Algebra trigonometry 8e ron larson

Algebra and Trigonometry
Eighth Edition

Ron Larson
The Pennsylvania State University
The Behrend College
With the assistance of

David C. Falvo
The Pennsylvania State University
The Behrend College

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Algebra and Trigonometry, Eighth Edition
Ron Larson
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Contents
A Word from the Author (Preface) vii

chapter P

Prerequisites

1

P.1
Review of Real Numbers and Their Properties 2
P.2
Exponents and Radicals 15
P.3
Polynomials and Special Products 28
P.4
Factoring Polynomials 37
P.5
Rational Expressions 45
P.6
The Rectangular Coordinate System and Graphs 55
Chapter Summary 66
Review Exercises 68
Chapter Test 71
Proofs in Mathematics 72
Problem Solving 73

chapter 1

Equations, Inequalities, and Mathematical Modeling

75

1.1
Graphs of Equations 76
1.2
Linear Equations in One Variable 87

1.3
Modeling with Linear Equations 96
1.4
Quadratic Equations and Applications 107
1.5
Complex Numbers 122
1.6
Other Types of Equations 129
1.7
Linear Inequalities in One Variable 140
1.8
Other Types of Inequalities 150
Chapter Summary 160
Review Exercises 162
Chapter Test 165
Proofs in Mathematics 166
Problem Solving 167

chapter 2

Functions and Their Graphs

169

2.1
Linear Equations in Two Variables 170
2.2
Functions 185
2.3
Analyzing Graphs of Functions 200

2.4
A Library of Parent Functions 212
2.5
Transformations of Functions 219
2.6
Combinations of Functions: Composite Functions 229
2.7
Inverse Functions 238
Chapter Summary 248
Review Exercises 250
Chapter Test 253
Cumulative Test for Chapters P–2 254
Proofs in Mathematics 256
Problem Solving 257

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iv

Contents

chapter 3

Polynomial Functions

259

3.1
Quadratic Functions and Models 260
3.2
Polynomial Functions of Higher Degree 270
3.3
Polynomial and Synthetic Division 284
3.4
Zeros of Polynomial Functions 293
3.5
Mathematical Modeling and Variation 308
Chapter Summary 320
Review Exercises 322
Chapter Test 326
Proofs in Mathematics 327
Problem Solving 329

chapter 4

Rational Functions and Conics

331

4.1
Rational Functions and Asymptotes 332
4.2
Graphs of Rational Functions 340
4.3
Conics 349
4.4
Translations of Conics 362

Chapter Summary 370
Review Exercises 372
Chapter Test 375
Proofs in Mathematics 376
Problem Solving 377

chapter 5

Exponential and Logarithmic Functions

379

5.1
Exponential Functions and Their Graphs 380
5.2
Logarithmic Functions and Their Graphs 391
5.3
Properties of Logarithms 401
5.4
Exponential and Logarithmic Equations 408
5.5
Exponential and Logarithmic Models 419
Chapter Summary 432
Review Exercises 434
Chapter Test 437
Cumulative Test for Chapters 3–5 438
Proofs in Mathematics 440
Problem Solving 441

chapter 6

Trigonometry
6.1
6.2
6.3
6.4
6.5
6.6

443

Angles and Their Measure 444
Right Triangle Trigonometry 456
Trigonometric Functions of Any Angle 467
Graphs of Sine and Cosine Functions 479
Graphs of Other Trigonometric Functions 490
Inverse Trigonometric Functions 501

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Contents

6.7
Applications and Models 511
Chapter Summary 522
Chapter Test 527
Problem Solving 529

chapter 7

Analytic Trigonometry

Review Exercises 524
Proofs in Mathematics 528

531

7.1
Using Fundamental Identities 532
7.2
Verifying Trigonometric Identities 540
7.3
Solving Trigonometric Equations 547
7.4
Sum and Difference Formulas 558
7.5
Multiple-Angle and Product-to-Sum Formulas 565
Chapter Summary 576
Review Exercises 578
Chapter Test 581
Proofs in Mathematics 582
Problem Solving 585

chapter 8

Additional Topics in Trigonometry

587

8.1
Law of Sines 588
8.2
Law of Cosines 597
8.3
Vectors in the Plane 605
8.4
Vectors and Dot Products 618
8.5
Trigonometric Form of a Complex Number 628
Chapter Summary 638
Review Exercises 640
Chapter Test 644
Cumulative Test for Chapters 6–8 645
Proofs in Mathematics 647
Problem Solving 651

chapter 9

Systems of Equations and Inequalities

653

9.1
Linear and Nonlinear Systems of Equations 654
9.2
Two-Variable Linear Systems 665
9.3
Multivariable Linear Systems 677
9.4

Partial Fractions 690
9.5
Systems of Inequalities 698
9.6
Linear Programming 709
Chapter Summary 718
Review Exercises 720
Chapter Test 725
Proofs in Mathematics 726
Problem Solving 727

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vi

Contents

chapter 10

Matrices and Determinants

729

10.1 Matrices and Systems of Equations 730
10.2 Operations with Matrices 744
10.3 The Inverse of a Square Matrix 759
10.4 The Determinant of a Square Matrix 768

10.5 Applications of Matrices and Determinants 776
Chapter Summary 788
Review Exercises 790
Chapter Test 795
Proofs in Mathematics 796
Problem Solving 797

chapter 11

Sequences, Series, and Probability

799

11.1 Sequences and Series 800
11.2 Arithmetic Sequences and Partial Sums 811
11.3 Geometric Sequences and Series 821
11.4 Mathematical Induction 831
11.5 The Binomial Theorem 841
11.6 Counting Principles 849
11.7 Probability 859
Chapter Summary 872
Review Exercises 874
Chapter Test 877
Cumulative Test for Chapters 9–11 878
Proofs in Mathematics 880
Problem Solving 883

Appendix A Errors and the Algebra of Calculus
Answers to Odd-Numbered Exercises and Tests
Index

A123

Index of Applications (web)
Appendix B Concepts in Statistics (web)
B.1
B.2
B.3

Representing Data
Measures of Central Tendency and Dispersion
Least Squares Regression

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A9

A1


A Word from
the Author
Welcome to the Eighth Edition of Algebra and Trigonometry! We are proud to offer you
a new and revised version of our textbook. With this edition, we have listened to you,
our users, and have incorporated many of your suggestions for improvement.

8th

4th

7th

3rd

6th

2nd

5th

1st

In the Eighth Edition, we continue to offer instructors and students a text that is
pedagogically sound, mathematically precise, and still comprehensible. There are many
changes in the mathematics, art, and design; the more significant changes are noted here.
• New Chapter Openers Each Chapter Opener has three parts, In Mathematics, In
Real Life, and In Careers. In Mathematics describes an important mathematical
topic taught in the chapter. In Real Life tells students where they will encounter this
topic in real-life situations. In Careers relates application exercises to a variety of
careers.
• New Study Tips and Warning/Cautions Insightful information is given to
students in two new features. The Study Tip provides students with useful
information or suggestions for learning the topic. The Warning/Caution points out
common mathematical errors made by students.
• New Algebra Helps Algebra Help directs students to sections of the textbook
where they can review algebra skills needed to master the current topic.
• New Side-by-Side Examples Throughout the text, we present solutions to many
examples from multiple perspectives—algebraically, graphically, and numerically.
The side-by-side format of this pedagogical feature helps students to see that a problem
can be solved in more than one way and to see that different methods yield the same

result. The side-by-side format also addresses many different learning styles.

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viii

A Word from the Author

• New Capstone Exercises Capstones are conceptual problems that synthesize key
topics and provide students with a better understanding of each section’s
concepts. Capstone exercises are excellent for classroom discussion or test prep, and
teachers may find value in integrating these problems into their reviews of the
section.
• New Chapter Summaries The Chapter Summary now includes an explanation
and/or example of each objective taught in the chapter.
• Revised Exercise Sets The exercise sets have been carefully and extensively
examined to ensure they are rigorous and cover all topics suggested by our users.
Many new skill-building and challenging exercises have been added.
For the past several years, we’ve maintained an independent website—
CalcChat.com—that provides free solutions to all odd-numbered exercises in the text.
Thousands of students using our textbooks have visited the site for practice and help
with their homework. For the Eighth Edition, we were able to use information from
CalcChat.com, including which solutions students accessed most often, to help guide
the revision of the exercises.
I hope you enjoy the Eighth Edition of Algebra and Trigonometry. As always, I
welcome comments and suggestions for continued improvements.

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Acknowledgments
I would like to thank the many people who have helped me prepare the text and the
supplements package. Their encouragement, criticisms, and suggestions have been
invaluable.
Thank you to all of the instructors who took the time to review the changes in this
edition and to provide suggestions for improving it. Without your help, this book would
not be possible.

Reviewers
Chad Pierson, University of Minnesota-Duluth; Sally Shao, Cleveland State University;
Ed Stumpf, Central Carolina Community College; Fuzhen Zhang, Nova Southeastern
University; Dennis Shepherd, University of Colorado, Denver; Rhonda Kilgo,
Jacksonville State University; C. Altay Özgener, Manatee Community College
Bradenton; William Forrest, Baton Rouge Community College; Tracy Cook, University
of Tennessee Knoxville; Charles Hale, California State Poly University Pomona; Samuel
Evers, University of Alabama; Seongchun Kwon, University of Toledo; Dr. Arun K.
Agarwal, Grambling State University; Hyounkyun Oh, Savannah State University;
Michael J. McConnell, Clarion University; Martha Chalhoub, Collin County
Community College; Angela Lee Everett, Chattanooga State Tech Community College;
Heather Van Dyke, Walla Walla Community College; Gregory Buthusiem, Burlington
County Community College; Ward Shaffer, College of Coastal Georgia; Carmen
Thomas, Chatham University; Emily J. Keaton
My thanks to David Falvo, The Behrend College, The Pennsylvania State
University, for his contributions to this project. My thanks also to Robert Hostetler, The
Behrend College, The Pennsylvania State University, and Bruce Edwards, University of
Florida, for their significant contributions to previous editions of this text.
I would also like to thank the staff at Larson Texts, Inc. who assisted with proofreading the manuscript, preparing and proofreading the art package, and checking and

typesetting the supplements.
On a personal level, I am grateful to my spouse, Deanna Gilbert Larson, for her
love, patience, and support. Also, a special thanks goes to R. Scott O’Neil. If you have
suggestions for improving this text, please feel free to write to me. Over the past two
decades I have received many useful comments from both instructors and students, and
I value these comments very highly.

Ron Larson

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Supplements
Supplements for the Instructor
Annotated Instructor’s Edition This AIE is the complete student text plus point-ofuse annotations for the instructor, including extra projects, classroom activities, teaching
strategies, and additional examples. Answers to even-numbered text exercises,
Vocabulary Checks, and Explorations are also provided.
Complete Solutions Manual This manual contains solutions to all exercises from the
text, including Chapter Review Exercises and Chapter Tests.
Instructor’s Companion Website
of instructor resources.

This free companion website contains an abundance

PowerLecture™ with ExamView® The CD-ROM provides the instructor with dynamic
media tools for teaching college algebra. PowerPoint® lecture slides and art slides of
the figures from the text, together with electronic files for the test bank and a link to the
Solution Builder, are available. The algorithmic ExamView allows you to create, deliver,

and customize tests (both print and online) in minutes with this easy-to-use assessment
system. Enhance how your students interact with you, your lecture, and each other.
Solutions Builder This is an electronic version of the complete solutions manual
available via the PowerLecture and Instructor’s Companion Website. It provides
instructors with an efficient method for creating solution sets to homework or exams
that can then be printed or posted.

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Supplements

xi

Supplements for the Student
Student Companion Website
student resources.

This free companion website contains an abundance of

Instructional DVDs Keyed to the text by section, these DVDs provide comprehensive
coverage of the course—along with additional explanations of concepts, sample
problems, and applications—to help students review essential topics.
Student Study and Solutions Manual This guide offers step-by-step solutions for all
odd-numbered text exercises, Chapter and Cumulative Tests, and Practice Tests with
solutions.
Premium eBook The Premium eBook offers an interactive version of the textbook
with search features, highlighting and note-making tools, and direct links to videos or

tutorials that elaborate on the text discussions.
Enhanced WebAssign Enhanced WebAssign is designed for you to do your
homework online. This proven and reliable system uses pedagogy and content found in
Larson’s text, and then enhances it to help you learn Algebra and Trigonometry more
effectively. Automatically graded homework allows you to focus on your learning and
get interactive study assistance outside of class.

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Prerequisites
P.1

Review of Real Numbers and Their Properties

P.2

Exponents and Radicals

P.3

Polynomials and Special Products

P.4

Factoring Polynomials

P.5

Rational Expressions

P.6

The Rectangular Coordinate System and Graphs

P

In Mathematics
Real numbers, exponents, radicals, and
polynomials are used in many different
branches of mathematics.

The concepts in this chapter are used to
model compound interest, volumes, rates
of change, and other real-life applications.
For instance, polynomials can be used
to model the stopping distance of an
automobile. (See Exercise 116, page 36.)

Darren McCollester/ Getty Images News /Getty Images

In Real Life

IN CAREERS
There are many careers that use prealgebra concepts. Several are listed below.
• Engineer
Exercise 115, page 35

• Financial Analyst
Exercises 99 and 100, page 54

• Chemist
Exercise 148, page 44

• Meteorologist
Exercise 114, page 70

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2

Chapter P

Prerequisites

P.1 REVIEW OF REAL NUMBERS AND THEIR PROPERTIES
What you should learn
• Represent and classify real numbers.
• Order real numbers and use
inequalities.
• Find the absolute values of real
numbers and find the distance
between two real numbers.
• Evaluate algebraic expressions.

• Use the basic rules and
properties of algebra.

Real Numbers
Real numbers are used in everyday life to describe quantities such as age, miles per
gallon, and population. Real numbers are represented by symbols such as
4
3 Ϫ32.
Ϫ5, 9, 0, , 0.666 . . . , 28.21, Ί2, ␲, and Ί
3
Here are some important subsets (each member of subset B is also a member of set A)
of the real numbers. The three dots, called ellipsis points, indicate that the pattern
continues indefinitely.

ͭ1, 2, 3, 4, . . .ͮ

Why you should learn it
Real numbers are used to represent
many real-life quantities. For example,
in Exercises 83–88 on page 13, you
will use real numbers to represent the
federal deficit.

Set of natural numbers

ͭ0, 1, 2, 3, 4, . . .ͮ

Set of whole numbers

ͭ. . . , Ϫ3, Ϫ2, Ϫ1, 0, 1, 2, 3, . . .ͮ

Set of integers

A real number is rational if it can be written as the ratio p͞q of two integers, where
q 0. For instance, the numbers
1
1
125
ϭ 0.3333 . . . ϭ 0.3, ϭ 0.125, and
ϭ 1.126126 . . . ϭ 1.126
3
8
111
are rational. The decimal representation of a rational number either repeats ͑as in
ϭ 3.145 ͒ or terminates ͑as in 12 ϭ 0.5͒. A real number that cannot be written as the
ratio of two integers is called irrational. Irrational numbers have infinite nonrepeating
decimal representations. For instance, the numbers

173
55

Ί2 ϭ 1.4142135 . . . Ϸ 1.41

␲ ϭ 3.1415926 . . . Ϸ 3.14

and

are irrational. (The symbol Ϸ means “is approximately equal to.”) Figure P.1 shows
subsets of real numbers and their relationships to each other.
Real

numbers

Example 1

Classifying Real Numbers

Determine which numbers in the set
Irrational
numbers

Rational
numbers

Integers

Negative
integers

Natural
numbers
FIGURE

ΆϪ13, Ϫ

Noninteger
fractions
(positive and
negative)

1

3

5
8

·

Ί5, Ϫ1, Ϫ , 0, , Ί2, ␲, 7

are (a) natural numbers, (b) whole numbers, (c) integers, (d) rational numbers, and
(e) irrational numbers.

Solution
a. Natural numbers: ͭ7ͮ
b. Whole numbers: ͭ0, 7ͮ
c. Integers: ͭϪ13, Ϫ1, 0, 7ͮ

Whole
numbers

Zero

P.1 Subsets of real numbers

Ά

·

1
5

d. Rational numbers: Ϫ13, Ϫ1, Ϫ , 0, , 7
3
8
e. Irrational numbers: ͭ Ϫ Ί5, Ί2, ␲ͮ
Now try Exercise 11.

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Section P.1

3

Review of Real Numbers and Their Properties

Real numbers are represented graphically on the real number line. When you
draw a point on the real number line that corresponds to a real number, you are plotting
the real number. The point 0 on the real number line is the origin. Numbers to the right
of 0 are positive, and numbers to the left of 0 are negative, as shown in Figure P.2. The
term nonnegative describes a number that is either positive or zero.
Origin
Negative
direction
FIGURE

−4

−3

−2

−1

0

1

2

3

Positive
direction

4

P.2 The real number line

As illustrated in Figure P.3, there is a one-to-one correspondence between real
numbers and points on the real number line.
− 53
−3

−2

−1

0

−2.4

π

0.75
1

2

−3

3

Every real number corresponds to exactly
one point on the real number line.
FIGURE

−2

2
−1

0

1

2

3

Every point on the real number line

corresponds to exactly one real number.

P.3 One-to-one correspondence

Example 2

Plotting Points on the Real Number Line

Plot the real numbers on the real number line.
a. Ϫ

7
4

b. 2.3
c.

2
3

d. Ϫ1.8

Solution
All four points are shown in Figure P.4.
− 1.8 − 74
−2
FIGURE

2
3

−1

0

2.3
1

2

3

P.4

7
a. The point representing the real number Ϫ 4 ϭ Ϫ1.75 lies between Ϫ2 and Ϫ1, but
closer to Ϫ2, on the real number line.
b. The point representing the real number 2.3 lies between 2 and 3, but closer to 2, on
the real number line.
2
c. The point representing the real number 3 ϭ 0.666 . . . lies between 0 and 1, but
closer to 1, on the real number line.
d. The point representing the real number Ϫ1.8 lies between Ϫ2 and Ϫ1, but closer to
Ϫ2, on the real number line. Note that the point representing Ϫ1.8 lies slightly to
7
the left of the point representing Ϫ 4.

Now try Exercise 17.

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4

Chapter P

Prerequisites

Ordering Real Numbers
One important property of real numbers is that they are ordered.

Definition of Order on the Real Number Line

a
−1

If a and b are real numbers, a is less than b if b Ϫ a is positive. The order of a
and b is denoted by the inequality a < b. This relationship can also be described
by saying that b is greater than a and writing b > a. The inequality a ≤ b means
that a is less than or equal to b, and the inequality b ≥ a means that b is greater
than or equal to a. The symbols <, >, Յ, and Ն are inequality symbols.

b

0

1

2

FIGURE P.5 a < b if and only if a lies to
the left of b.

Geometrically, this definition implies that a < b if and only if a lies to the left of
b on the real number line, as shown in Figure P.5.

Example 3
−4

−3

FIGURE

−4

−2

a. Ϫ3, 0
−2

−1

0

1

1 1
,
4 3

1 1
d. Ϫ , Ϫ
5 2

1

c. Because 14 lies to the left of 3 on the real number line, as shown in Figure P.8, you
1
1
1
can say that 4 is less than 3, and write 14 < 3.

P.8
− 12 − 15
−1

FIGURE

c.

a. Because Ϫ3 lies to the left of 0 on the real number line, as shown in Figure P.6, you
can say that Ϫ3 is less than 0, and write Ϫ3 < 0.
b. Because Ϫ2 lies to the right of Ϫ4 on the real number line, as shown in Figure P.7,
you can say that Ϫ2 is greater than Ϫ4, and write Ϫ2 > Ϫ4.

1
3

0

b. Ϫ2, Ϫ4

Solution

P.7
1
4

FIGURE

Place the appropriate inequality symbol ͑< or >͒ between the pair of real numbers.

0

P.6
−3

FIGURE

−1

Ordering Real Numbers

1
1
d. Because Ϫ 5 lies to the right of Ϫ 2 on the real number line, as shown in Figure P.9,
1
1
1
1

you can say that Ϫ 5 is greater than Ϫ 2, and write Ϫ 5 > Ϫ 2.

0

Now try Exercise 25.

P.9

Example 4

Interpreting Inequalities

Describe the subset of real numbers represented by each inequality.
a. x Յ 2

x≤2
x
0
FIGURE

1

2

3

4

P.10
−2 ≤ x < 3

x

−2

−1

FIGURE

P.11

0

1

2

3

b. Ϫ2 Յ x < 3

Solution
a. The inequality x ≤ 2 denotes all real numbers less than or equal to 2, as shown in
Figure P.10.
b. The inequality Ϫ2 ≤ x < 3 means that x ≥ Ϫ2 and x < 3. This “double inequality”
denotes all real numbers between Ϫ2 and 3, including Ϫ2 but not including 3, as
shown in Figure P.11.
Now try Exercise 31.

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Section P.1

5

Review of Real Numbers and Their Properties

Inequalities can be used to describe subsets of real numbers called intervals. In the
bounded intervals below, the real numbers a and b are the endpoints of each interval.
The endpoints of a closed interval are included in the interval, whereas the endpoints of
an open interval are not included in the interval.

Bounded Intervals on the Real Number Line
Notation

Interval Type
Closed

͓a, b͔
The reason that the four types
of intervals at the right are called
bounded is that each has a finite
length. An interval that does not
have a finite length is unbounded
(see below).

WARNING / CAUTION
Whenever you write an interval
containing ϱ or Ϫ ϱ, always
use a parenthesis and never a

bracket. This is because ϱ and
Ϫ ϱ are never an endpoint of an
interval and therefore are not
included in the interval.

͑a, b͒

Open

͓a, b͒

Inequality

Graph

a Յ x Յ b

x

a

b

a

b

a

b

a

b

a < x < b

x

a Յ x < b

͑a, b͔

x

a < x Յ b

x

The symbols ϱ, positive infinity, and Ϫ ϱ, negative infinity, do not represent
real numbers. They are simply convenient symbols used to describe the unboundedness
of an interval such as ͑1, ϱ͒ or ͑Ϫ ϱ, 3͔.

Unbounded Intervals on the Real Number Line
Notation
͓a, ϱ͒

Interval Type

͑a, ϱ͒

Open

Inequality
x Ն a

Graph
x

a

x > a

x

a

͑Ϫ ϱ, b͔

x Յ b

x

b

͑Ϫ ϱ, b͒

Open

͑Ϫ ϱ, ϱ͒

Entire real line

x < b

x

b

Example 5

Ϫϱ < x <

ϱ

Using Inequalities to Represent Intervals

Use inequality notation to describe each of the following.
a. c is at most 2.

b. m is at least Ϫ3.

c. All x in the interval ͑Ϫ3, 5͔

Solution
a. The statement “c is at most 2” can be represented by c ≤ 2.
b. The statement “m is at least Ϫ3” can be represented by m ≥ Ϫ3.
c. “All x in the interval ͑Ϫ3, 5͔” can be represented by Ϫ3 < x ≤ 5.
Now try Exercise 45.

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6

Chapter P

Prerequisites

Example 6

Interpreting Intervals

Give a verbal description of each interval.
a. ͑Ϫ1, 0͒

b. ͓ 2, ϱ͒

c. ͑Ϫ ϱ, 0͒

Solution
a. This interval consists of all real numbers that are greater than Ϫ1 and less than 0.
b. This interval consists of all real numbers that are greater than or equal to 2.
c. This interval consists of all negative real numbers.
Now try Exercise 41.

Absolute Value and Distance
The absolute value of a real number is its magnitude, or the distance between the

origin and the point representing the real number on the real number line.

Definition of Absolute Value
If a is a real number, then the absolute value of a is

ԽaԽ ϭ ΆϪa,
a,

if a ≥ 0
.
if a < 0

Notice in this definition that the absolute value of a real number is never negative.
For instance, if a ϭ Ϫ5, then Ϫ5 ϭ Ϫ ͑Ϫ5͒ ϭ 5. The absolute value of a real
number is either positive or zero. Moreover, 0 is the only real number whose absolute
value is 0. So, 0 ϭ 0.

Խ Խ

ԽԽ

Example 7

Finding Absolute Values

Խ Խ
ԽϪ4.3Խ ϭ 4.3

ԽԽ

2
2
ϭ
3
3

a. Ϫ15 ϭ 15

b.

c.

d. Ϫ Ϫ6 ϭ Ϫ ͑6͒ ϭ Ϫ6

Խ Խ

Now try Exercise 51.

Example 8
Evaluate

Evaluating the Absolute Value of a Number

ԽxԽ for (a) x > 0 and (b) x < 0.
x

Solution

ԽԽ

a. If x > 0, then x ϭ x and

ԽԽ

ԽxԽ ϭ x ϭ 1.
x

b. If x < 0, then x ϭ Ϫx and

x

ԽxԽ ϭ Ϫx ϭ Ϫ1.
x

Now try Exercise 59.

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x


Section P.1

Review of Real Numbers and Their Properties

7

The Law of Trichotomy states that for any two real numbers a and b, precisely one
of three relationships is possible:
a ϭ b,

a < b,

Example 9

or

a > b.

Law of Trichotomy

Comparing Real Numbers

Place the appropriate symbol (<, >, or =) between the pair of real numbers.

Խ Խ᭿Խ3Խ

Խ

a. Ϫ4

Խ᭿Խ10Խ

Խ Խ᭿ԽϪ7Խ

b. Ϫ10

c. Ϫ Ϫ7

Solution

Խ Խ ԽԽ
Խ Խ Խ Խ
Խ Խ Խ Խ

Խ Խ
ԽԽ
Խ Խ
Խ Խ
Խ Խ
Խ Խ

a. Ϫ4 > 3 because Ϫ4 ϭ 4 and 3 ϭ 3, and 4 is greater than 3.
b. Ϫ10 ϭ 10 because Ϫ10 ϭ 10 and 10 ϭ 10.
c. Ϫ Ϫ7 < Ϫ7 because Ϫ Ϫ7 ϭ Ϫ7 and Ϫ7 ϭ 7, and Ϫ7 is less than 7.
Now try Exercise 61.

Properties of Absolute Values

ԽԽ

2. Ϫa ϭ a

Խ Խ Խ ԽԽ Խ

4.

1. a Ն 0
3. ab ϭ a b

−2

−1

0

ԽԽ

0

Absolute value can be used to define the distance between two points on the real
number line. For instance, the distance between Ϫ3 and 4 is

7
−3

Խ Խ ԽԽ
a
ԽaԽ, b
ϭ
b
ԽbԽ

1

2

3

4

P.12 The distance between Ϫ3
and 4 is 7.

ԽϪ3 Ϫ 4Խ ϭ ԽϪ7Խ
ϭ7

FIGURE

as shown in Figure P.12.

Distance Between Two Points on the Real Number Line
Let a and b be real numbers. The distance between a and b is

Խ

Խ Խ

Խ

d͑a, b͒ ϭ b Ϫ a ϭ a Ϫ b .

Example 10

Finding a Distance

Find the distance between Ϫ25 and 13.

Solution
The distance between Ϫ25 and 13 is given by

ԽϪ25 Ϫ 13Խ ϭ ԽϪ38Խ ϭ 38.

Distance between Ϫ25 and 13

The distance can also be found as follows.

Խ13 Ϫ ͑Ϫ25͒Խ ϭ Խ38Խ ϭ 38
Now try Exercise 67.

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Distance between Ϫ25 and 13


8

Chapter P

Prerequisites

Algebraic Expressions
One characteristic of algebra is the use of letters to represent numbers. The letters are
variables, and combinations of letters and numbers are algebraic expressions. Here
are a few examples of algebraic expressions.
5x,

2x Ϫ 3,

x2

4
,
ϩ2

7x ϩ y

Definition of an Algebraic Expression
An algebraic expression is a collection of letters (variables) and real numbers
(constants) combined using the operations of addition, subtraction, multiplication,
division, and exponentiation.

The terms of an algebraic expression are those parts that are separated by addition.
For example,
x 2 Ϫ 5x ϩ 8 ϭ x 2 ϩ ͑Ϫ5x͒ ϩ 8
has three terms: x 2 and Ϫ5x are the variable terms and 8 is the constant term. The
numerical factor of a term is called the coefficient. For instance, the coefficient of Ϫ5x
is Ϫ5, and the coefficient of x 2 is 1.

Example 11

Identifying Terms and Coefficients

Algebraic Expression
1
7
b. 2x2 Ϫ 6x ϩ 9
3 1
c. ϩ x4 Ϫ y
x

2
a. 5x Ϫ

Terms

Coefficients

1
7
2x2, Ϫ6x, 9
3 1 4
, x , Ϫy
x 2

1
7
2, Ϫ6, 9
1
3, , Ϫ1
2

5x, Ϫ

5, Ϫ

Now try Exercise 89.
To evaluate an algebraic expression, substitute numerical values for each of the
variables in the expression, as shown in the next example.

Example 12

Evaluating Algebraic Expressions
Value of
Variable

Expression
a. Ϫ3x ϩ 5
b. 3x 2 ϩ 2x Ϫ 1
2x
c.
xϩ1

xϭ3
x ϭ Ϫ1
x ϭ Ϫ3

Substitute

Value of
Expression

Ϫ3͑3͒ ϩ 5
3͑Ϫ1͒2 ϩ 2͑Ϫ1͒ Ϫ 1
2͑Ϫ3͒
Ϫ3 ϩ 1

Ϫ9 ϩ 5 ϭ Ϫ4
3Ϫ2Ϫ1ϭ0
Ϫ6
ϭ3

Ϫ2

Note that you must substitute the value for each occurrence of the variable.
Now try Exercise 95.
When an algebraic expression is evaluated, the Substitution Principle is used. It
states that “If a ϭ b, then a can be replaced by b in any expression involving a.” In
Example 12(a), for instance, 3 is substituted for x in the expression Ϫ3x ϩ 5.

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Section P.1

Review of Real Numbers and Their Properties

9

Basic Rules of Algebra
There are four arithmetic operations with real numbers: addition, multiplication,
subtraction, and division, denoted by the symbols ϩ, ϫ or и , Ϫ, and Ϭ or /. Of
these, addition and multiplication are the two primary operations. Subtraction and
division are the inverse operations of addition and multiplication, respectively.

Definitions of Subtraction and Division
Subtraction: Add the opposite.
a Ϫ b ϭ a ϩ ͑Ϫb͒

Division: Multiply by the reciprocal.
If b

0, then a͞b ϭ a

΂b΃ ϭ b .
1

a

In these definitions, Ϫb is the additive inverse (or opposite) of b, and 1͞b is
the multiplicative inverse (or reciprocal) of b. In the fractional form a͞b,
a is the numerator of the fraction and b is the denominator.

Because the properties of real numbers below are true for variables and
algebraic expressions as well as for real numbers, they are often called the Basic Rules
of Algebra. Try to formulate a verbal description of each property. For instance, the
first property states that the order in which two real numbers are added does not affect
their sum.

Basic Rules of Algebra
Let a, b, and c be real numbers, variables, or algebraic expressions.
Property
Commutative Property of Addition:
Commutative Property of Multiplication:
Associative Property of Addition:
Associative Property of Multiplication:
Distributive Properties:

Example

Additive Identity Property:
Multiplicative Identity Property:

Additive Inverse Property:

aϩbϭbϩa
ab ϭ ba
͑a ϩ b͒ ϩ c ϭ a ϩ ͑b ϩ c͒
͑ab͒ c ϭ a͑bc͒
a͑b ϩ c͒ ϭ ab ϩ ac
͑a ϩ b͒c ϭ ac ϩ bc
aϩ0ϭa
aи1ϭa
a ϩ ͑Ϫa͒ ϭ 0

Multiplicative Inverse Property:

aи

1
ϭ 1, a
a

4x ϩ x ϭ x 2 ϩ 4x
͑4 Ϫ x͒ x 2 ϭ x 2͑4 Ϫ x͒
͑x ϩ 5͒ ϩ x 2 ϭ x ϩ ͑5 ϩ x 2͒
͑2x и 3y͒͑8͒ ϭ ͑2x͒͑3y и 8͒
3x͑5 ϩ 2x͒ ϭ 3x и 5 ϩ 3x и 2x
͑ y ϩ 8͒ y ϭ y и y ϩ 8 и y
5y 2 ϩ 0 ϭ 5y 2
͑4x 2͒͑1͒ ϭ 4x 2
5x 3 ϩ ͑Ϫ5x 3͒ ϭ 0
2

͑x 2 ϩ 4͒

0

΂x

2

΃

1
ϭ1
ϩ4

Because subtraction is defined as “adding the opposite,” the Distributive Properties
are also true for subtraction. For instance, the “subtraction form” of
a͑b ϩ c͒ ϭ ab ϩ ac is a͑b Ϫ c͒ ϭ ab Ϫ ac. Note that the operations of subtraction
and division are neither commutative nor associative. The examples
7Ϫ3

3 Ϫ 7 and

20 Ϭ 4

4 Ϭ 20

show that subtraction and division are not commutative. Similarly
5 Ϫ ͑3 Ϫ 2͒

͑5 Ϫ 3͒ Ϫ 2 and 16 Ϭ ͑4 Ϭ 2)

͑16 Ϭ 4) Ϭ 2

demonstrate that subtraction and division are not associative.

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10

Chapter P

Prerequisites

Example 13

Identifying Rules of Algebra

Identify the rule of algebra illustrated by the statement.
a. ͑5x3͒2 ϭ 2͑5x3͒
b.

΂4x ϩ 31΃ Ϫ ΂4x ϩ 31΃ ϭ 0

c. 7x и

1
ϭ 1,
7x

x

0

d. ͑2 ϩ 5x2͒ ϩ x2 ϭ 2 ϩ ͑5x2 ϩ x2͒

Solution
a. This statement illustrates the Commutative Property of Multiplication. In other
words, you obtain the same result whether you multiply 5x3 by 2, or 2 by 5x3.
b. This statement illustrates the Additive Inverse Property. In terms of subtraction, this
property simply states that when any expression is subtracted from itself the result
is 0.
c. This statement illustrates the Multiplicative Inverse Property. Note that it is
important that x be a nonzero number. If x were 0, the reciprocal of x would be
undefined.
d. This statement illustrates the Associative Property of Addition. In other words, to
form the sum
2 ϩ 5x2 ϩ x2
it does not matter whether 2 and 5x2, or 5x2 and x2 are added first.
Now try Exercise 101.

Properties of Negation and Equality
Let a, b, and c be real numbers, variables, or algebraic expressions.
Notice the difference between
the opposite of a number and a
negative number. If a is already
negative, then its opposite, Ϫa,
is positive. For instance, if
a ϭ Ϫ5, then

Ϫa ϭ Ϫ(Ϫ5) ϭ 5.

Property
1. ͑Ϫ1͒ a ϭ Ϫa

Example
͑Ϫ1͒7 ϭ Ϫ7

2. Ϫ ͑Ϫa͒ ϭ a

Ϫ ͑Ϫ6͒ ϭ 6

3. ͑Ϫa͒b ϭ Ϫ ͑ab͒ ϭ a͑Ϫb͒

͑Ϫ5͒3 ϭ Ϫ ͑5 и 3͒ ϭ 5͑Ϫ3͒

4. ͑Ϫa͒͑Ϫb͒ ϭ ab

͑Ϫ2͒͑Ϫx͒ ϭ 2x

5. Ϫ ͑a ϩ b͒ ϭ ͑Ϫa͒ ϩ ͑Ϫb͒

Ϫ ͑x ϩ 8͒ ϭ ͑Ϫx͒ ϩ ͑Ϫ8͒

6. If a ϭ b, then a ± c ϭ b ± c.

1
2

7. If a ϭ b, then ac ϭ bc.

42

8. If a ± c ϭ b ± c, then a ϭ b.

1.4 Ϫ 1 ϭ 75 Ϫ 1 ⇒ 1.4 ϭ 75

9. If ac ϭ bc and c

3x ϭ 3

ϭ Ϫx Ϫ 8

0, then a ϭ b.

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ϩ 3 ϭ 0.5 ϩ 3

и 2 ϭ 16 и 2
и4

⇒ xϭ4


Section P.1

Review of Real Numbers and Their Properties

11

Properties of Zero
The “or” in the Zero-Factor
Property includes the possibility
that either or both factors may be
zero. This is an inclusive or, and
it is the way the word “or” is
generally used in mathematics.

Let a and b be real numbers, variables, or algebraic expressions.
1. a ϩ 0 ϭ a and a Ϫ 0 ϭ a
3.

0
ϭ 0,
a

a

2. a

0

4.

и0ϭ0

a
is undefined.
0

5. Zero-Factor Property: If ab ϭ 0, then a ϭ 0 or b ϭ 0.

Properties and Operations of Fractions
Let a, b, c, and d be real numbers, variables, or algebraic expressions such that
b 0 and d 0.
1. Equivalent Fractions:
2. Rules of Signs: Ϫ

a
c
ϭ if and only if ad ϭ bc.
b d

a Ϫa
a
Ϫa a
and
ϭ
ϭ
ϭ
b
b
Ϫb
Ϫb b

3. Generate Equivalent Fractions:

a ac
ϭ , c

b bc

4. Add or Subtract with Like Denominators:

a c
a±c
± ϭ
b b
b

5. Add or Subtract with Unlike Denominators:
In Property 1 of fractions, the
phrase “if and only if” implies
two statements. One statement
is: If a͞b ϭ c͞d, then ad ϭ bc.
The other statement is: If
ad ϭ bc, where b 0 and
d 0, then a͞b ϭ c͞d.

6. Multiply Fractions:
7. Divide Fractions:

Example 14

a
b

c

0

a c
ad ± bc
± ϭ
b d
bd

ac

и d ϭ bd

a
c
a
Ϭ ϭ
b d b

d

ad

и c ϭ bc ,

c

0

Properties and Operations of Fractions

a. Equivalent fractions:

x 3 и x 3x
ϭ
ϭ
5 3 и 5 15

c. Add fractions with unlike denominators:

b. Divide fractions:

7 3 7 2 14
Ϭ ϭ и ϭ
x 2 x 3 3x

2x 5 и x ϩ 3 и 2x 11x
x
ϩ
ϭ
ϭ
3
5
3и5
15

Now try Exercise 119.
If a, b, and c are integers such that ab ϭ c, then a and b are factors or divisors of c.
A prime number is an integer that has exactly two positive factors—itself and 1—such
as 2, 3, 5, 7, and 11. The numbers 4, 6, 8, 9, and 10 are composite because each can be
written as the product of two or more prime numbers. The number 1 is neither prime
nor composite. The Fundamental Theorem of Arithmetic states that every

positive integer greater than 1 can be written as the product of prime numbers in
precisely one way (disregarding order). For instance, the prime factorization of 24 is
24 ϭ 2 и 2 и 2 и 3.

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12

Chapter P

P.1

Prerequisites

EXERCISES

See www.CalcChat.com for worked-out solutions to odd-numbered exercises.

VOCABULARY: Fill in the blanks.
p
of two integers, where q 0.
q
________ numbers have infinite nonrepeating decimal representations.
The point 0 on the real number line is called the ________.
The distance between the origin and a point representing a real number on the real number line is
the ________ ________ of the real number.
A number that can be written as the product of two or more prime numbers is called a ________ number.
An integer that has exactly two positive factors, the integer itself and 1, is called a ________ number.
An algebraic expression is a collection of letters called ________ and real numbers called ________.

The ________ of an algebraic expression are those parts separated by addition.
The numerical factor of a variable term is the ________ of the variable term.
The ________ ________ states that if ab ϭ 0, then a ϭ 0 or b ϭ 0.

1. A real number is ________ if it can be written as the ratio
2.
3.
4.
5.
6.
7.
8.
9.
10.

SKILLS AND APPLICATIONS
In Exercises 11–16, determine which numbers in the set are
(a) natural numbers, (b) whole numbers, (c) integers,
(d) rational numbers, and (e) irrational numbers.
11.
12.
13.
14.
15.
16.

ͭϪ9, Ϫ 72, 5, 23, Ί2, 0, 1, Ϫ4, 2, Ϫ11ͮ
ͭΊ5, Ϫ7, Ϫ 73, 0, 3.12, 54 , Ϫ3, 12, 5ͮ

ͭ2.01, 0.666 . . . , Ϫ13, 0.010110111 . . . , 1, Ϫ6ͮ

ͭ2.3030030003 . . . , 0.7575, Ϫ4.63, Ί10, Ϫ75, 4ͮ

ͭϪ ␲, Ϫ 13, 63, 12Ί2, Ϫ7.5, Ϫ1, 8, Ϫ22ͮ
ͭ25, Ϫ17, Ϫ 125, Ί9, 3.12, 12␲, 7, Ϫ11.1, 13ͮ

In Exercises 17 and 18, plot the real numbers on the real
number line.
7

17. (a) 3 (b) 2
18. (a) 8.5 (b)

4
3

5
(c) Ϫ 2 (d) Ϫ5.2
8
(c) Ϫ4.75 (d) Ϫ 3

In Exercises 19–22, use a calculator to find the decimal form
of the rational number. If it is a nonterminating decimal,
write the repeating pattern.
19.
21.

5
8
41
333

20.
22.

1
3
6
11

24.

−3
−7

−2
−6

−1
−5

0
−4

1
−3

−2

25. Ϫ4, Ϫ8
3

27. 2, 7

26. Ϫ3.5, 1
16
28. 1, 3

5 2
29. 6, 3

8
3
30. Ϫ 7, Ϫ 7

In Exercises 31– 42, (a) give a verbal description of the
subset of real numbers represented by the inequality or the
interval, (b) sketch the subset on the real number line, and
(c) state whether the interval is bounded or unbounded.
31.
33.
35.
37.
39.
41.

x Յ 5
x < 0
͓4, ϱ͒
Ϫ2 < x < 2
Ϫ1 ≤ x < 0
͓Ϫ2, 5͒

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

x Ն Ϫ2
x > 3
͑Ϫ ϱ, 2͒
0 ≤ x ≤ 5
0 < x ≤ 6
͑Ϫ1, 2͔

In Exercises 43–50, use inequality notation and interval
notation to describe the set.

In Exercises 23 and 24, approximate the numbers and place
the correct symbol ͧ< or >ͨ between them.
23.

In Exercises 25–30, plot the two real numbers on the real
number line. Then place the appropriate inequality symbol
ͧ< or >ͨ between them.

2

3

−1

0

43.
44.
45.
46.
47.
48.
49.
50.

y is nonnegative.
y is no more than 25.
x is greater than Ϫ2 and at most 4.
y is at least Ϫ6 and less than 0.
t is at least 10 and at most 22.
k is less than 5 but no less than Ϫ3.
The dog’s weight W is more than 65 pounds.
The annual rate of inflation r is expected to be at least
2.5% but no more than 5%.

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