Preview Algebra for College Students (9th Edition) by Margaret L. Lial, John Hornsby, Terry McGinnis (2019)

A LG E B RA FO R
CO LLE G E S T U D E NT S
LIAL / HORNSBY / McGINNIS

ninth edition


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EDITION

9

Algebra
for College
Students
Margaret L. Lial
American River College

John Hornsby

University of New Orleans

Terry McGinnis


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Library of Congress Cataloging-in-Publication Data
Names: Lial, Margaret L., author. | Hornsby, John, 1949- author. | McGinnis,
  Terry, author.
Title: Algebra for college students.
Description: 9th edition / Margaret L. Lial (American River College), John
  Hornsby (University of New Orleans), Terry McGinnis. | Boston : Pearson,
  [2020] | Includes index.
Identifiers: LCCN 2019000106 | ISBN 9780135160664 (student edition) | ISBN
  0135160669 (student edition)
Subjects: LCSH: Algebra--Textbooks.
Classification: LCC QA154.3 .L53 2020 | DDC 512.9--dc23
LC record available at />1 19

ISBN 13: 978-0-13-516066-4
ISBN 10: 0-13-516066-9


CONTENTS
Preface vii

Study Skills  S-1
STUDY SKILL 1  Using Your Math Text  S-1

STUDY SKILL 6  Managing Your Time  S-6

STUDY SKILL 2  Reading Your Math Text  S-2

STUDY SKILL 7  Reviewing a Chapter  S-7

STUDY SKILL 3  Taking Lecture Notes  S-3

STUDY SKILL 8  Taking Math Tests  S-8

STUDY SKILL 4  Completing Your Homework  S-4

STUDY SKILL 9  Analyzing Your Test Results  S-9

STUDY SKILL 5  Using Study Cards  S-5

STUDY SKILL 10 Preparing for Your Math Final

Exam S-10

R

Review of the Real Number System  1

R.1 Fractions, Decimals, and Percents  1
R.2 Basic Concepts from Algebra  14
R.3 Operations on Real Numbers  26

R.5 Properties of Real Numbers  45
Chapter R Summary  52
Chapter R Test  54

R.4 Exponents, Roots, and Order of Operations  36

1

Linear Equations, Inequalities, and Applications  55

1.1 Linear Equations in One Variable  56

1.7 Absolute Value Equations and Inequalities  125

1.2 Formulas and Percent  65

SUMMARY EXERCISES  Solving Linear and Absolute Value

1.3 Applications of Linear Equations  78
1.4 Further Applications of Linear Equations  92
SUMMARY EXERCISES  Applying Problem-Solving

Techniques 101
1.5 Linear Inequalities in One Variable  103
1.6 Set Operations and Compound Inequalities  116

2

Equations and Inequalities  136
Chapter 1 Summary  137
Chapter 1 Review Exercises  142
Chapter 1 Mixed Review Exercises  145
Chapter 1 Test  146
Chapters R and 1 Cumulative Review
Exercises 148

Linear Equations, Graphs, and Functions  149

2.1 Linear Equations in Two Variables  150

2.5 Introduction to Relations and Functions  199

2.2 The Slope of a Line  161

2.6 Function Notation and Linear Functions  210

2.3 Writing Equations of Lines  176
SUMMARY EXERCISES  Finding Slopes and Equations

of Lines  191
2.4 Linear Inequalities in Two Variables  192

Chapter 2 Summary  219
Chapter 2 Review Exercises  222
Chapter 2 Mixed Review Exercises  224
Chapter 2 Test  225
Chapters R–2 Cumulative Review Exercises  227

iii


ivContents

3

Systems of Linear Equations  229

3.1 Systems of Linear Equations in Two
Variables 230
3.2 Systems of Linear Equations in Three
Variables 245
3.3 Applications of Systems of Linear
Equations 254

4

Exponents, Polynomials, and Polynomial Functions  279

4.1 Integer Exponents  280
4.2 Scientific Notation  290
4.3 Adding and Subtracting Polynomials  296
4.4 Polynomial Functions, Graphs, and
Composition 302
4.5 Multiplying Polynomials  315

5

4.6 Dividing Polynomials  324
Chapter 4 Summary  331
Chapter 4 Review Exercises  334
Chapter 4 Mixed Review Exercises  337
Chapter 4 Test  337
Chapters R–4 Cumulative Review
Exercises 338

Factoring 341

5.1 Greatest Common Factors and Factoring
by Grouping  z 4

- 26

- k = - 1k

-1

- 26

r = 1r

1

3x
8

3
8

x
3

=
=

3

8
1x
3

x
=

1
3

x

1
3

5p and -21p

-6x2 and 9x2

Like terms

3m and 16x

7y 3 and -3y 2

Unlike terms

Different variables

Different exponents

on the same variable

OBJECTIVE 4  Use the commutative and associative properties.
Simplifying expressions as in Examples 2(a) and (b) is called combining like terms.
Only like terms may be combined. To combine like terms in an expression such as
-2m + 5m + 3 - 6m + 8,
we need two more properties. From arithmetic, we know that the following are true.
3 + 9 = 12

9 + 3 = 12

and

3 # 9 = 27

9 # 3 = 27

and

The order of the numbers being added or multiplied does not matter. The same
answers result. The following computations are also true.
15 + 72 + 2 = 12 + 2 = 14

5 + 17 + 22 = 5 + 9 = 14

15 # 72 # 2 = 35 # 2 = 70

5 # 17 # 22 = 5 # 14 = 70

The grouping of the numbers being added or multiplied does not matter. The same

answers result.
These arithmetic examples can be extended to algebra.
Commutative and Associative Properties

For any real numbers a, b, and c, the following hold true.
a+b=b+a
ab = ba

 

Commutative properties

(The order of the two terms or factors changes.)
Examples:  9 + 1-32 = -3 + 9, 91 -32 = 1-329
a + 1 b + c2 = 1 a + b 2 + c
a 1 bc2 = 1 ab 2 c

(1)1*

Numerical
Coefficient

Term

The inverse properties “undo” addition or multiplication. Putting on your shoes
when you get up in the morning and then taking them off before you go to bed at night
are inverse operations that undo each other.
Expressions such as 12m and 5n from Example 2 are examples of terms. A term
is a number or the product of a number and one or more variables raised to powers.
The numerical factor in a term is the numerical coefficient, or just the coefficient.

Terms with exactly the same variables raised to exactly the same powers are like
terms. Otherwise, they are unlike terms.

(1)1*

▼▼ Terms and Their
Coefficients

Associative properties

 

(The grouping among the three terms or factors changes, but the order stays
the same.)
Examples:  7 + 18 + 92 = 17 + 82 + 9, 7 # 18 # 92 = 17 # 82 # 9




SECTION R.5   Properties of Real Numbers

49

The commutative properties are used to change the order of the terms or factors in
an expression. Think of commuting from home to work and then from work to home.
The associative properties are used to regroup the terms or factors of an expression.
Think of associating the grouped terms or factors.
NOW TRY
EXERCISE 3

Simplify.
-7x + 10 - 3x - 4 + x

EXAMPLE 3 Using the Commutative and Associative Properties

Simplify.
-2m + 5m + 3 - 6m + 8
= 1-2m + 5m2 + 3 - 6m + 8

Associative property

= 3m + 3 - 6m + 8

Add inside parentheses.

= 1-2 + 52m + 3 - 6m + 8

Distributive property

The next step would be to add 3m and 3, but they are unlike terms. To combine 3m
and -6m, we use the associative and commutative properties, inserting parentheses
and brackets according to the rules for order of operations.
= 33m + 13 - 6m24 + 8

Associative property

= 313m + 3 -6m42 + 34 + 8

Associative property

= -3m + 13 + 82

Associative property

= 33m + 1-6m + 324 + 8

Commutative property

= 1-3m + 32 + 8

Combine like terms.

= -3m + 11

Add.

In practice, many of these steps are not written down, but it is important to realize
that the commutative and associative properties are used whenever the terms in an
expression are rearranged and regrouped to combine like terms.
NOW TRY
EXAMPLE 4 Using the Properties of Real Numbers

Simplify each expression.
(a) 5y - 8y - 6y + 11y
= 15 - 8 - 6 + 112y
= 2y

(b) 3x + 4 - 51x + 12 - 8

Combine like terms.

Be careful with signs.

= 3x + 4 - 5x - 5 - 8

Distributive property

= 3x - 5x + 4 - 5 - 8

Commutative property

= -2x - 9

Combine like terms.

(c) 8 - 13m + 22
NOW TRY ANSWER
3. -9x + 6

Distributive property

= 8 - 113m + 22

Identity property

= 8 - 3m - 2

Distributive property

= 6 - 3m

Combine like terms.


50

CHAPTER R    Review of the Real Number System

NOW TRY
EXERCISE 4

(d) 3x1521y2

Simplify each expression.
(a) -31t - 42 - t + 15
(b) 7x - 14x - 22
(c) 5x16y2
(d) 315x - 72 - 81x + 42

= 33x1524y

Order of operations

= 3315x24y

Commutative property

= 115x2y

Multiply.

= 331x # 524y
= 313 # 52x4y
= 151xy2

Associative property

Associative property

Associative property

= 15xy
As previously mentioned, many of these steps are not usually written out.
(e) 413x - 52 - 214x + 72
= 12x - 20 - 8x - 14

Distributive property

= 12x - 8x - 20 - 14

Commutative property

= 4x - 34

Combine like terms.

Like terms may be combined by adding or subtracting the coefficients of the terms
and keeping the same variable factors.
NOW TRY

NOW TRY ANSWERS

4. (a)  -4t + 27  (b)  3x + 2
(c)  30xy  (d)  7x - 53

R.5 Exercises
Video solutions for select
problems available in MyLab
Math

! CAUTION Be careful. The distributive property does not apply in Example 4(d) because
there is no addition or subtraction involved.
13x21521y2 ≠ 13x2152 # 13x21y2

FOR
EXTRA
HELP

MyLab Math

Concept Check  Choose the correct response.
1.The identity element for addition is
1
A.  -a  B.  0  C.  1  D.  .
a
3.The additive inverse of a is
1
A.  -a  B.  0  C.  1  D.  .
a

2.The identity element for multiplication is
1

A.  -a  B.  0  C.  1  D.  .
a
4.The multiplicative inverse of a, where
a ≠ 0, is
1
A.  -a  B.  0  C.  1  D.  .
a

Concept Check  Complete each statement.
5. The distributive property provides a way to rewrite a product such as a1b + c2 as
the sum _______ .
6. The commutative property is used to change the _______ of two terms or factors.
7.The associative property is used to change the _______ of three terms or factors.
8.Like terms are terms with the _______ variables raised to the _______ powers.
9.When simplifying an expression, only _______ terms can be combined.
10. The numerical coefficient in the term -7yz 2 is _______ .




SECTION R.5   Properties of Real Numbers

51

Simplify each expression. See Examples 1 and 2.
11. 21m + p2

12. 31a + b2

13. -121x - y2

14. -101p - q2

15. 5k + 3k

16. 6a + 5a

17. 7r - 9r

18. 4n - 6n

19. -8z + 4w

20. -12k + 3r

21. a + 7a

22. s + 9s

23. x + x

24. a + a

26. -13m - n2

27. -1 -x - y2

28. -1-3x - 4y2

25. -12d - f 2

29. 21x - 3y + 2z2

30. 813x + y - 5z2

Simplify each expression. See Examples 1– 4.
31. -12y + 4y + 3y + 2y

32. -5r - 9r + 8r - 5r

33. -6p + 5 - 4p + 6 + 11p

34. -8x - 12 + 3x - 5x + 9

35. 31k + 22 - 5k + 6 + 3

36. 51r - 32 + 6r - 2r + 4

37. 10 - 14y + 82
39. 10x1321y2

38. 6 - 19y + 52

40. 8x1621y2

2
41. - 112w217z2
3

5

42. - 118w215z2
6

43. 31m - 42 - 21m + 12

44. 61a - 52 - 41a + 62

45. 0.2518 + 4p2 - 0.516 + 2p2

46. 0.4110 - 5x2 - 0.815 + 10x2

47. -12p + 52 + 312p + 42 - 2p

48. -17m - 122 + 214m + 72 - 6m

49. 2 + 312z - 52 - 314z + 62 - 8

50. -4 + 414k - 32 - 612k + 82 + 7

Complete each statement so that the indicated property is illustrated. Simplify each answer if
possible. See Examples 1– 4.
51. 5x + 8x =

52. 9y - 6y =
(distributive property)

53. 519r2 =
(associative property)
55. 5x + 9y =
57. 1 # 7 =

(distributive property)
54. -4 + 112 + 82 =
(associative property)
56. -5 # 7 =

(commutative property)

(commutative property)
58. -12x + 0 =

(identity property)
1
1
59. - ty + ty =
4
4

(identity property)
60. -

(inverse property)

61. 81 -4 + x2 =
(distributive property)
63. 010.875x + 9y2 =

9
8
a- b =

8
9

(inverse property)

62. 31x - y + z2 =
(distributive property)
64. 010.35t + 12u2 =

(multiplication
property of 0)

(multiplication
property of 0)

65. Concept Check  Give an “everyday” example of a commutative operation.
66. Concept Check  Give an “everyday” example of inverse operations.


52

CHAPTER R    Review of the Real Number System

The distributive property can be used to mentally perform calculations.
38 # 17 + 38 # 3
= 38117 + 32

Distributive property

= 381202

Add inside the parentheses.

= 760

Multiply.

Use the distributive property to calculate each value mentally.
67. 96 # 19 + 4 # 19
8
8
70. 1172 + 1132
5
5

#

#

68. 27 # 60 + 27 # 40

69. 58

71. 8.751152 - 8.75152

72. 4.311692 + 4.311312

3
-8
2

3
2

RELATING CONCEPTS For Individual or Group Work  (Exercises 73 –78)
When simplifying an expression, we usually do some steps mentally. Work Exer­
cises 73–78 in order, providing the property that justifies each statement in the
given simplification. (These steps could be done in other orders.)
3x + 4 + 2x + 7
73.
74.
75.
76.
77.
78.

Chapter R

= 13x + 42 + 12x + 72_______________________________

= 3x + 14 + 2x2 + 7 _______________________________
= 3x + 12x + 42 + 7 _______________________________
= 13x + 2x2 + 14 + 72_______________________________

= 13 + 22x + 14 + 72 _______________________________

= 5x + 11 _______________________________

Summary

Key Terms
R.1
fractions
numerator
denominator
proper fraction
improper fraction
lowest terms
mixed number
reciprocals
decimal
terminating decimal
repeating decimal
percent
R.2
set
elements (members)
finite set

natural (counting) numbers
infinite set
whole numbers
empty (null) set
variable
number line
integers
coordinate
graph
rational numbers
irrational numbers

real numbers
additive inverse
(opposite, negative)
signed numbers
absolute value
equation
inequality

R.3
sum
difference
product
quotient
reciprocal
(multiplicative inverse)
dividend
divisor
R.4
factors
exponent (power)
base
exponential expression
square root
positive (principal)
square root

negative square root
constant
algebraic expression

R.5
identity element for
addition
(additive identity)
identity element for
multiplication
(multiplicative
identity)
term
coefficient
(numerical coefficient)
like terms
unlike terms




53

CHAPTER R  Summary

New Symbols
0.6bar notation that signifies
  repeating digit(s)
%percent
5a, b6 set containing the elements
  a and b
∅
empty set
{

is an element of (a set)

o
is not an element of
3 is not equal to
5x∣ x has property P6

set-builder notation
∣ x ∣
absolute value of x
*
is less than
"
is less than or equal to

+
is greater than
#is greater than or equal to
am
m factors of a
! radical symbol
!apositive (principal) square
  root of a

Test Your Word Power
See how well you have learned the vocabulary in this chapter.
1. The denominator of a fraction
A.  is the number above the
­fraction bar
B.  gives the total number of equal

parts in the whole
C. gives the number of shaded
parts in the whole
D. is the smaller number in the
fraction.
2. A proper fraction is a fraction that
has
A. numerator greater than
denominator
B.  numerator equal to
denominator
C. numerator less than
denominator
D. denominator less than
numerator.
3. The empty set is a set
A.  with 0 as its only element
B.  with an infinite number of
elements
C.  with no elements
D.  of ideas.
4. A variable is
A.  a symbol used to represent an
unknown number
B.  a value that makes an equation
true
C.  a solution of an equation
D.  the answer in a division
problem.

5. An integer is
A.  a positive or negative number
B.  a natural number, its opposite,
or zero
C.  any number that can be
graphed
D. the quotient of two numbers.
6. The absolute value of a number is
A.  the graph of the number
B.  the reciprocal of the number
C.  the opposite of the number
D. the distance between 0 and the
­number on a number line.
7. The reciprocal of a nonzero
­number a is
1
A.  a  B.  a  C.  -a   D.  1.
8. A factor is
A.  the answer in an addition
problem
B.  the answer in a multiplication
problem
C. one of two or more numbers
that are added to get another
number
D. any number that divides evenly
into a given number.
9. An exponent is
A. a symbol that tells how many
­numbers are being multiplied

B.  a number raised to a power
C. a number that tells how many
times a factor is repeated
D. a number that is multiplied.

10. An exponential expression is
A. a number that is a repeated
factor in a product
B.  a number or a variable written
with an exponent
C. a number that tells how many
times a factor is repeated in a
product
D. an expression that involves
addition.
11. A term is
A.  a numerical factor
B.  a number or a product
of a number and one or
more ­variables raised to
powers
C. one of several variables with
the same exponents
D. a sum of numbers and
­variables raised to powers.
12. A numerical coefficient is
A. the numerical factor in a term
B.  the number of terms in an
expression
C. a variable raised to a power

D. the variable factor in a term.

ANSWERS
3

1 2 5

1.  B; Example: In the fraction 4 , the denominator is 4.  2.  C; Examples: 2 , 7 , 12   3.  C; Example: The set of whole numbers less than 0 is the empty set,
1

written ∅.  4.  A; Examples: x, y, z  5.  B; Examples: -9, 0, 6  6.  D; Examples: ͉ 2 ͉ = 2 and ͉ -2 ͉ = 2  7.  B; Examples: 3 is the reciprocal of 3 ;
5

2

- 2 is the reciprocal of - 5 .  8.  D; Example: 2 and 5 are factors of 10 because both divide evenly (without remainder) into 10.  9.  C; Example:

In 23, the number 3 is the exponent (or power), so 2 is a factor three times, and 23 = 2 # 2 # 2 = 8.  10.  B; Examples: 34 and x10  11.  B; Examples:
x

6, 2 , -4ab2  12.  A; Examples: The term 8z has numerical coefficient 8, and the term -10x3y has numerical coefficient -10.


54

CHAPTER R    Review of the Real Number System

Chapter R

Test

View the complete solutions
to all Chapter Test exercises in
MyLab Math.

FOR
EXTRA
HELP

Step-by-step test solutions are found on the Chapter Test Prep Videos available in
MyLab Math.

Perform the indicated operations.
3 1
1. +
4 6

2.

3
9
,
7 14

4. 0.7 * 0.04

3. 13.25 - 6.417

Complete the table of fraction, decimal, and percent equivalents.
Fraction in Lowest Terms

5.
6.

Decimal

Percent
4%

5
6

7.

1.5

5
8.Graph e -3, 0.75, , 5, 6.3 f on a number line.
3
Let A =

5 - 26, -1, -0.5, 0, 3, 225, 7.5, 242 , 2 -4 6. Simplify the elements of A as neces-

sary, and then list those elements of A that belong to the specified set.
9.Whole numbers

10. Integers

11. Rational numbers

12. Real numbers

Perform the indicated operations.
13. -6 + 14 + 1 -112 - 1-32

14. -

15. 10 - 4 # 3 + 61-42
17.

5
10 2
- a+ b
7
9
3

16. 7 - 42 + 2162 + 1-42 2

-233 - 1 -1 - 22 + 24

18.

291 -32 - 1-22

8 # 4 - 32 # 5 - 21-12
-3 # 23 + 24

Find each square root. If it is not a real number, say so.
19. 2196

22. Evaluate

8k +
r-2

2m2

20. - 2225

for k = -3, m = -3, and r = 25.

21. 2 -16

Simplify each expression.
23. -312k - 42 + 413k - 52 - 2 + 4k

24. 13r + 82 - 1-4r + 62

25. Match each statement in Column I with the appropriate property in Column II. Answers
may be used more than once.
I

STUDY SKILLS REMINDER
You will increase your chance
of success in this course if you
fully utilize your text. Review
Study Skill 1, Using Your
Math Text.

II

(a)  6 + 1 -62 = 0

A.  Distributive property

(c)  5x + 15x = 15 + 152x

C.  Identity property

(e)  -9 + 0 = -9

E.  Commutative property

(b)  -2 + 13 + 62 = 1-2 + 32 + 6
(d)  13 # 0 = 0
(f)  4 # 1 = 4

(g)  1a + b2 + c = 1b + a2 + c

B.  Inverse property
D.  Associative property
F.  Multiplication property of 0