Intermediate Algebra

661 Pages·2010·10.83 MB·English

by  Jerome E. Kaufmann

INTERMEDIATE
ALGEBRA
N I N T H E D I T I O N
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INTERMEDIATE
N I N T H
ALGEBRA
E D I T I O N
Jerome E. Kaufmann
Karen L. Schwitters
Seminole State College of Florida
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Intermediate Algebra,Ninth Edition © 2011, 2007Brooks/Cole, Cengage Learning
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CONTENTS
1 Basic Concepts and Properties 1
1.1 Sets, Real Numbers, and Numerical Expressions 2
1.2 Operations with Real Numbers 10
1.3 Properties of Real Numbers and the Use of Exponents 20
1.4 Algebraic Expressions 27
Chapter 1 Summary 36
Chapter 1 Review Problem Set 38
Chapter 1 Test 40
2 Equations, Inequalities, and Problem Solving 41
2.1 Solving First-Degree Equations 42
2.2 Equations Involving Fractional Forms 49
2.3 Equations Involving Decimals and Problem Solving 57
2.4 Formulas 64
2.5 Inequalities 74
2.6 More on Inequalities and Problem Solving 81
2.7 Equations and Inequalities Involving Absolute Value 90
Chapter 2 Summary 97
Chapter 2 Review Problem Set 101
Chapter 2 Test 104
Chapters 1 – 2 Cumulative Test 105
3 Polynomials 107
3.1 Polynomials: Sums and Differences 108
3.2 Products and Quotients of Monomials 114
3.3 Multiplying Polynomials 119
3.4 Factoring: Greatest Common Factor and Common Binomial Factor 127
3.5 Factoring: Difference of Two Squares and Sum or Difference of Two Cubes 135
3.6 Factoring Trinomials 141
3.7 Equations and Problem Solving 149
Chapter 3 Summary 155
Chapter 3 Review Problem Set 158
Chapter 3 Test 161
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vi Contents
4 Rational Expressions 163
4.1 Simplifying Rational Expressions 164
4.2 Multiplying and Dividing Rational Expressions 169
4.3 Adding and Subtracting Rational Expressions 175
4.4 More on Rational Expressions and Complex Fractions 182
4.5 Dividing Polynomials 190
4.6 Fractional Equations 196
4.7 More Fractional Equations and Applications 202
Chapter 4 Summary 211
Chapter 4 Review Problem Set 216
Chapter 4 Test 218
Chapters 1 – 4 Cumulative Review Problem Set 219
5 Exponents and Radicals 221
5.1 Using Integers as Exponents 222
5.2 Roots and Radicals 229
5.3 Combining Radicals and Simplifying Radicals That Contain Variables 238
5.4 Products and Quotients Involving Radicals 243
5.5 Equations Involving Radicals 249
5.6 Merging Exponents and Roots 254
5.7 Scientific Notation 259
Chapter 5 Summary 265
Chapter 5 Review Problem Set 269
Chapter 5 Test 271
6 Quadratic Equations and Inequalities 273
6.1 Complex Numbers 274
6.2 Quadratic Equations 281
6.3 Completing the Square 289
6.4 Quadratic Formula 293
6.5 More Quadratic Equations and Applications 300
6.6 Quadratic and Other Nonlinear Inequalities 308
Chapter 6 Summary 314
Chapter 6 Review Problem Set 318
Chapter 6 Test 320
Chapters 1 – 6 Cumulative Review Problem Set 321
7 Linear Equations and Inequalities in Two Variables 323
7.1 Rectangular Coordinate System and Linear Equations 324
7.2 Linear Inequalities in Two Variables 337
Contents vii
7.3 Distance and Slope 342
7.4 Determining the Equation of a Line 353
7.5 Graphing Nonlinear Equations 363
Chapter 7 Summary 371
Chapter 7 Review Problem Set 376
Chapter 7 Test 379
8 Conic Sections 381
8.1 Graphing Parabolas 382
8.2 More Parabolas and Some Circles 390
8.3 Graphing Ellipses 397
8.4 Graphing Hyperbolas 401
Chapter 8 Summary 408
Chapter 8 Review Problem Set 411
Chapter 8 Test 413
Chapters 1 – 8 Cumulative Review Problem Set 414
9 Functions 417
9.1 Relations and Functions 418
9.2 Functions: Their Graphs and Applications 425
9.3 Graphing Made Easy via Transformations 436
9.4 Composition of Functions 445
9.5 Inverse Functions 450
9.6 Direct and Inverse Variations 457
Chapter 9 Summary 465
Chapter 9 Review Problem Set 473
Chapter 9 Test 476
10 Systems of Equations 477
10.1 Systems of Two Linear Equations and Linear Inequalities in Two Variables 478
10.2 Substitution Method 483
10.3 Elimination-by-Addition Method 489
10.4 Systems of Three Linear Equations in Three Variables 498
10.5 Matrix Approach to Solving Systems 506
10.6 Determinants 511
10.7 3 (cid:2)3 Determinants and Systems of Three Linear Equations in Three Variables 516
10.8 Systems Involving Nonlinear Equations 523
Chapter 10 Summary 528
Chapter 10 Review Problem Set 534
Chapter 10 Test 536
Chapters 1 – 10 Cumulative Review Problem Set 537
viii Contents
11 Exponential and Logarithmic Functions 541
11.1 Exponents and Exponential Functions 542
11.2 Applications of Exponential Functions 548
11.3 Logarithms 557
11.4 Logarithmic Functions 566
11.5 Exponential Equations, Logarithmic Equations, and Problem Solving 571
Chapter 11 Summary 580
Chapter 11 Review Problem Set 585
Chapter 11 Test 587
Appendix A Prime Numbers and Operations with Fractions 589
Appendix B Binomial Expansions 597
Answers to Odd-Numbered Problems and All Chapter Review, Chapter Test,
and Cumulative Review Problems 601
Index I-1
PREFACE
When preparing Intermediate Algebra, Ninth Edition, we wanted to preserve the fea-
tures that made the previous editions successful and, at the same time, incorporate improve-
ments suggested by reviewers.
This text was written for college students who need an algebra course that bridges the gap
between elementary algebra and the more advanced courses in precalculus mathematics. It
covers topics that are usually classified as intermediate algebra topics.
The basic concepts of intermediate algebra are presented in this text in a simple, straight-
forward way. Algebraic ideas are developed in a logical sequence and in an easy-to-read man-
ner without excessive formalism. Concepts are developed through examples, reinforced
through additional examples, and then applied in a variety of problem-solving situations.
There is a common thread that runs throughout the book:
1. Learn a skill
2. Practice the skillto help solve equations, and
3. Apply the skillto solve application problems
This thread influenced some of the decisions we made in preparing the text.
• When appropriate, problem sets contain an ample number of word problems. Approxi-
mately 450 word problems are scattered throughout the text. These problems deal with a
variety of applications that show the connection between mathematics and its use in the
real world.
• Many problem-solving suggestions are offered throughout the text, and there are special
discussions on problem solving in several sections. And when different methods can be
used to solve the same problem, those methods are presented for both word problems and
other skill problems.
• Newly acquired skills are used as soon as possible to solve equations and inequalities,
which, in turn, are used to solve word problems. Therefore, the concept of solving equa-
tions and inequalities is introduced early and reinforced throughout the text. The concepts
of factoring, solving equations, and solving word problems are tied together in Chapter 3.
In approximately 500 worked-out examples, we demonstrate a wide variety of situations, but
we leave some things for students to think about in the problem sets. We also use examples to
guide students in organizing their work and to help them decide when they may try a shortcut.
The progression from showing all steps to demonstrating a suggested shortcut format is
gradual.
As recommended by the American Mathematical Association of Two-Year Colleges,
many basic geometry concepts are integrated into a problem-solving setting. This book
contains worked-out examples and problems that connect algebra, geometry, and real-world
applications. Specific discussions of geometric concepts are contained in the following
sections:
Section 2.2Complementary and supplementary angles; the sum of the measurements of the
angles of a triangle equals 180°
Section 2.4Area and volume formulas
Section 3.4The Pythagorean theorem
Section 6.2More on the Pythagorean theorem, including work with isosceles right triangles
and 30°–60° right triangles
ix