Algebra demystified

ALGEBRA DEMYSTIFIED

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ALGEBRA DEMYSTIFIED

RHONDA HUETTENMUELLER

McGRAW-HILL

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otherwise.

DOI: 10.1036/0071412107

To all those who struggle with math

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vii

CONTENTS

Preface ix

CHAPTER 1 Fractions 1

CHAPTER 2Introduction to Variables 37

CHAPTER 3 Decimals 55

CHAPTER 4 Negative Numbers 65

CHAPTER 5 Exponents and Roots 79

CHAPTER 6 Factoring 113

CHAPTER 7 Linear Equations 163

CHAPTER 8 Linear Applications 197

CHAPTER 9 Linear Inequalities 285

CHAPTER 10 Quadratic Equations 319

CHAPTER 11 Quadratic Applications 353

Appendix 417

Final Review 423

Index 437

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ix

PREFACE

This book is designed to take the mystery out of algebra. Each section con￾tains exactly one new idea—unlike most math books, which cover several

ideas at once. Clear, brief explanations are followed by detailed examples.

Each section ends with a few Practice problems, most similar to the examples.

Solutions to the Practice problems are also given in great detail. The goal is

to help you understand the algebra concepts while building your skills and

confidence.

Each chapter ends with a Chapter Review, a multiple-choice test designed

to measure your mastery of the material. The Chapter Review could also be

used as a pretest. If you think you understand the material in a chapter, take

the Chapter Review test. If you answer all of the questions correctly, then

you can safely skip that chapter. When taking any multiple-choice test, work

the problems before looking at the answers. Sometimes incorrect answers

look reasonable and can throw you off. Once you have finished the book,

take the Final Review, which is a multiple-choice test based on material from

each chapter.

Spend as much time in each section as you need. Try not to rush, but do

make a commitment to learning on a schedule. If you find a concept difficult,

you might need to work the problems and examples several times. Try not to

jump around from section to section as most sections extend topics from

previous sections.

Not many shortcuts are used in this book. Does that mean you shouldn’t

use them? No. What you should do is try to find the shortcuts yourself. Once

you have found a method that seems to be a shortcut, try to figure out why it

works. If you understand how a shortcut works, you are less likely to use it

incorrectly (a common problem with algebra students).

Because many find fraction arithmetic difficult, the first chapter is devoted

almost exclusively to fractions. Make sure you understand the steps in this

chapter because they are the same steps used in much of the rest of the book.

For example, the steps used to compute 7

36 þ 5

16 are exactly those used to

compute

2x

x2 þ x 2 þ

6

x þ 2

.

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Even those who find algebra easy are stumped by word problems (also

called ‘‘applications’’). In this book, word problems are treated very care￾fully. Two important skills needed to solve word problems are discussed

earlier than the word problems themselves. First, you will learn how to

find quantitative relationships in word problems and how to represent

them using variables. Second, you will learn how to represent multiple quan￾tities using only one variable.

Most application problems come in ‘‘families’’—distance problems, work

problems, mixture problems, coin problems, and geometry problems, to

name a few. As in the rest of the book, exactly one topic is covered in

each section. If you take one section at a time and really make sure you

understand why the steps work, you will find yourself able to solve a great

many applied problems—even those not covered in this book.

Good luck.

RHONDA HUETTENMUELLER

x PREFACE

ACKNOWLEDGMENTS

I want to thank my husband and family for their patience during the many

months I worked on this project. I am also grateful to my students through

the years for their thoughtful questions. Finally, I want to express my appre￾ciation to Stan Gibilisco for his welcome advice.

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CHAPTER 1

Fractions

Fraction Multiplication

Multiplication of fractions is the easiest of all fraction operations. All you

have to do is multiply straight across—multiply the numerators (the top

numbers) and the denominators (the bottom numbers).

Example

2

3

4

5 ¼ 2 4

3 5 ¼ 8

15 :

Practice

1: 7

6

1

4 ¼

2: 8

15

6

5 ¼

1

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3: 5

3 9

10 ¼

4: 40

9

2

3 ¼

5: 3

7

30

4 ¼

Solutions

1: 7

6

1

4 ¼ 7 1

6 4 ¼ 7

24

2: 8

15

6

5 ¼ 8 6

15 5 ¼ 48

75

3: 5

3 9

10 ¼ 5 9

3 10 ¼ 45

30

4: 40

9

2

3 ¼ 40 2

9 3 ¼ 80

27

5: 3

7

30

4 ¼ 3 30

7 4 ¼ 90

28

Multiplying Fractions and Whole Numbers

You can multiply fractions by whole numbers in one of two ways:

1. The numerator of the product will be the whole number times the

fraction’s numerator, and the denominator will be the fraction’s

denominator.

2. Treat the whole number as a fraction—the whole number over

one—then multiply as you would any two fractions.

2 CHAPTER 1 Fractions