 2>^ = -2

22.

/ - 2x = 7 7 + 3x=2(-i,5)

2a 4b = 6 -a-3b = l

23.

x + / = -2 2x + y = 4

(-1,-2)

practice, sfee

page 424.

24.

x

-

2x

= y =

5/

-\-

' '

'

2

4

(2,0)

((6.-8)

^ more

y - x = 3 7 = 2^ + 4 (-1,2)

-

(2,-2) For

20.

V

c

\J

y?

iSELF-TESTi Solve by the substitution method. 1.

y = 2x X

-{-

3y

=

2.

\

U^^'

X

=

3.

3y

2x=y +

^^

5^^''^^

4p

=

3q

p-q = 2 (-6, -8)

Solve by the addition method. 4.

3x

X

-/ = -\-y =

2 6^^''^^

5.

2a

+

3b

= -I

a-3b =

_

6.

^'^'

^^

4

3r

+

2^

=

4

4^-2^=10 (2,

-1)

Solve by the subtraction method.

'7'^^y = ;c-37=

^

,0 2) -6^°'^^

^-^ +

2^

+

2/

35

= ^-2 = 4^

5) '

*

9.3x-y=\0

x-y =

4

(3,-1)

249

5

•

Word Problems, Two Variables

You know how

to solve

Now you

problems by using one variable.

can

use two variables in problem solving.

EXAMPLE

Art and Lynn were partners in a bowling tournament. In the first game, Lynn's score was 10 more than Art's.

1

Their combined score was 330. Find each person's score.

= =

Let a Let b

Art's score

Lynn's score

Lynn's score was 10 more than 4^

4'

b

=

4^

sb

10

+

Art's. •i'

a

Their combined score was 330. a

Now

solve the two equations.

Take your second equation.

=330

b

-\-

We'll use the substitution method.

'b

=\0

^a

-\-

-^

a

b'= 330 "x Then

a

+(10 + 2a

+

fl)

10

2a

Now

a

use one of your

original equations.

10 10

substitute.

= 330^ = 330 = 320 = 160 + +

Then 160

substitute.

x

+7 = 8 -2y = 5(3,

+ 5/ = 3 + 2y = 13(-59,

3x

2.

X

2)

2x 3x

3.

36)

+7 = 5 - 2y = 4 (2,

4. l)

- ^b = 5 3fl + 26 = 4

a

(2,-1) 5.

+ = - 2/? =

3« fl

/>

4

+/ = 7 3x - 2j =

6.

6(2,

-2)

_c +

3c

7.

jc

11(5, 2)

= -7 5^ = 6(-i,

g.

4^-

2a

2

•

—

f

a^

1

4

«

,0/, s

-^-^ - 2a^ 4

x2

24.

8m 2^m

2

-5

*

5

6

5

(ab)^

/^

—-

2

^

-^i^y

9„ 9x - 9

^ -2a 8^2 b

7^-77+12 •

d

/ — 3

^"^

29

a

——~ ^

4ab

C

2

y + 3

^: Ix

2

^ ^-^— — 9

——

-^"^ X

2

a

3/2

15r^

(-^y)^

^^

''^-^

fif2

£

d^

^

^~

7^

2

^

_

fl^

1

-y

,

I

^^^ +

1

^^±J-

14.

3ib

16.

47 "

+

4

_ -

'

;

7- ^ ^— L^^ 1

4

19

'4 + _?:

^

-

2

_

-4

r

16

8

4

6x

17.

1

4x •"

+

2

3

^^

8

8

~

'

" 2

-

_

3

^

4

-

^

12.

7x + 9 12

6 ^

-

^

+

4

x-2-

::

1

3a - 11

15.

^^^^ +

5

+

12

4

*

3x +

1

8

^L±^ 3

b

-4

3

-2

2b

^^-^5 s^^

18^

~"4 3~A +

^

20

4

16

+

3fl

6

2

21. 5

,^

.

3y

-^

9

4

2

+

lOx

3

- 3a

3

-

2/

+

7^ '

^^±1 - 2 ^1^ 5

2

^ 12f- 7

8

+

-

3r

5 1.

12

4

2

-

_ 15

iijf

2

-

4/c

18

3

8

5x + 7/ ox

1 ^

~r

2x

^

4

8.

1

9

5.

3

13.

~

^

6

12

10

+

^

1

6

_ 3

a

-

2A:

2.

T5~~

5

3

7.

x +

4

X 8x + 10

2

-\-

1

15

22 "

^

"^

8x

_ 1^

3

,

18

^

-

^

3x

—

1

2x

1

5a + 33

36

12

1

6x

3x 8x

1

295

5

14

•

Equations with Fractions

Sometimes

how

see

want

you'll

these equations can be solved.

^+^= 2

Think about the first.

them. Let's

to solve equations with fractions in

Rename

left

10

3

side

3x

the fractions

,

2x

10

same

so that they have the

denominator.

3x

+

2x 10

Now

you have a proportion. All you have to do is to

6^^^^1 5x

X

il + iA

Check:

2

60 \2

10

3

6

+

4

10

/

10

It is

J:

= =

cross-multiply.

easy to do just one step at a time, and simplify one side of the

equation

That way you can get a proportion which

at a time.

not

is

difficult to solve.

10

5

10

+

3

4

]_

^+ 4

20

3a_

4

2

+

20

1_

a

«

296

= =

70 10

always wise to check your answers.

,^a

^^^^2

+ 20) = 2a + 40 = 40 = 4 =

2ia

In

2

4

n

10

3a

n

n

10

4

It is

3^

a

EXAMPLES

4(3«)

\2a \0a

a

f/A

Written Exercises

W/M/M/M/M/M/M/M/M/M/M/M/M/M/M/M/A

Solve.

1.

4.

10.

4-4=

1+ 4

4 30

^=^ 12

3

2

3

7

5

10

«

+

i?-

^=

2.

:^

+^=

6

7

4

6

8.

4_A=6^ 5

11.

7 6

3.

£+^= 6

3

5

3

«

_

15

X

2«

^1

14.

i« -

=

«

467 1

_9

12. -^

15.

4 42

+

9 16.

^^Ltl

+ ^Lizl =

5

17.

^

~

^

-

^^

4

X 19.

+

X

3

—

=

1

22.

6x

-

4

^

=5

18.

3

1

2c

6

15

2_A^23 3x

--

-^

4

9

+

^

=

6

-92

3

-2 J_

20.

2c

C

~ 4

-7 2

^=1

X

%

B

15

9.1_A = l-60

c

- = l^ «-f An + n

612

1 = ^2

5.

1

13 42

2=ii^^ + ;c4

c

+

3

_

5

X

4c

23. 2

7x-l ^

3^

6

Diophantus was an ancient Greek mathematician. According problem was written on his tombstone:

3x

_

19x

+

3

3

_3

4

to legend, this

HERE LIE THE REMAINS OF DIOPHANTUS. HE WAS A CHILD FOR ONE SIXTH OF HIS LIFE. AFTER ONE TWELFTH MORE, HE BECAME A MAN. AFTER ONE SEVENTH MORE, HE MARRIED. FIVE YEARS LATER HIS SON WAS BORN. THE SON LIVED HALF AS LONG AS HIS FATHER AND DIED FOUR YEARS BEFORE HIS FATHER.

How

old did Diophantus live to be? 84 years

297

15

•

Work Problems

Suppose an long

it

office

manager wants

will take to get a report

who work

to figure

how

typed by two typists

at different speeds.

Suppose your mother helps you to paint a room and you wonder how long the job will take.

Algebra can help solve both problems. Let's consider the painting

EXAMPLE

problem.

and her mother plan to paint the living room. it would take her 10 hours alone. Her mother says she could do it herself in 5 hours. How long would it take them to do the job together? Ella

Ella thinks

r/i

A

Written Exercises 1.

W/M/M/M/M/M/M/M/M/M/M/M/M/M/M/M/A

Mr. Macy's two grandchildren offer

to paint his kitchen.

Sandy thinks it would take her 10 hours alone. Meg says it would take her 15 hours alone. How long would the job take if they worked together?

6 hours

5.

Taddy and Rose deliver groceries for a neighborhood store. Today Taddy needs 4 hours to deliver the groceries alone. 2— Rose says it would take her 5 hours. ob together?

hours

B

9.

Judy Rogers needs 6 hours correct

some

papers.

to

When

her assistant helps, the job is How finished in 4 hours.

long would

take the assis12 hours alone? tant to work it

B 16

•

Binomial Denominators (Optional)

When you want to add

or subtract fractions with

more than one term

in

the denominator, there will probably be quite a few steps.

EXAMPLE

Think:

+

1

which x(x +

x{x 4{x

+ +

+ 2) 2)

+

x{x

+

I

is

must

new denominator by both x and x + 2.

find a

divisible

2) will work.

2)

3jc

+ 2) 4jc + 8 + 3.x x(x + 2) 7jc + 8 x(x + 2) xix

Think:

which

EXAMPLE

2

4 2(«

+

+

a

^

+

'

C2>

-3

X

+

Ci>

4){a

+

jc

302

-

4){a

+

2) will work.

2)

Think: I must find a new denominator"^ which is divisible by both x — 3 and X + \. {x — 3)(j>c + 1) will work.

1

5(.x + 1) 4(x - 3) {X 3)(x + {x 3)(x + 1) 5(jc + 1) - 4{x 3) U - 3)(x + 1) 5 — 4x + 12 5x {X - 3)(x + 1) + 17 {X 3)U + 1) -I-

(a

new denominator by both a — 4 and

find a

2)

- 4){a + 2) + {a2(a + 2) + ?>{a - 4) (« - 4)(a + 2) 2fl + 4 + 3(3 - 12 (^ _ 4)(« + 2) 5fl - 8 {a - 4){a + 2) 3

2.

must

divisible

2

{a

EXAMPLE

+

I

is

1)

Exercises W/M/M/M/M/M/M/M/M/M/M/M/M/M/M/M/A -54 x2 + lOx - 21 4(a - 6) + 2a(a + 6) — :^79- -: 15. (b + 2)(b (x + 3)(x (a + /))(a - b) 3) 3)

r/i \Nritten

-zrT-.

Simplify.

A

1 '

3

_)_

k

k

+ +

8/c

^\

/c(/f

5

A

2 •

1)

/

/

+

+ +

5(f

3 "^

'(f

5

2)

3.

4.

r

+

5

r

(r

+ 7

(y

7.

3

5){,r

(3

ja +

5.

"

a

4

c

—

a

A

-\-

—

a

C

5

—

4

X

y

X

3

+

X

6

;c2

—

+

6

-y

r

1

19.

^2

_^2

3(x

^ X

6^^

-

/7

r

+

4- 3

-

+

1

a

+ 3x + 3/ + y)(x -

^^

3

a

—

3

r 2a

1)^^

-

a

1

+ 2s - s)(r +

4 - a

a

r

_|_

—

s

'

r

(r

2)

4

j^

4

a

2

-\-

8

(c

5

+

+

+

la

_^

—

a

b

c^)2

b

4_^

_12

jg '

+

+^

c

6^)2

/

2c - 3

l(c

c

c—

c

+

+ 3)(c-

3)^^

21.

3

3

23. JC

+

1

.X

+

^

+ 5 4-^2 +

j^

X

5

1

5

,

+

ts

-\-

t"^

2«

2_^ J_

—

- 1) + 5)

4(3(

5f/

(a

^ = l3 +

X

a

Solve for the variable.

a

3

11)

^

_^

-

5)

2)(a

-

3(a

22.^

—

7

-

*

(c

—y

+

6)(£)

8 + 5c

36 1

+

2Z?(ib

3x^

16.

b

3

r2

14

11(^

+

X

^

- X X + / - 29 + 6)(x -

5)

5 6(3/)

Z)

—y

+ 20 - 4)(c +

b

9.

+

c

Z?

\

X -\-

2)

5)

_

9

4)^^ '

(c

7

13.

-

4)(a

4

2)

3(a + 4)(a- 1)^^

+

+

^(>f

3

c2

7

6

jQ

2)

+ 3/ - 1 + 3){y - 2) g

3(3a +

_

8a

+

2 /^

7 +

+

-3(x +

5 ;C

-3r r

2.

5)

2 2)

303

'/a

Reviewing Antiemetic

Recall that decimals are another

tens

way of

SIciiis

W/M/M/M/M/M/i

writing fractions.

CAREER IMGTEBC9K Environmental Sciences

O

Society is becoming more and more aware of the need to protect our environment against pollution, land abuse, and destruction of wildlife. Here are some careers in the field of pollution control.

o Air pollution engineers study the effects

of

air pollution,

controlling

and develop plans

for

it.

Chemical engineers can recycle waste materials, converting old newspapers

o

into sugar, for example.

Water pollution technicians anaand bacterial

lyze the chemical

content of water samples.

305

g ER CCRIVER f Tipping

V)

People in a restaurant

2

who

wait on tables

usually get tips. When you are the customer, do you know how to figure the tip? The amount you leave depends on you, but people usually leave about a 10%, 15%, or the figuring

First think

is

done

in

20% tip.

All

your head.

about percents and what they mean.

100%

50%

25%

10%

the whole

half

one fourth

one tenth

Suppose you want to give a 10% tip. All you have to do is round the amount on your restaurant check, then move the decimal point one place to the

left.

Your check

Once you know how your

to figure 10%,

result.

Your check

then you can find

20% by

doubling

7i

Reviewing

{See pp. 268-273.)

Simplify.

lAl

2^-J5x •

3

24

A 3k

187 ^'

VM/M/M/M/M/M/M/A

C/iapter

t/ie

_27>'2

3 •

I5x

^

7'•

18a:/

1

8

6

•

ll£

"L

22

2

4

-"^^

^1

12«

3

'

^^ ^

•

2.^

10

Sx^

'

2/7

^^

^^^'^'

9

b

-ab^

iM 5„

5

'

28«^2j_ 46

3^-15 a

11.

.

—

— y ^ y - X X

a 20.

1

a-3

1—

21.-

-1

2a

2 jc^

3m/7 - 2/nn

^m^n^

1

19.-

1^-

r>

Ai -1

^2 a - b /?

- 3/2 — y^2~

3^2

12m2^2 17.

-^

_ +

^

—a ^2-49 1

-22

22.

1

Solve.

23.

^

{See pp. 274-277.)

The

ratio of blue to

How much

yellow in a paint mixture

blue paint

ratio of

How many Solve.

coaches to players

—X = «.

'3

33.

—=— 2n

4

26.

7

_

n

+

1

^

2Q

6

*4+ 3a

18 oranges cost $2.70.

Simplify.

^ 21

is

3 to 2.

10 cans of the mixture? 6 cans

is

2 to

9.

•

\2

6

How much

35.

make

{See pp. 278-283.)

14

2o

to

players are there? 18 players

2

25.

needed

team has 22 members, including coaches.

24. Central High's soccer

The

is

^

"^

will 36

^

'

21

27.

1

_

-5 = —10A

9

a

—

2

_

^

^.

6 34. 3

oranges cost? $5.40

{See pp. 284-285.)

63

4

x

=—^ 7+11 3y

10

28.

_L 4

32

20

—

2

_

*x+23

dozen apples

How much

2/?

b

'

2

_

J_

-1

cost $2.25.

will 2

dozen

cost? $1.50

4

^

2

_

1

Simplify.

43

47

(See pp. 286-287.)

1 ^ _L x^

—

y^

44

9

L- ^

X

+

y

-^

^-A

X - y 1-

45

3

b a __I^

5

a

-\-

4«

^- - ^

^

—

b

a

^!-

+ b

46

3a

49

-^ - ^- — —

a

3 :

—

\

3 —

Here's what you II learn in this chapter: 1.

2.

To solve equations with decimals. To use equations with decimals to

solve

word

problems. 3. 4. 5.

To To To

solve equations with percents.

solve

word problems involving percents. and mixture

solve interest, investment,

problems.

Chapter 10 Decimals

and Percents

1 •

Many

Decimals

equations contain decimals.

and how

to

work with them.

Fraction

First, take a

new look

at

decimals

EXAMPLE

4

Multiply: 0.42

x

6.65

^ 2.7930

2 • Division Witii Decimals In dividing by a decimal, the

first

step

is

make

to

the divisor an integer,

or a whole number.

EXAMPLE

1

The Problem

First Step

Then Divide

1.6)1.76

1.6)1.7 6

16)17.6

1.1

16

The

16 16

division could be written in

Moving

fractional form.

the dec-

1.76

imal points one place to the right in the division

like multiplying the

is

numerator and denominator by

EXAMPLE

2

^

17.6

16

1.6

10.

Then Divide

First Step

The Problem

170 0.02 JXia

0.02 Jl4

The

2J34O

division could be written in

Moving the mal points two places to the fractional form.

in the division

is

^xioo-^

deciright

3.4

like multiplying the

numerator and denominator by

^

340

0.02

2

100.

Often divisions do not come out "even." In those cases, you will have to

round the answer.

EXAMPLE

3

Find 3.75

^

Round

0.07.

to

one decimal place.

Keep 53.57




=

5.4

5.4 5.4

last year.

Let X

=

Then

1.2jc

5.4

28 1.4

4

mg mg

iron in the chicken

iron in the spinach

/

5.4

ball player's batting average

average

X

10

54

What was

is

0.330 this year. This

is

his average last year?

batting average last year.

=

batting average this year.

1.2jc

1000

X

\.2x

1200JC JC

= = = =

0.330

1000

X

0.330

330 0.275


\fi

X X

EXAMPLE

^

1!

2>

root.

Some

quadratic equations have no real-number solutions.

EXAMPLE

3

+

;c2

= = -9f

2.

(IOjc

-1 + 5

5.

-

+

8.

+

6x + 10 (4x - 7)

-

3)

/— -^4

0.9m

6.

5 \/x

9.

3m2

—

3.

0.03m o.87m

4 V^ 20x

•

16

Factor. 7.

10.

a^ H- la a{a

+

x2

13.

/

+

7x

+

(X

+

7)

12

3)(x

+

127

+

11.

4)

/

-

(y

_

6)2 14. ^2

36{K +

+

8^3

X^ x^(8x +

-

3/

Z)2 (a

6)(a

+

15. ;c2

6)

16. (4x

-

16jc2

-

25

+

5)(4x

17.

+ 6m —

m^

40

(m + 10)(/n -

5)

18. x^

+

ii„

-

-

{x

3m(m +

1

(n

3)

-

+ 2m

_

12. n2

18

+

6)(K

1)

-

5x

8)(x

-6x

4)

18

-

9){n

4)

2)

24

+

3)

-\-9

{X

-

3

=4a - 9-4

3)2

Solve.

+

4

=

22.

5(2m

+

3)

25.

-(x -

19. 6;c

28 4

=

-7 = -28-7

20. 3y

35 2

23.

+

-{a

=

3)

3fl

+

1-1

+

21.

7a

24.

-4{y- \)=y-6 2

10)

=

4

+

X

26.

-3x +

12

=

-jc

-

3 28.

—=15

^^ ^^'

34.

— y = \0 X = +y = \\y=

is

=

O.Oly

2.4

^+^21

30 30.

5

(3x \4x 12

27.

8

29.^ =

3

4

OAy -

3:

^-

-^

'^^'

[2^ [ a

— +

b

3b

= \0 a = -2b

30% of what number? 40

= 4; = -2 35.

f + 1 = 910 4x

33. 3jc

What

+

3y 27

= =

\5x =

percent of 18

3;

= -

7 /

12*^

is

661^% 36.

-

x(x

4)

=

00, 4

37.

6m(m +

3)

=

Oo,

-3

38.

(y

-

4)(y

+

3)

=1 -3

4,

39.

m^

- 5m =

=

42. «2 _^ « 45. 2x2

+

3jc

Oo, 5

40. x^

-\-

6-3, 2

43. 2jc2

-

46. ^2

2

=

4x

4S.

X

2y

=

=0-^. -3 41.

+

5;c

+

5

=

y'-y -6 = 3,

-2

44. 4(x

+

1)2

=

362,

47. 4x2

-

6x

+

2

-5 ± VB

49.

-4

= 1

''2

2

the graph of the equation.

+

3

= 50±5

f-^ ^ Draw

-\-

Check student's graphs.

y =

x^

404 A more extensive cumulative review

-2 for

50.

7 =

x2

-

8x

+

16

Chapters 1-12 can be found on page 444.

Extra Practice Exercises CHAPTER

1

For use with pages 2- 7 Find the value of the expression.

A

1.

7

6 42

•

2.

If

/I

+

2(3

=

13.

9n 27

If

=

jc

2) 10

3, find

11.

X

-\-

If

=

2, find

j;

21.

+

1)

24

4 24

7 63

4.

6

7.(6-l)-3i5

8.

(4

+

2)

12.

10

+

(6

9

3.

•

8

-

(2

15. 9

-

« 6

11.

3) 2

•

•

-

32

1)16

•

the value of the expression.

y =

5, find

y^5

+

6(y

40

8

10. 4(5

14.

10 and

•

6.5-2-64

5.2-3+410 9.

5

+

9

« 12

16. 9

^

« 3

x

+

5

the value of the expression.

18.

-7

^

5

ig^

x

23.

(j

3.

2n

7m + 3m lOm

x

-\-

-\-

25

y

20.

-7

i

the value of the expression.

2)

24

22.

+

6y

2 14

-

1)

-\-

n3n

-

4 4

-

24. 2/

3

1

For use with pages 8-11 Simplify.

A

n2n

\.

n

5.

3x

-

X 2x

9.

4a

+

5a 9a

13.

3x

-\-

Sx

2x

-

17.

-\-

-

+

5

X ^0x

3a:

+

+

«

«

3/7

2.

/?

6.

5/

-

2y 3y

7.

10.

6x

+

3x 9x

11.

4a

14.

n

-

15.

a

-\-

5x - 5

6n

"7^-71

18.

6«

-

6

+

+

-\-

2a

2a

-

-

+

4.

«

8.

lOx

2«

-

3/?

x 9x

+

5x

b^a - 616. 3n

-

n + 2n +

2« 8n - 6

19.

12.

5m +

2

+

7«

-

6a?

+ 7n

+ 13

21. 8

+

6y

+

5y

-

7i1k +

1

22. 7

+

3;c

+ 1

23.

406

5x

+

3/

-

y + 1 Ix ^6x + 2/

24.

2x

+

5y

-

y + 9a: 11x + 4/

25.

4«

+

r

3x 7x

m

m + 6m 1

20. 6

-

5x

3a 3a

+

Im +

+

9x

2x + r

n

+

9 1

6

3n 2r

1 1

For use with pages 12-15 Find the value of the expression.

A

1.

329

2.

42 16

3.

224

4.

62 36

5.

41 4

6.

2^8

7.

33 27

8.

5225

Simplify. 9.

yyyy^

10.

X'X

S' S

14.

a

'

a' b b

18. 2

•

c

22. 6

'SwiSn

13. r

17. 3

21.

'

'

rs^

a' a 3a^

2'{4r)8r

x2

'

a-b-

c 2c2

•

11.

a' a' a

\5.

X

'

y y y xy^

19. 5

•

2

•

23. (4«)

•

ly

25. (2r)(4r)8r2

26. (10;c)(5;c)50x2

27.

29. (2a){4b)8ab

30. (3jc)(47)i2x/

31. 2

•

12. c

a^

«

•

n

1

0n2

4 1 6n

Sy 56/2

'

«

5

•

•

a 1 0a2

c

'

c

a

'

b

20. 6

'

a^a'

\6.

6«

24.

28. 65

•

•

'

b

-

c c*

'

c • c ab^c^

'

b sa^b

30n

5

5^ 30s2

x-x-A-xax^

32.

For use with pages 16-17 State the expression without parentheses.

A

1.

3(;c

+

3)3x + 9

2.

l(y

+

1)7/ +.7

3.

5{a

-

l)5a - 5

4.

3{n

+

2)3/7

5.

8(«

+

4)8n + 32

6.

6(c

-

2)6c - 12

7.

4{x

+

3)4x + 12

8.

aia

+

l)a2 + a

9.

4x -

2)x2 - 2x

y(y

+

7)/^ +

n.

«(«

+

2)n^ + in

12. c{c

+

73,

6{2n

-

8)12/7

4(5;c

-

4)20x - I6l5. 3(2a

-

17. 3(2«

-

3b)6a - 9618. 2i4x

-

2y)8x - 4/19. ai6a

+

13.

Use

B

10.

- 4814.

the distributive property.

21. 2(a7

+

3)

+

1

24. 4{x

+

3)

+

2x6x + 12

27. 4{2a

30. 5(2

+

+

3)

3x)

-

2/7

+

7

7« + 1 a + 13

-4x -1 1

1x

+

3

Then combine

+

1)

+

25. 2{x

+

4)

- 2x8

28. 3(5x

+

7)

3 4/

+

2x

+

-

20

17x + 31. 6{a

+

1)

6a - 3 16. 5(3^ + 2) 1 5x + 10

-

b)6a^ + ab20. x(4x 2y) 4x2 _ 2x/

+

23. 3(«

7

4{a + 2) 10a + 14

+

1)

+

83/7

+

26. 8(3«

+ 0-224/7

29. 2(37

+

5)

-

1

32. l{n

+

3)

+

11

+ 6

57 + 4 / + 14

2(« 9/7

-

6

+ 9c

9)c2

like terms.

4(7

22.

1)

+

6)

+ 9 407

For use with pages 18-21 Find the value

A

1.

if

possible.

6-0o

2.

If not, write impossible.

5-15

8-0o

3.

4. -^ o

4

5.

—

6.

—

7.

6

11.

^

15.

3x

impossible

8.

1

-t-

—

impossible

7

9.-^-^0

10.

—7

3

=

I.

Find the value of the expression.

13. l{x

-

1)0

Let X

"*"

^

impossible

12.

-

12

impossible

1

14.

X

XI

--

--

4

x3

16. -7-^1

4 Let a 17.

=

(fl

Find the value of the expression.

6.

-

1)

-

5

2x

-

3 2x

18. {2a

1

-

2)

-

10

1

19.

^-^

23.

4«

-

•

li

5 24. 2(x

+

20.

i

Simplify.

21. 3

+

22. 7

+

47

-

7 4/

+

5

+

-

2a

6a 25.

-|-«/7

26.

I

-cc

^•3x

27.

28.

/or use with pages 22-25 Tell

A

which of the numbers shown in color

3=4

a solution.

UorO

2.

x-2 =

4.

«

_

4

6.

«

-

5

1.

«

+

3.

5

+7 =

8

2' 2,

or

5.

6

-

4

4, 3,

or 2

a

=

Tell which of the 7.

9.

11.

408

is

«

+

1

>

1,

4, 5

16,

19,

3, 4,

10

15,

3

1

numbers shown

1^

9

12.

2c

>

3

10

6, 7, 9,

6, 8,

10,

4^,

1,

9

12

^

n

4)

-

8

2x

CHAPTER

2

For use with pages 34-37 Solve.

A

1.

x-2=46

2.

7-4 = 26

3.

«-3 =

5.

7-4

=

711

6.

x-3=69

1.

n

-1 =

9.

«

+

2

=

6 4

10.

jc

+

1

=

76

11.

;c

+

7

=

8

1

12.

7 +

13.

X

+

5

=

20 15

14.

7 +

6

=

14 8

15.

A7

+

9

=

15 6

16.

m +

17.7-5=2025

18.

X

-6 =

1319

19.

JC

+

6

=

1913

20. «

+

7

16 21

23.

jc

+

14

21 7

24. r

-

11

25 7

27.

jc

-

6

=

7 +

3

=

36

4.

r-4 = 48

6^3

8.

m-3

=

=

6

8ii

104

=

12

18 6

Solve.

+

21. «

+

25.

^

29.

15

9 16

= =

=7 +

17 8

22. «

-

5

22 6

26. «

+

18

30.

-

4

87

For use with pages

JC

= =

= 04

31.

=

18 24

27 24

=

14 7

=

22 33

28. 30

=

«

-

5 35

16

=

jc

-

1430

32.

For use with pages 42-45 Solve.

A

,

1.

2x

+

4

=

6

1

2.

4y

-

5.

5/

+

5

=

15 2

6.

6;c

+

9.

3fl

+

6

=

3610

10.

In

-% =

13.

7a

+

14

=

211

14.

2m +

17.

6x

-

12

=

307

18. 4fl

+

=

3

18

=

=

1

=

2

136 461

5«

-

3

=

22 5

8.

2«

-

6

=

20i3

137

12.

8x

-

5

=

355

51i2

16.

9«

-

40

=

328

20.

5/?

-

13

=

176

3«

-

7

=

7.

4«

+

5

=

45io

11.

3jc

-

8

=

15.

4«

+

3

=

19. 7jc

+

12

36 3

4l7

4.

3.

93

1 1

=

6

75 9

For use with pages 48-49 Solve.

A

1.

2x

+

3x

=

15 3

4.

2x

+

3x

-

5

=

7.

2«

-

5

8«

=

-

+ 6

-

10.

15«

13.

28

=7 -

16.

46

=

2;c

4

+

7«

+ 5x

2.

4a

-2a =

307

5.

5c

-

2c

45 5

8.

7x

+

3

-

11.

8/

-

3

+7 =

24 3

3j 8

14.

40

=

13x

-

6x

-

-

17.

19

=

8a

-

6a

=

34

5

37

+

10 5 8

= =

2;c

+

7

204 18 3

67

+

3/

6.

9«

+

a

-

2

9.

4«

+

7

+

3a2

1.

Three times a number plus

What 2.

is

the

number?

Cynthia's father

3.

How

old

Amy

has twice as

is

total 42.

Cynthia? 8

yrs. old

much money

as Barb.

Together they have $15. How much does Amy have?$io 4.

410

Our class has 90 students. There are 4 more boys than girls. How many boys are there? 47

=

7a

-

16

+

2a 8

=

6x

+

3x

-

4

6

18.

21

26 years older than Cynthia.

is

Together their ages

21 2

30 8

56

64.

= =

15.

is

28 3

7a

2 6

number

5 times the

=

-

10«

8

18 2

6

12.

For use with pages 54-57

A

=

3.

+

-

4x 5

For use with pages 58-59 Solve.

A

1.

2«

=

«io

2.

4x

=

X

4.

\6

-2y = 6y2

5.

5x

-

20

7.

7x

-

8.

3/

+

9

=

4^ 9

10.

67

-

13.

2«

+

10

12

+

= 4x4

=

2

1 1

4^

=

7

+ =

85

11.

3x

+

1

=

X

+

6«i

14.

5x

-

8

=

10

2.

l{a

-

+

73

+

2x6

+

2x

8x

6.

lOx

9.

14

-

2x

12.

5fl

-

4

=

8

15. 7c

-

4

=

2c

3x lo

+

=

3.

15 5

-

61

=

14

=

3x

2

5x2

- ^2 +

62

For use with pages 60-61 Solve.

A

B

1.

2{n

-

4)

=

I612

3)

=

^al

3.

5(z

+

2)

=

10z2

4.

6(7

-

3)

= O3

5.

8(x

-

2)

=

O2

6.

5(m

7.

4(x

+

3)

=

6x6

8.

6(7

+

3)

=

I80

9.

3(2

-

x)

=

10.

4(x

-

2)

=

3x

+

11.

3(x

+

1)

=

2x

+

10?

12.

2(5

-

x)

= 3x2

13. 2(c

+

3)

=

3(c

-1)9

14. 4(r

-

=

2(r

+1)5

15.

4(x

-

4)

=

money

as Bob.

2io

2)

+

2)

=

,

253 3xi

2(x

-

1) 7

For use with pages 62-63 Solve.

A

1.

2.

Al has 3 times as much Bob has $10 more than Together the three boys How much money does

Ed.

have $65. each have?

Ed: $5; Bob: $15; Ai:

$45

Jo, May, and Sara save dimes. Jo has twice as many dimes as May. Sara has 4 more dimes than May.

Together the three have 100 dimes. How many dimes has Jo? 48 dimes

B

3.

Ned is 1 years older than Rob. Next year Ned will be twice as old How old is Ned?21 yrs. old 1

as

Rob.

411

CHAPTER

3

For use with pages 72-75 Compare

A

the numbers. Write

-2

1.

5.6

-7>

?

Graph the

/or

-5>

?

9.

«

+

1

13.

X




3.

0> line.

3

=

2

7.

?

-4>

4.

-6

?

-1


8.-7

?

-3
)4(4«4M)

23. i4rsf (2s

-xV

36x4/4

(3m«)3(-3m«)

26. (2x2^2)(2;c3y3)2

-8I/774/74

8xV

wiV/r

(xY

12.

14. (x7)2 x2/2

-a^b* 25.

11. (j;5)2 yio

27.

8000/r7i2ni2

-BAa^/b^

pages 144-147

Multiply.

A

+

1.

2>{n

4.

— 2(a —

1.

-3{2a

5)

3/7

3Z))

+

- 6a -

2.

-4(x

-2a + Gb

5.

«(x

8.

- \(5x +

+

46 1

6c)

26 -

1

-

-

a)

20 - 9a +

a2

a)(4

-

16. (Sa

B

19. (a

-

+

\)(a

3a2

l)(a^ a^ +

+

+ n^

+ a

—

_

s^^s

-

5)(«

+

5;c2

-

jc ^^2

-

y)(x^

-

\){d^

9.

4x(x^y + xy + Sy) 4xV + 4xV + 1 2x)^

+

1) 1

-

3)

+ 2mn +

+

4){x

«)

8)

-

6)(n

9)

- 3n - 54

-

n){m'^

-

n^)

m^ — mn^ — nm- + 24. {n

-

3/) -Bx^ - Qxy

+ 12x + 32

n^

(m

e/,^

+ 2m^n + m^n

m'^

21.

/72)

m2(w2

j^

18. (n

2xy -^2)

+

a4

— 3j>c(2x +

+

x^ — 3x^y + xy^ + y^

\

+

6.

15. (x x^

3)

g^a

6{a^

1)12.

+

+

3.

z

+ 2n - ^5

-

20. (x

1)

+ 7)(x2 + 2xy + /) 23. {d^ x3 + 3x2/ + 3jf^2 + ^

22. {x

z)

{5x - 2)(5x - 4) 25x2 - 30x + 8

17.

1

—

+

2y

-3x4 + 14. (n

2a

a"^

ax - a/

_

-x{3x^

11^

1)

+ 2ai-

-4x + 4/

-5x - 2y -

^2 ^ _^ 1) 10. 5a(a3 Sa^ + 5a^ - 5a2 + 5a 13. (5

— J^)

8c

_

+

- y)

+ 15

+

2)(«2

+

n3

+

+

6a72

4„

n^

+

12/1

4)

+ 8

For use with pages 148-153 You may use

Multiply.

A

1.

(«

+ /j2

4.

(3x

3)(«

+

-

7/7

(2jc

FOIL

method. 2.

4)

+ 12

+

2)(2x

6x2 7.

+

the

+

5jf

+y)(4x

_

x2 5.

3)

A

- y)

8.

is its

-

\){5y

area?

(2«

6/72

+

+

17/7

1)

+

3.

+

b)(a

-

2b)

(2n

+

5)

5)(m - 2) - 7/77 + 10

6.

(la

+ 3)(3a - 1) 21a2 + 2a - 3

9.

(5x

- y)(3x - y)

1

- 3a6 - 2b^

cm by

-

(m

/n2

/

1)

25/2 _

2a2

rectangle measures (3«

What

(5/

6)(;c

+ 3x - 18

6^

8x2 + 2x/ - /2 10.

+

(x

15x2

cm.

- 8x/ +

/2

v

+ 5 cm2 417

Express as a trinomial. 11. («

2)2

12.

+ 4n + 4

n^

15. {X x^

+ +

^2

- \{n +

— 0^ —

a2

2)2

23.

/72

a2

+7)2

-2(4jc

A =\bh

/I\ \. /

^

=

A =

or

+

8/7

+

+

7)2

-

+

21. 4(2r

+

16rs

(;c

-

10)2

- 20x + 100

x2

\Obf

- 20a6 +

16r2

14.

+ 49

14a)

17. («

- 4a6 + 46^

20.

Find the area. Use

B

13. {n

-32x2 _ i6xK - 2/2

— 4

4/7

3)2

- Ibf

16. (a

37)2

+ 6xy + 9y'

19.

-

(7

- 10/ + 25

18.

(3m

+

2«)2

9,772

+ '\2mn +

4/72

5)2

22.

-3(2x -

3^)2

+

- 1 2x2 + ^Qxy - 27/2

100/>2

4s2

Iw.

24.

1

/

=

3«

+

1

1/7

1/1/

I6/72

H

/br

iis^ M'lVA

+

/:

I8/72

2/7

pages 154-155, 158-159

Divide.

3

1.

3.—.

2.

4

X

M 4/j

6x^

5.

+ 6n

-?^

n

3x2

-49a^b^

63/2^

2j!c

4x

-

9.

Sy —^2x-4/

10.

-

6a

—

6b

x^y^

xy

6 20/2^

10^2

_

13.

5^

"I

J

^

xy ~|~

^X

8x2j

+

- 2n -

12x^2

_

-X -

1

iDX

3x2

4.

5^3

16^3

16.

.^^

17.

4y 2x2 + 2xy - Ay^

-

1

+

6a2

-

18.

8a4

Simplify.

B

-

— x'^

5/2 4aj2

Zxy

15/72

12.

-a + b

-

—

\2xy

IO73

-2\ab

4^2

4x

5/^

lb

a

19.

+

30/72/2

CHAPTER

6

For use with pages 170-173 Find the prime factors.

A

1.

20 2

40

2.

5

2,

2, 2, 2,

5

2502,

3.

5, 5,

5

4.

no

11

2, 5,

140

5.

2, 2, 5,

common

Find the greatest

6.10,25 10.

factor of each pair.

5

90 15

15,

7

7.32,284

8.

40,48 8

12.

11.

26,

39

9.63,14?

1

70,105 35

13. 55,

242

ii

Factor.

-

14.

14

17.

2^2

20.

lOxy

2x

+

-

2(7

6x

2x{x

15.

20n

18.

y^

21.

1

X)

+

- /2 3,(iox

3)

-

y)

-

—

4 4(5n -

3y y(y -

_

1^2

i)

3)

44n ^^n(n -

4)

-

-

16.

12

19.

14^^

+

7x 7x(2x +

22.

6ab^

+

24^2^

36jci2(1

6ab{b + 4a)

_

23. 3n2

+

9„

12

24.

B

26. 5

28. a'^b

-

30.

+

a^b^

ab(a^

-

a^b^

a^b

+

-

aA^

-

+

4^2

- 3n + 4) - 25« + 75«2 + \25n^ 5(1 - 5n + 15n2 + 25^3)

12^7 + 24/2 + 3x/ + 6/2)

4(x2

3{/72

'

6xy

27.

25.

7a

+

7(a

+

+

\4b

+

2/)

70c 10c)

+ 42^2/ - 66jcy + llx^

+ 7x/ - llxV^ + 12xV) 36^^2 ^ 43^2 29. 12^2^ 1 26(a2 - 3a6 + 46) 6x/(1

_

ab^ b^)

lOO^y _ 75^:^ + 50^2/ _^ 25xy 25xy(4xV - 3xV + 2x/ + 1)

15m2«4

31.

_|_

45m3«5 - 60m^n^ + 3/77/7 - 4m^n^)

15/772/74(1

For use with pages 1 76-181 Factor.

A

1.

j2 (y

4. ;c2 (X

+

7^

+

3){y

+ +

+

1)(x

33

+

rt2

5.

^2

+

+

\9b

(b

+

9)(b

10. ^2

+

14^

(a

+

12)(a

+

2)

13. jc2

+

14x

+

13

90

8.

^

24

(x+13)(x+1)

13a:

+

11.

14.

24

+

4)

+

36

+

4)(x

12)(/77

n2

+

i9„

(n

+

17)(/7

+

lOy

72

(y

+

8){y

+

+ +

36

+

+

3.

6.

9.

3)

34

12.

2)

+

16ai

+

6)(/7

+ +

n^

+

16«

(n

+

^^)(n

+

13«

+

12)(/j

n^ (/7

60 10)

+

55

+

+

5)

12

+

1)

m2 + 17m +

16

{m + ^6)(m +

2)

16

n^ (/7

9)

m2 + 15m + (m +

+10)

6){n

+ (x

+

io«

+

(n

+3)

b^

7.

2.

4)

+

\4x 1

12

+

15.

x^ (x

+ +

16a:

14)(x

+ +

1)

28 2)

419

3x)

1)

-

16. b"^

19.

_

y2

+

\0b

-

(b

14^

16

17.

20. «2

45

_^

(K-9)(K-5)

-

22. ;c2

-

(X

B

25.

+

(x

_

^2

80

+

40)(x

+

i/s^

with pages

Many

Factor.

A

1.

4.

_

j2

5^

-

36

-

9)(n

-

1)

_

i8„

+

-

93

29.

-

10)(n

21.

24.

100

+

(/I

5)

_

{-jy

+

60

27.

x^

2)

{y

40« + 144 - 36)(n - 4)

33. n^ (/?

^^){y

-

101;c

+

lly

9

_^

_

(y

x^

-{-

+

5„

-

4

(6

(„

13.

-

y2

7y

^2

+

lOx

Watch

Factor.

1)

72

-

18)(/

-

4)

+

21«

+

68

+

4)(n

+

-

\4n

+

49

(n

-

7)2

17)

182-187

3)2

-

2.

x)^

5.

x2

4-

+

«2

+

+

4)(„

1)

8.

-

^2

+

16jc

64

+

20n

(x

100

(5

-

11.

x)2

m2«2

+

13x

+

(n

3.

S)^

+

10)^

42

-

7)

_ 4^„

_^

-

(X

10. 25

100

- 100)(x -

are trinomial squares.

12;c

+

«2

4)

+

6.

n^

-

81

18x

+

x^

-

x)2

(9

7.

5)

44

-

-

-

y^

+

15^

(x

30.

-

^2){y

_

4)

(m - 30)(/n -

_

-

-

/^

15

-\-

3)(n

(/

m^ - 32m + 60

32. «2

^^

Sn

-

{y

8)

+

-

n^ {n

gO

29a: (X + 25)(x

2)

(K

/or

9

+

26. jc^

24y - 31)(K - 3) 31. ^2 + 20x + 64 (x + 16)(x + 4) 28.

+

{n

9)

+

42x

23. «2

18.

5)

-10/2

(n

90

-

10)(x

+

x^

+

19x

m2 - 9m + 20 {m - 4)(m -

-2)

8)(b

6)(X

(mn -

9.

_

j2

22/ (y

4

_

12. /,2

+

121

-

11)2

+

iqZ? {b

2)2

-

25 5)2

the signs.

-

-

8 (K

8)(k

+

i)

14.

-

n^

-

2n

\5(n

-

5){n

+

3)15.

+

y^

6y

-

40

f/-4)(/+10)

_

16. «2 {n

19.

-

_

n2 (n

_

8„

-

17.

m2 - 5m - 50 (m - 10)(m +

2)

_

8«

9)(n

20

+

10)(n

9

20. n^

+1)

(n

13«

-

-

^)(n

+

-

«)(m

+

18.

-

^2 (jf

5)

-

_

21. «2

14

-

7;c

12)(x

3^

60

+

_

5)

4

{n

-

3.

(1

-

5x)(l

+

^

(^ _^ 2b){a

-

9.

(4«

+

3)(4« - 3) 1 6n2 - 9

12.

(8x

-

3)(8x + 3) 64x2 _ ^

14)

4)(n

+

1)

For use with pages 188-191 Multiply at sight.

A

+

-

4)(x

1.

(jc

4.

Oy +

7.

(5x

+

1)(37

4) x2

-

2;;)(5jc

1)

-

- 16 9/2

2y)

-

1

2.

(m

5.

(2x

8.

(8m

(1

-

6a){\ 1

420

+

6a)

- 36a^

11. (9

«)

m2 -

- y){2x + 7) 4x2

25x2-4/2 10.

+

-

-

«)(8m + n) 64m2 - n2

2Z7)(9

+

81 -

2b) 4/)2

n2

_

yt

5x^ 2Zj}

Factor. 13.

^2 -

(X

B

-

25

5)(x

+

5)

CHAPTER

7

For use with pages 204-207

The

table below

shows the average lifespan of several animals.

Bat,

6

yr.

Find three solution pairs for the equation. Answers may vary.

(-1, -4);

=

13.

7

(0,

-3);

+

17. 2jc

(-1,

—

X (1,

y

3

-2)

-1,

=

2); (0, 0); (1,

-2)

Guess an equation

B

14. (

y + X = 4x

18.

(-1, 16); to

5

(-1,

6); (0, 5); (1, 4)

(0,

+

=12

y

12); (1, 8)

go with the

15. y

+

3); (0,

1); (1,

-

y

19.

(-1, -2);

2x

=

1

-1)

=

table.

y = 3x -

y = 2x

For use with pages 220-223

Draw

the graph of the equation. Check student's graphs.

\.

y = 5x

4.

X

1.

y = —Ix

What 10.

-\-

is

y =

1

the slope of the line

y = 1x1

11.

when

2.

y = X +

5.

y — 2x =

6

8.

7 = 3x —

1

its

graph

7 = -Zx -3

is

5

drawn? 12.

y

=

\6.

^

y + 4x =

(-I, 11); 20.

(0, 0); (1, 2)

X

21.

y = X — 5

A

2x

y

(-1,

1

(0, 7); (1, 3)

-5x =

\0

5); (0, 10); (1,

15)

CHAPTER

8

For use with pages 236-239 Solve by the graphing method.

A

\.

B

5.

y = -X 7 = x + 2(-i,i)

7=x-3 7 =

3a:

+

1.

6.

y -Ix 7 = 3x-3(3,

2x+7 =

X— 7 =

2

3. 6)

3

7.

7

+7 = 3 — 2x +7 =

x

4.

-3(2,

3x-4 = — — 3x

/ = j;

+ —

3;c

1)2a:

x+7

8.

1

=7

1

=7 (-2, =

3x-7 =

_1_

6

2

(-!-¥)

(f-¥)

(l-l)

(l-l)

For use with pages 240-243 Find the slope of the line whose equation

A

1.

Do

7 = 3x

3

1.

is

given.

y = 2x + \2

3.

7 =

-2>x

- 1-3

4.

7 =

1

1

^x + 5^

the equations share one solution pair, no solution pair, or all solution

pairs?

S.

y = 2>x 7 = 3;c— 2no

6.

x

y =

-\-

X— 7 =

d

1.

2 one

x -\- y = Q 3x + 37 =

8. all

^+7

3

X

12 no

= +7 =

For use with pages 244-249 Solve by the substitution method.

A

1.

7

=

3a: (3, 9)

a:

-f

7 =

2.

12

7 = Ax (2, 8) 7 - a: = 6

3.

- 2y = -4 3a: + 7 = 2 (0,

X

4. 2)

-7 = 3 + 7 = 5 (2,

3a: a:

Solve by the addition or subtraction method. 5.

a: a:

(

+7 = -7 =

4 6

6. (5,

-

1

)

2a: 2a:

+ +

27 37

= =

5

7.

4

3a: a:

- 27 = - 27 = 4

- 27 = 18 5a: + 27 =

8.

x

12.

x

Solve by the method easiest for you 9.

3a:

4a:

+

7

=7

2=7

(-9, -34)

424

10.

7

=

3a:

4x

11.

-7 = (0,0)

a:

=7 + 57 =

3a:

-

=

47

3

8

a:-67=1

(1, 1)

(7, 1)

3)

-5)

B

13.

-

X X

5y

4y

= =

14.

3x

+

4/

X

-^

4y

9

(45. 9)

= =

6

5x

15.

\5x

7

(4-f)

-/ = -\-y =

5

16.

10

+

4x 4x

(l-f)

= 3 = -2

\2y \0y

\44

22

/

For use with pages 250-257 Solve. 1.

= 5 + 37 = 4

4-

jc

2jc

2y

2.

-7 = 4 + 27 = 6

2x 3x

(-7,6) 5.

3a: — j^ = X + 2/ =

9.

a:

- 47 = 2 - IO7 =

3x

6

6.

+

2a;

X

2

37

=

7.

1

+7 = (11,

4 -7)

4.

x - 37 = 3a; — 7 = (1.

57

8.

9

-1)

(1-

4 4

=

+7 = - 57 = 2 — 27 = 5

3jc a;

(-21, -13)

-1)

variables in solving these problems.

The sum of two numbers is 15. One number is twice the other.

What

are the

numbers?

5.

10.

10

Two hats and 3 ties cost $15.50. One hat and 2 ties cost $9.00. What does each item cost? hat: $4.00; tie:

11.

-

4jc

X

4 (-2, -1)

(2.0)

(2,0)

Use two

3.

Two bats and ball cost $10. One bat and 2 balls cost $8. What does a bat cost? $4.00 1

13. Al's score is 6

Their scores

Find their

more than

Jo's.

12.

total 16.

scores. Jo: 5; Ai:

The sum of two numbers is 15. One number plus twice the other Find the numbers.

14.

1

$2.50

5,

25.

is

10

One egg and 2 coffees cost $1. Two eggs and 1 coffee cost $1.25. What does one egg plus one coffee

cost? $.75

Solve.

B

15.

2x 3;>c

18. 3;c 2a:

21.

+ + -

37

47

47 57

= = = =

16.

6(18. -12) 5

19.

1(3, 1)

4x 3x

—

37

2x

-

47

3a:

27

57

= = = =

1

0(-2, -3) 4 l(-8, -5)

17. 2a: 5a:

20. 4a: 3a:

at 25C each and some pens at 75c The total number of pens plus pencils was 25 and the total was $6.75. How many of each did the club buy? 24 pencils;

+ —

27

= =

+

37 4y

= -2 =

57

(0, 0)

1

Tom's club buys some pencils each. cost

1

pen

425

CHAPTER

9

For use

pages 268-273

after

Simplify.

l6 4

6.

p^ 25a 5a -y^

12x2

3x

+ 2« +

^

^

x'y'

30;cv

6

*^-

1

^^

17 *'•

2

2

SxY

iO;c

-

1

«

21.

-1(-1 +

r

7^2

"-"•

-

9.'

r

8

5„3^2

„5

-^

18 *•

10

;,2

^'^

20 5

3

18

—-20xy^_^ -^—

10.

:r

lOm^n 5m

'

'^''*'

_

40^25

^

^^y + lOy

^^

66^ •

->^

_^2

1

24x4;;3 •

1

24^4^4

i2;c3y2

10

X + y

•

^^-^^ I X2a - Mb 3

— 1.

Write the expression with a factor 20.

—

8.•

7

14a:

^-

4

12

7.*

-4xy^

"^

16

•^-

5

15

X -i(-x)

-

22. 6

4y

-1(-6 +

n)

23.

-^2 +

^2

-1(n2-z2)

4k)

Simplify.

m— 24.

4a?2

20

—

l-«-

-An

g

a:2

^y

•;c2

/or Ms^

+ +

wiV/r

25.

1

—m

n

z

4

29

26.

a

—

_

3

4^ — 8 48 _ i2„2_

^Q

1

3

+ x

2a:

6x

— y2 — y ~ ^

a:2

27.

-,|

1

—« _3

3«2

1

-x -

y

1

3n + 3

6 + 3n

+ x + 5^ 1

+ +

1

^^

x^

—

Ix

-^

12

x - 3 + 2

•;c2-2jc-8'

5

— 3a: — 10 x - 5 •^•;c2-6;c-16^-8 ^^

a:2

pages 274-283 in simplest form.

18 points to 12 points

3.

$6 to $24

3 ^

1

^

ratio of girls to

How many

2.

4 days to 40 days jq

4.

9 students to 36 students ;J;

There are 21 students

The

— 3y -3 y ~ ^

3a:

-1

4

1.

5.

426

-^

z

9-a:2

Write the ratios

A

— —

d

n

in a

boys

is

math

class.

3 to 4.

boys in the class?

How many

girls? 12; 9

Solve the proportion. 6.

^ = -^ X

10.

—=—

7.

X

+

«

Five apples cost

How much

X + 2 -— —=— X

4

1

15.

30 < 22.-2 ? ,

,

or

=

in

place of the

-4

23. 3

5x

+

25

15.

3n

-

n

Can you

-

(2x

18

-e

=

-5

= - 16 -

1)

=

-8 X

-

1

^^=-39 —

21.

J

?.

-(-4)

> 25.

3

12.

9

Write

= 3o

x

10 8 18. 4

-81

+

6.

?

-3(-4)

24. 6

-(-2)

?

(-2)(-4)

< give a value for

—?

for

—? U

O ,

State the value if possible. 0; impossible

Problems 1.

A number is doubled. the

2.

3.

5.

is

added. The

sum

is

Find two consecutive whole numbers whose sum

Mary has dimes.

4.

Then 4

80.

What is

number? 38

twice as

is

65. 32,

many dimes

How many

as Don. Together they have 105 dimes does each have? Mary: 70, Don: 35

South High School played 32 games. They won three times many as they lost. How many did they win? 24

Tom

says "If

subtract

5,

I

I

33

triple

get 25."

the number I'm thinking What is the number? 10

as

of and then

439

2

Cumulative Review for Ciiapters 1-6 Basic Skills Simplify.

1.

15.JL0

2.

-(-3)2-9

3.

6.

-4a +

a -la-'iOa

\5x 4.

2(x

7.

(jc

-

-

8)

+

2x 4x - 16

5.

10x2 _\x2 9^2

27)

+

(3x

+

8.

(x2

27) 4x

-

+

2;c

-

1)

(^2

^

(_^)2

+

2x

-

+

4a

-4x +

1)

\

a^

+ 4a +

2

Multiply. 9.

Aa-a^Aa'

13. {x

+

10.

-

6){x

1) x2

(-3xv)(4x2y2)-i2xV H. (-3^6)3

+ 5x -

14. {n

6

-

4)(« «2

+

15.

4)

-7)

-27a3/,3 12. 2x(5

- 16

lOx - 2x/

{y

+

2){y

-3>)y^-

y-Q

Express as a trinomial. 16. {X x2

-

3)2

17.

- 6x + 9

x2

(jc

+

+ 8x +

4)2

18. (2;c

4x2

1

-

1

-

3)2

19.

(3x

-

9x2-1 2x/ +

2x + 9

2^)2 4/2

Divide.

-24^2^2

8«3

6«4

-^^^

24.

-

—5x

a

+ 3x - 5x2

1

^

26. -^^^

25.

— x'^

-72xv2

5a _ ^2 _ ^3

_j,

+ 2xy^

Factor.

27.

3x/

-

6x2j2 3xK(i - 2xK)

28.

a:2

(X

30.

jc2

+

4x

+

4

(X

+

2)(x

+

2)

+ +

5x

2)(x

+ +

6

+

72

+

7^

3r'je2

-

9^2

29.

6

(/

_|_ 9„ _ 22 (/7+11)(/7-2)

31. «2

(x

+

Solve.

440

X

-

5

=

20 25

2.

40

=

20

-

1)(k

+

-

3/)

3/)(x

.^

Equations and Formulas

1.

+

3)

«

-20

3.

4fl

+

1

=

13 3

6)

^

4.

-=

-4-16

-

6.

5x

+

30

8.

3x

-

{2x

Solve for 10.

6jc

2(x

7.

-10 = -3x + 24

9.

{2a

1)

3;c

+

2)

-

1

=

5x

+ 6-2

+

1)

-

14 8

(5a

-

1)

= -11

x.

=

Solve for

=

=

5.

4

24 4

=y^

11.

6x

\4.

y + X =

12.

kx

=y

j;.

13.

7 +

16.

Solve

7

=

10 3

P =

^x for

x.

22-

17.

g

Solve

15. 4jc

D=

rt

1.

Find the perimeter of a 4-cm by 4-cm square. 16 cm

2.

Find the area of a rectangle with length 2x and width Ax.

3.

A

triangle has

an 8-cm base and a 4-cm height. Find

its

for

5.

Find two consecutive numbers whose sum

6.

Wendy bought some cupcakes at did she buy? 20

Her

profit

\2


3.

3


'2(3a:

2.

8

3.

+

3x 5v)

4. 6.

2m« 9mn^(2

— m)

7.

Page 187

U

1.

+ 9)(x + 2) 2. (>' + 12)(_>' + 2) - 4)(fl - 2) 4. (m + 13)(m - 2) (x - 6)(x - 6) 6. (x + 12)(x - 4) {a - 3)(fl - 9) 8. (/ - 6)( F - 7) {b + 27)(^ - 3) 10. (m - 14)(m - 4) (jc + 10)(x + 10) 12. (m + 8)(m + 8)

3. (« 5. 7.

9.

11

,

Page 193 169 2. 9 - 16x2 3. 25^2 _ 49 - 8)(fl + 8) 5. (m - 6n){m + 6n) {Ax - 3v)(4x + 3v) 3(x + 3)(x + 2) 8. 2( V - SXj + 3) 2(m — 2«)(m + In)

1.

m2 -

4.

{a

6. 7.

9.

CHAPTER

Page 257

(-ll ll^

2.

(2,0)

4.

Yvette: 15; Lauren: 18

95c

milkshake:

:

(-2,1)

3.

1.

hamburger:

5.

60

21.

1.

23.

7

33. 5ar

Page

23

Page 23

Not possible 31. Not possible 35. Not possible

29.

+

31. 8x

Impossible

7.

5.

13.

27. Subtract 9.

1.

3.

17. 4fl

23.

II.

20 and 14

40 and 68 23. 12 and 18 lx,2x + 7 29. 11,22

18

15.

Page 9 I. 6x

17. 9

9.

21.

4

+ 5y +4

206

Page 21

14

II.

43.

8.

35

3.

I.

31. 96

21.

13.

27.

17

ba

37.

29. 36

35.

Page 19

11.

21.

29. 26

10

27.

39. Subtract 4.

Page 7 1. 24

24

35. In

7

10

7.

17.

10

25.

+

3

5.

24

15.

+ 15 + 32x

27. 7jc 33. 8

lOfl

+

37.

9.

6

19.

+

2

3

35

11.

21.

24

29. 6c

+

3

b 16

39.

125

\0x 47. \6b 53^ 6^3 55^ 30^2

45.

49.

6P

57.

2x

+

10

59. 20jc

61. 8c

-

\2d

63. 3a

51. 6x2

- \0y - \W 1

65. 3«

+

+

15m

69.

bb

8x

81.

83. a

93. 4

91.

41. 6

43. 3

45. 5

28A:

53. 4

55.

+

77. Impossible

79.

57. 2n

89. 5n

59.

+ +

18a:

36^

20w +

71.

5/j

75. Impossible

-

73.

67.

6x

85.

95. 3

10^

87.

97. 8

101. 2

10

99.

^n

16

2

23. 8

25.

43. 9

13

35. 6

45.

12

5.

13

7.

24

9.

11.

17. 9

19.

30

21. 25

27. 5

29. 35

31. 51

15. 9

10

33.

10

3.

15

37.

39. 21

11

41.

19

4

12

8

5.

17

15.

12

17.

5

23.

13

25.

10

27.

13

Mixed

18

7.

14

9.

29.

15

11.

12

21.

17

19.

31. 8

15

18

/?

11.

5

27.

17

29. 27

31. 46

25. 4jc

11. 3

2x 31. 2x

+ + +

12

25.

180

33. 54

35.

12

17.

5

7.

9

21.

12

27. 25

37.

10

9.

19.

18

29.

39. 56

15

44 41. 48 31.

35. 27

14

45. 3 55. 21

19

53.

5.

14

25. 2

23. 21

43. 5

5

15.

3.

23

4

7

7.

20

9.

6

19.

1

27. 21

29.

10

17.

«

Ax

c

37. 51

47. 21 57.

39.

8

5.

= 63; 9 = 73; 34 = 24; 8

first

equation

is

5 8

in

them. Multiplying by

amount,

2,

+

jc

3

5

3.

y

+

11. 3jc

+

6;

21.

X X

Answers may

2

13.

+ +

5. jc

1.

a:

9.

+

+

D

61. 6

same amount or doubling the

= 2 X 3b = h =\ = 9 7. 4f = 12; c = 3 = 3 11. 3 13. 3 15.

5

3>;

4

17. 2

19.

5

21. 5

23. 6

25. 5

27.

29. 5

31. 2

33. 5

35. 3

37. 3

39. 7

10

Curto

7.

15. r

+

2; c

4

vary.

13

results in a full glass in either case.

5x = 10; X la =\%\ a 6« = 18; «

I

19. c

11 2;

+

Z?

+

Annie

Pages 43-44 5.

2x

29.

Puzzles and Things, page 53

true only in the sense

that both glasses have the

13. 2

1

23. 7

3x

Puzzles and Things, page 41

The

11.

5

9.

21. 5

41. 30

51.

59. 28

to

doesn't matter

It

8

7.

19. 5

31. 5

18

49. 20 19

11. 6

21. 21

1

+

1.

9.

17.

Practice Exercises, page 40

II

20.

=

9

Pages 51-53

13

43. 72

33.

3.

27.

14

1.

4

25. 4

5.

+

n

always obtain 20.

will

17. 2

23.

13.

20

6

1.

Page 39

Mixed

+

15. 2

7

12

5.

with.

21. 33

3

9

3.

number from

29

13.

80, \QQn

Subtracting the original

19.

7

172

11.

what number you begin

19

15.

4329

5.

Page 48

6

9.

is

17.

5.

+

I00«

4,

17

3.

302

3.

you use algebra to solve this problem, you will notice that the next

7.

15.

23. 30

4

+

11

3.

13. 6

1.

15

If

7.

Practice Exercises, page 37

9

1.

2>

Page 47 5n

15

47. 25

15

13.

3.

19

25.

the last step

10

\1;

51.

12

Arithmetic: 40, 44, 880, 800; Algebra: 5«,

Page 37 1.

16;

=

«

2042 27. 500 Down 7. 27 9. 41 15. 3204 3. 3921 19. 46 21. 925 23. 10 17. 1120

Page 35 13.

5

1.

865

13.

1.

= =

4

49. 6

47. 7

= 32; « = 6 n =

8

Page 45 Across

CHAPTER

2a7

3x

+

1

1;

25; 8

1

15. 2

17.

25. 7

27. 3

29. 2

37. In

=^n +

13.

19.

1

1

31. 9

21. 2

23. 2

33.

35.

+

39. 2«

12; 3

1

=

24

23.^

-3-2-1

10

1

2

3

3

4

5

1

2

3

2

3

8«; 4 25.

Page 61 1. 6x 7.

+

24

15

3.

48/?

16jc

-

2

11.

9.

17. 2

19.

4

29.

31.

15

39. c

41. a

21. 3

40 3

33. 3

4

13.

23.

+

48/

5.

-3-2-1 29-

c

42,.

.._4

27.

27. 5

37. 4

14

35.

10

15.

25. 3

-10 12

24

4

^

I

I

I

-3-2-1

I

1

Pages 62-63 calories

31.

-4 -6 or -6< -1 5. 5 > -4 or -4 < 5 7. > -7 or -7 < - 9. 8 > -8 or -8 < 8 11.-17. Check to be sure you I.

-2




7.

>

3.

=

2

>

5.

^''-

4-hH -3-2-1

13

37.

15.

>

7.




= -6

a:

—h-^H—

h-^

\

19.

-13

27.

-6

35.

-13

43.

71. 17.

83. 91.

-3

23.

1

-11

31.

18

21. 29.

-12

37.

39.

3

2

1

25.-11 33. 4

-5

41.

1

45.-15 47. Ix - y 49. W^ -6a: + 7>y 53. -4 55. -9 -7 59. -17 61. -24 63. 24 -72 67. / - 5/ 69. -2a: - 6 -2,a + 3b 73. -x^ 75. -y + 1 X + 9 79. / 81. -y -6x + 9 -9 85. -4 87. -10 89. -4 -5 93. -2 95. -8 97. 9 99. 2 -3a:

101. 5

CHAPTER

4

Page 103 I. 22

9x

II.

15

3.

5.

The width

3a:

is

10

16

m2

+

3

lla

7.

cm and

9.

14a:

the length

is

cm.

7.

49cm2 16cm2




b.

lOx;;

23.

31. 6

15.

-2 25. 11 or -6 33. 14

tamer must be bald; since Geraldine and Amos have hair, Christopher is the lion tamer. Then, by lion


a

7^

_,_

_,_

1

11.

_4x'y

17.

-Ax'^y

Melba and Fran are both wearing black would know he must be

hats, then Brent

wearing red (since there are only 2 black hats). Since Brent does not know, either Melba or Fran (or both) must be wearing red. If Melba is wearing a black hat, Fran

23. 3r3/2

-6x

25.

21.

27. 4fl^c2

'f29.

-Id^b

1

or 59 minutes.

Page 159 a

8/?7

21.

-\-

17.

25.

Page 153

+ 4r + 4 9 7. x2 - 4x + 4 5. x2 16 11. / _ I4y + 49 8x 9. ;c2 15. m2 + 14m + 49 13. x2 4- 18x + 81 19. 9^2 + \2a +4 17. «2 _ 10/7 + 25 23. 9x2 ^ ^^y +^2 21. 16x2 - 16x + 4 25. 100^2 _ 20ab + f^ II. 16x2 - 24x7 + 29. 2fl2 + 4a + 2 31. 3x2 _ i2;c + 12 33. -2x2 + 8x - 8 8x 6x

+ + +

16

3.

3/) 5. 4x - ly 3. 2x - _y + 9/7 9. a + 9 11. b"^ - ab b'^ 15. 'imn^ — a^ + ab + 2m X - 3x2v + 2 19. Irs + 3^2 - 6^2 -Ad^b + fl2/,2 _ 2^3 23. -2 - 6« 3x2 _ 8;c_y + 9/2 -3x^ + 4x3 _ 6^2

+

1.

7.

wearing a red

hat.

number doubles every minute,

and the basket is full after 1 hour, the minute, basket was half full in 1 hour —

13.

+ + -

3jcyz

19.

31. 27fl3c4

would thus know she must be wearing red. Since Fran does not know, Melba must be

x2

——

7/74

Since the Puzzles and Things, page 151

I.

i5_

Puzzles and Things, page 157

39. 6x2

If

6m2

2x3

13.

27.

1

/•2

V

Page 161 1.

x

9.

a

+ -

2

3.

5

11.'

v

+4 -6

x

+

2

13.

X

-

5.

X

7.

x-4

2

Puzzles and Things, page 161 Yes; "Nothing split" really

is

better than a

means "There

than a banana

split."

is

banana

nothing better

If there

is

better than a banana split, cannot be better than a banana

nothing

then crackers split.

1

Page 162 1.

^2

^4

3.

13. 4^

^3

15.

33.

^2

^2

7.

17.

^7

5.

4 23.

-^

-^

35.

^5

9

^ 4

41.

11.

29. (x

33. (2

Green Mouthwash: $.12/fl. oz.; Fresh Mouthwash: $.115/fl. oz. 3. Dazzle shampoo: about $.2l/fl. oz.; Dew Drops shampoo: about $.20/fl. oz. 1.

Pages 166-167 7.

-

2

3«

9.

23.

+ 8^2 _a2 + 8 Am - 5n

29.

flS

13. 6^2 19.

+ +8

2

5x2 11. 4«2

^ 7^

2x

-\2y

5.

15, 5

i-j

Ix

25.

-

2y

33, 28jc4

-a^

35.

I.

31. {n 35. (x2

+ y){m - x) - A){y + 2)

_ Zj2)^ +5(x + 3) 3.

7r)r2

5

1,

3.

II. 3, 7

-56/

41.

-jc^/

-lOx

_,.

1

-6ab^

75.

+1

83. a^

+ 5a + 6 + 4/ + 5

3x

b) b)

(a2

5,

72

Pages 178-179

37.

43.

4I.

79. 6n

87.

x2(l -tt)

-7x

73.

65.

7.

4

2,

5.

13.- 3, 8

4

3,

7.

3,

5

9.

3,

6

15. 6, 6

+ 6)(fl+ 1) 19. (X + 6)(x + 3) + 3)(r + 7) 23. {n + 1)(« + 16) {a + l)(fl + 4) 27. (x + l)(x + 10) {n + A){n + 2) 31. (x + 7)(x + 2) + 3)(^ + 9) (7+ 1)(7+ 11) 35. (x + 4)(x + 8) 39. (x + 3)(x + 10) {a + l){a + 10) 43. (x + 25)(x + 2) (m + 25)(m + 3) 47. (a + 21)(fl + 3)

29.

69.

57.

61.

(4

33.

+ 5n 18x - 27/ 51. n^ -5n +6 6x2 _ 5;^^ _y2 55, loa - 31a «2 _ 9„ 18 59. 30«2 _^ i6„ + 2 3/ + 13^ + 4 63. «2 4. 8« + 16 67. 4x2 + 4x7 + 72 x2 + 2x + 9m2 - 12mw - 4«2 71. 3n

53.

-

1.

25.

27.

m^n^ 47. ^2 + 4a

39.

45. 30 49.

+

2){a

a){a

21. (r

- 2/

-jc2

21.

31. ^4

37. a^

+ -

17. (a

7«

3.

-lax

-I-

Page 175

Page 165

2x

7)

25(-2a2 ^b'^\zab) 8x^7x2/ - 9xy - 8)

25.

^3

+

2x

1(5/

23. 6(x2

^2

31.

29. |-

1

21. 4(fl2

^2

21.

39.

^3

11.

7

37. 42

13

^3

9.

19.

27. I5

25. 4?13

12

1.

- 3;;) 17. 3(x2 _ + 27 + 4) + ^ab - 4^2) X + 4xv + 7)

15. 9x2(1 19.

81.

77.

-

2x

11«2

1

85. 4a/?

-

3

(Z,

45.

Puzzles and Things, page 179

No,

not possible.

is

it

14

Page 181 I. -3,

-2

3.

-6,

-3

5.

-6,

-4

9. (x - 3)(x - 2) 7. (x - 5)(x - 2) 13. (x - 3)(x - 8) II. {y 1) 5){y 17. (« - A)in - 1) 15. {n + 6)(« + 3) 21. iy - l){y - A) 19. (X + 4)(x + 2) 23. (« + 5)(« + 5) 25. {y - e){y - 5) 27. (x - 7)(x - 7) 29. (m - 5)(m - 8)

Page 183

CHAPTER Page

6

171 7

3.

1.

2, 3,

7.

2,2,2,3,5

13.

7

23.

14

33. 20

15.

2, 2, 3,

1

25. 25

35. 6

3

5.

2, 5, 3

2,2,2,2,3, 13

9.

17.

11

19.

27. 4

37.

16

29. 8

21.

11.

3

12

31. 4

100

+ x) 3. 2(x - 5) 5. x(3x - 3) 9. ln(4n - 1) 4x(x — 2) 13. 5mn{5 — mn)

I.

3(3

2x(x

II.

yes (X

+ +

yes

4)2

5.

7. (a + - 2)2 13. (x - 9)(x (1 - 10x)2

no

11. (x

9)2

17.

19. {a

-bf

21.

5)2

{y

+

7)2

10)

Page 186 1.

-5,2

9.

7,

17. 1)

3.

15. {n

13.

Page 173 7.

1.

9.

21. 25.

29.

-4 (x + (b (X {b + (X +

3.

-4,

11.

14,

- 3) + A) 5)(x + 4) 6)ib - A) 14)(x - 2) 7)(x

3)(b

5.

1

-2,

1

7.

-10,3

-1

+ 9)(x - 2) - 6){n + 3) 23. V - \)iy + 15) 27. {b - 10)(^ + 3) 31. iy + S)iy - A) 15. (X

19. (n (

33.

(>'

35.

(m

39. iy

Mixed

+ 9)(;^ - 3) - 4)(m + 16) - 6)(y + 1)

Page 193 37. {n

-

9){n

+

7)

Practice Exercises, page 187

3. (« - 6)(« - 2) + 4) 7. (y - l)(y + 3) 9. (X 3)(x + 7) 11. (b- \)ib + 5) 13. {b -l){b - 1) 15. (y - A)(y + 5) 19. (n - l)(n - 8) 17. {X + l){x + 5) 21. (x + 9){x + 6) 23. (n + 9)(n + 5) 25. (« - \0){n + 5) 27. (7 + 26)(/ - 2) 29. Ax -A 1.

(jc

5.

{y

+ +

31. Ay

+1)

9)(jc

3){y

7.

n2 JC^

- 100 - 4x2

/2

3.

9.

_4

m"

5.

9^2

11^

21.

- «2 25^2 36 19. 25x2 _ gyz 100x2 - 25/ 23. fl^ - 4

27.

2496

13. 17.

25/

35. 3575

15.

37.

— A2^ _4

81m2

31. 8091

29. 896

6384

39.

25.

33.

399

864

2496

Page 191 1.

yes

9. (;c

13. 15. 17. 19.

21.

23. 25. 29.

31. 33. 35. 37. 39. 41. 43.

3.

no

5.

yes

7.

no

- 4)(x + 4) 11. (« - 3)(« + 3) a - 3b)ia + 3b) x - 2y)(x + 2y) b - Sc){b + 8c) 2a - 5b)(2a + 5b) X - ll)(x + 11) 8x - 37)(8x + 3y) 2 + x)(2 - X) 27. (8 - x)(8 + x) X - y)(x + y) 2x3 _ 2)(2x^ + 3) 12 - llx)(12 + Ux) 6bc - 2a)(6bc + 2a) / + 9){y - 3){y + 3) 4^2 + \){2a + \){2a - 1) 16 + x2/)(4 + xy){A - xy) xy + l)(x2;; + \){x'^y - 1)

a-b

b

1.

(3x

5.

{2b

Page 197 1.

1.

3.

19. ab(b

37. 41.

49. 53.

+b

57. 59.

63. 67.

5.

4

+

a)

14

7.

6

17. 2(n^

21.

-4x)

3y

9.

+

11.

x2

2)

2y\y +

25. 6(6x

A)

+

1)

+ 4) 29. (n + 3)2 31. (x - 5)2 (n + A)(n + 5) 35. (y - S)(y - 1) {n - \\)(n -2) 39. (n + 7)(« +5) (a - 9)2 43. (m + 4)2 {n + 25)(« + 3) 47. (y - 5)(_y - 8) (y + 6)(7 + 3) 51. (X + 6)(x + 5) (n + A)(n + 16) 55. (x + 3)(x + 9) (x - 4)(x - 20) (X + 16)(x + 2) 61. (y - 9)(j - 7) (n - 9)(n + 2) 65. (m + 8)(w - 2) (y - \2)(y + 2) 69. (y - 6)2

27. x(x

33.

3

15.

23. 2x2(1

a-b b

98-199 3

13. 6

45.

a

186,487 kg

Pages

a-b \

45.

5(a

+ l)(x + 1) 3. (2a + 3)(a + 3) + 3)(b + 1) 7. (3a - 5)(a - 1) 9. (3x - 4)(x - 2) 11. (3^ + l)(y - 2) 13. (2x - l)(x + 3) 17. (Aa + 9)(a - 1) 15. (2x + 3)(x - 3) 19. (3b + 5)(2b - 5) 21. (Aa - 5)(a + I) 23. (2x - 5)(2x + 1) 25. (Ar + 3)(2r - 3) 27. (4x + 3)(2x - 1) 29. 2(2c + l)(c + 1) 31. 3(3/ + 2)(y + 1) 33. 3(2x + l)(x + 2) 35. (5/ - 2)(2_y - 5) 37. (3x - ll)(2x + 11) 39. (3x + 4)(4x - 3) 41. (5n - l)(3n + 4)

-A

_49 — J'^

3{y

Page 195

Page 189 1.

+ A){y + 2) 3. 2{y - 5)(y - 3) - 6)ia + 2) 7. 4(x - 3)(x + 5) 11. 8(x - 2)(x + 2) 9. 2(x - 4)2 13. 4(x - 3)(x + 3) 15. 6(x - 2y){x + 2y) 17. -l(x -3)(x - 1) 19. 2(x - 9)(x + 9) 21. -A(a - 2)(a + 1) 23. 2(a - I0b)(a + lOb) 1.

5.

+ 5){a - 2) 73. {b + 6){b - 1) _ 16 77. - 4^2 79. 16«2 _ 81. 9x2 _y2 83_ ^2y2 _ 85. 16jc2 - 9/ 87. {a - b){a + b) 89. (5 - «)(5 + n) 91. (2x - y){lx + /) b){Aa + 93. (4a 95. {la - b){la + 6) 97. (1 -ab){\ -^ ab) 99. 3(a + 1)^ 101. 4(jc - 2){x + 2) 103. 3(a + 3)(a + 2) 105. 3(x - 4)(x + 3) 107. 2{n + 7)(« - 3) 109. a{b - 2c)(b + 2c) 111. 2(a + 4)(fl + 2) 71. (a

75.

/

1

l

1

Z?)

Page 200

= 14 3. m = 9 5. X = -5 = -1 9. X = -3 11. X = 6 17. 2^2+15 a = -5 15. a = 3^2 4x + 21. 3^2 - 4 25. 4m2 — 2«2 5jc2 + -2m2 +3m -3 29. a - 3b a2/,4 33^ 2x^ + 3x2 _ 4^ «2 - 14/7 + 48 37. 3^2 ^ ^ _ 14 6/ - 77 - 3 41. 25fl2 _ b^ a{a + 6b'^) 45. 3mn(\ + 4m) (a - 2){a - 1) 49. (« - 3)(« - 4) (x + 8)(x - 2) 53. 4(jc - 3)(jc + 2)

1.

jc

7.

jc

13. 19.

23. 27. 31.

35. 39. 43. 47. 51.

1

.

1

jc

CHAPTER

7

Pages 206-207 1.

July,

3.

2,500

August

AVERAGE WEEKLY PAY OF IN MANUFACTURING

PERSONS

11.

Pages 218-219 I.

a, c

9.

3; 5;

3.

5.

c

-1;

5.

8

II.

y = J =

25. 2jc

7.

11.

+X - 4jc — 5 =y 6 9

6;

2;

1;

10; 0;

14;

16

4

=9 19.'/ = 5 - 4x 23. 7 = 9x - 7

15.

13. 6; 3; 9;

21.

b

v

jc

Note: Answers to exercises 27-37

may

vary. 27. (1,0), (2,1),

(-1,-2)

29. (1,-1), (2,-2),

(-1,1)

31. (0,6), (1,7), (-1,5) 33. (0,4), (2,8), (-2,0) 35. (0,7), (1,11), (-1,3) 37. (0,2), (1,0), (-1,4) 41.

7 =

Page 221 1.

4jc

+

1

39.

43. a, b

y = 4x 45. a, b

13.

Page 231

NEWSPAPER SALES FOR ONE YEAR

1.

33

17.

Page 249

'(

23.

15. (3,3)

(9,3)

33.

17.

(5,

13. (3,5)

11. (3,4)

9. (4,3)

(2,-2)

21.

5.

3

(-20,11)

7.

(-2,-1)

3.

3

29.

41.

(-1,5)

19.

1)

45.

(6,-8)

23.

37.

47.

Pages 252-253 4

17,

1.

24, 19

3.

Porky: 65 kg 9.

Bugs: 35 kg;

5.

Bud: 170; Jan: 158

7.

11. 5c

length: 18; width: 6

Puzzles and Things, page 253

Let b = the number of balls, c = the number of cups, and p = the number of pennies. Then b + c = \2p and

Ab

-\-

=

c;

b

-

c)

+

2/7

=

50/7

= 12/7 — c, so = c; 48/7 - 4c + = c. 10 pennies

2p

4(12/7

5c;

10/7

2/7

=

c;

49.

Page 255 (-2,3)

1.

(6,3)

3.

9.

(3,0)

11.

(2,4)

7.

(-1,-1)

(-2,6) 19. (1,-1)

13.

(-5,-3)

17.

15. (3,2)

5.

(2,-3)

(-1,1)

21.

Pages 256-257 hotdog: 50C; cola: 40C

1.

student: $2

milkshake:

X

=

5.

ice

65'

+72

-

3.

17.

25x2

5^

1.

4^2

.

.

- 9 11. 81 (x - 5)(x + 5) 9(x - 27)(x + 27

1

15.

5(fl

19. A{n

-

23. /(jc2

27. 29.

b)(a 6)(«

+ +

- j)

4(x

b)

17.

4)

21. x{\

25.

10(fl

+

(Sa - 2b)(3a + 2^) 4(r - 25)(r - 4^) 31.

33. (X ->^)(x

+y)(x^

-

3)^ 3jc)

2b){a

5(x

-

-

15)(x

+/)

Page 422 (pages 204-207)

AVERAGE LIFESPAN

1.

50

OF SEVERAL ANIMALS

40 30

>

20 10

==

3b)

+

4)

5.

9.

1

1

Page 429

13.

(pages 318-321) 5.

-355.4

1.

10

6000 9. 2000

3.

20 and 30

7.

_4,

5.

(pages 322-327)

1.

80%; 0.80

0.05;

9.

$0.63

11. $3.07

13.

'

'''^-

$3.10

15.

M

7.

3.

\^, 4

- V2

-2\/2 -

1.

1.20% 3.350 5.60

,

2y%

3,

V^

1

+

1

+ V33

-

1

5.

$150

9%

at

1.

500 student

tickets

ii

6

15.

7

1.

5

3.

5.

7.

1

25. irrational

23. irrational

Page 432 (pages 354-355) 5.

7V3~

10

1.

27

7.

V5"

9x

9.

3.

80

x^yV^

11.

Page 432 (pages 356-357) 3.

6 or

9.

±4.5

-6

5.

1.

7 or

-7

2V7"or -2y/T

±4.2

7.

Page 433 (pages 358-361)

1.7.2

cm

1.

3.

3.7.5

cm

Page 433 (pages 362-367)

-|

^

5.

10 7.

6

9.

12

17. 0.79

13.

11. 90jc

15

5 15.

x^V^

19. 0.26

Page 433 (pages 368-369)

1.

-6\/r

V2

9.

-2^/3^

(pages 380-385)

1.

5.

2V6

7.

3.

3\^ H.

-^—

Page 434 5.

4,4

5,2

7.

0,

-1

9.

0,

-5

0,9

3.

3.

0,2

11.

-3,

-5

1,

_

1 '

4 17.

1,2

-8

1.

Page 435

l.a.

y =x

7

and 2 or -14, -15

14, 15 or

2,3,4 or -2, -3,

(pages 396-399)

-3.464 13. -1.732 -8.718 17. 9.9 19. 7.9 11.

21. rational

11.

4 15.

-7 and -2 5.

Page 432 (pages 348-353)

4,

03

(pages 394-395)

(pages 336-339)

5.

Page 435

Page 431

9.

-2,7

13.

$390 3. $6000 and $50 at 18%

2

^/2\

1

4

Page 43 1.

±2\/2

2

9.

(pages 332-335)

1

i±^. A^^ 2

9.

no solution

1.

+

4

1

(pages 390-393) 3.

$0.98

I

Pages 434-435

$10

$140

(pages 328-331)

17. 6,

11.

Pages 430-43

7.

-5,2

9.

m

0.09;

5.

8

2\^20

3.

15.

1

(pages 386-389)

Page 430

17.

2,

Page 434

-4

3.a.

5.a.

b.

7.a.

y =

5.-1 15. -8

6

7.

-6

9.

8

17.

13. 9

7

11.

19.

21. 9

1

15.