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CliffsQuickReview

Algebra I

By Jerry Bobrow, PhD

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CliffsQuickReviewAlgebra I

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Table of Contents

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .1

Chapter 1: Preliminaries and Basic Operations . . . . . . . . . . . . . . . . . . . .3

Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .3

Groups of numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .3

Ways to show multiplication . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .5

Common math symbols . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .5

Properties of Basic Mathematical Operations . . . . . . . . . . . . . . . . . . . . . . .6

Some properties (axioms) of addition . . . . . . . . . . . . . . . . . . . . . . . . . .6

Some properties (axioms) of multiplication . . . . . . . . . . . . . . . . . . . . . .7

A property of two operations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .8

Multiplying and Dividing Using Zero . . . . . . . . . . . . . . . . . . . . . . . . . . . .9

Powers and Exponents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .9

Squares and cubes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .10

Operations with powers and exponents . . . . . . . . . . . . . . . . . . . . . . . .10

Square Roots and Cube Roots . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .12

Cube roots . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .12

Approximating square roots . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .13

Simplifying square roots . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .13

Grouping Symbols . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .14

Parentheses ( ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .14

Brackets [ ] and braces { } . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .14

Order of operations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .15

Divisibility Rules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .16

Chapter 2: Signed Numbers, Fractions, and Percents . . . . . . . . . . . . .18

Signed Numbers (Positive Numbers and Negative Numbers) . . . . . . . . . .18

Number lines . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .18

Addition of signed numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .19

Subtraction of signed numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .20

Minus preceding parenthesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .20

Multiplying and dividing signed numbers . . . . . . . . . . . . . . . . . . . . . .21

Fractions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .21

Negative fractions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .21

Adding positive and negative fractions . . . . . . . . . . . . . . . . . . . . . . . .21

Subtracting positive and negative fractions . . . . . . . . . . . . . . . . . . . . .22

Multiplying fractions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .22

Canceling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .22

Multiplying mixed numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .23

Dividing fractions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .23

Dividing complex fractions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .23

Dividing mixed numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .24

Simplifying Fractions and Complex Fractions . . . . . . . . . . . . . . . . . . . . .24

Decimals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .25

Changing terminating decimals to fractions . . . . . . . . . . . . . . . . . . . .25

Changing fractions to decimals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .25

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Changing inﬁnite repeating decimals to fractions . . . . . . . . . . . . . . . .26

Important equivalents that can save you time . . . . . . . . . . . . . . . . . . .28

Percent . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .28

Changing decimals to percents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .28

Changing percents to decimals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .29

Changing fractions to percents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .29

Changing percents to fractions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .29

Finding the percent of a number . . . . . . . . . . . . . . . . . . . . . . . . . . . . .30

Other applications of percent . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .30

Percent–proportion method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .31

Scientiﬁc Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .32

Multiplication in scientiﬁc notation . . . . . . . . . . . . . . . . . . . . . . . . . .32

Division in scientiﬁc notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .33

Chapter 3: Terminology, Sets, and Expressions . . . . . . . . . . . . . . . . . . .35

Some Basic Language . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .35

Understood multiplication . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .35

Letters to be aware of . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .35

Set Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .36

Special sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .36

Describing sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .36

Types of sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .36

Comparing sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .37

Operations with sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .37

Variables and Algebraic Expressions . . . . . . . . . . . . . . . . . . . . . . . . . . . . .37

Evaluating Expressions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .38

Chapter 4: Equations, Ratios, and Proportions . . . . . . . . . . . . . . . . . . .40

Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .40

Axioms of equality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .40

Solving equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .41

Literal equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .44

Ratios and Proportions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .46

Ratios . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .46

Proportions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .46

Solving proportions for value . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .47

Chapter 5: Equations with Two Variables . . . . . . . . . . . . . . . . . . . . . . . .49

Solving Systems of Equations (Simultaneous Equations) . . . . . . . . . . . . .49

Addition/subtraction method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .49

Substitution method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .52

Graphing method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .53

Chapter 6: Monomials, Polynomials, and Factoring . . . . . . . . . . . . . . .55

Monomials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .55

Adding and subtracting monomials . . . . . . . . . . . . . . . . . . . . . . . . . .55

Multiplying monomials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .56

iv CliffsQuickReview Algebra I

6370-X FM.F 4/27/01 9:34 AM Page iv

Table of Contents v

Dividing monomials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .57

Working with negative exponents . . . . . . . . . . . . . . . . . . . . . . . . . . . .58

Polynomials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .58

Adding and subtracting polynomials . . . . . . . . . . . . . . . . . . . . . . . . . .58

Multiplying polynomials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .59

Dividing polynomials by monomials . . . . . . . . . . . . . . . . . . . . . . . . .61

Dividing polynomials by polynomials . . . . . . . . . . . . . . . . . . . . . . . . .61

Factoring . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .64

Factoring out a common factor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .64

Factoring the difference between two squares . . . . . . . . . . . . . . . . . . .64

Factoring polynomials having three terms of the form ax2+ bx + c . . . .65

Factoring by grouping . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .69

Chapter 7: Algebraic Fractions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .71

Operations with Algebraic Fractions . . . . . . . . . . . . . . . . . . . . . . . . . . . .72

Reducing algebraic fractions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .72

Multiplying algebraic fractions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .72

Dividing algebraic fractions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .73

Adding or subtracting algebraic fractions . . . . . . . . . . . . . . . . . . . . . .73

Chapter 8: Inequalities, Graphing, and Absolute Value . . . . . . . . . . . .77

Inequalities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .77

Axioms and properties of inequalities . . . . . . . . . . . . . . . . . . . . . . . . .77

Solving inequalities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .78

Graphing on a Number Line . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .79

Graphing inequalities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .80

Intervals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .81

Absolute Value . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .82

Solving equations containing absolute value . . . . . . . . . . . . . . . . . . . .82

Solving inequalities containing absolute value and graphing . . . . . . . .83

Chapter 9: Coordinate Geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .87

Coordinate Graphs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .87

Graphing equations on the coordinate plane . . . . . . . . . . . . . . . . . . . .89

Slope and intercept of linear equations . . . . . . . . . . . . . . . . . . . . . . . .92

Graphing linear equations using slope and intercept . . . . . . . . . . . . . .95

Finding the equation of a line . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .98

Linear Inequalities and Half-Planes . . . . . . . . . . . . . . . . . . . . . . . . . . . .100

Open half-plane . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .100

Closed half-plane . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .102

Chapter 10: Functions and Variations . . . . . . . . . . . . . . . . . . . . . . . . . .106

Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .106

Relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .106

Domain and range . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .107

Deﬁning a function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .108

Graphs of functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .108

6370-X FM.F 4/27/01 9:34 AM Page v

Graphs of relationships that are not functions . . . . . . . . . . . . . . . . . .109

Determining domain, range, and if the relation is a function . . . . . .110

Finding the values of functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . .112

Variations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .113

Direct variation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .113

Inverse variation (indirect variation) . . . . . . . . . . . . . . . . . . . . . . . . .115

Chapter 11: Roots and Radicals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .117

Simplifying Square Roots . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .117

Operations with Square Roots . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .119

Under a single radical sign . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .119

When radical values are alike . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .119

When radical values are different . . . . . . . . . . . . . . . . . . . . . . . . . . .119

Addition and subtraction of square roots after simplifying . . . . . . . .120

Products of nonnegative roots . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .120

Quotients of nonnegative roots . . . . . . . . . . . . . . . . . . . . . . . . . . . . .121

Chapter 12: Quadratic Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . .124

Solving Quadratic Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .124

Factoring . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .124

The quadratic formula . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .127

Completing the square . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .129

Chapter 13: Word Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .133

Solving Technique . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .133

Key Words and Phrases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .134

Simple Interest . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .135

Compound Interest . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .136

Ratio and Proportion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .137

Percent . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .137

Percent Change . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .138

Number Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .139

Age Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .140

Motion Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .142

Coin Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .143

Mixture Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .145

Work Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .146

Number Problems with Two Variables . . . . . . . . . . . . . . . . . . . . . . . . . .147

CQR Review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .150

CQR Resource Center . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .159

Glossary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .161

Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .166

vi CliffsQuickReview Algebra I

6370-X FM.F 4/27/01 9:34 AM Page vi

Introduction

CliffsQuickReview Algebra I is designed to give a clear, concise, easy-

to-use review of the basics of algebra. Introducing each topic, defin-

ing key terms, and carefully “walking” through each sample problem type

in a step-by-step manner gives the student insight and understanding to

important algebraic concepts.

The prerequisite to get the most out of this book are an understanding of

the important concepts of basic math—working with fractions, decimals,

percents, and signed numbers. CliffsQuickReview Algebra I starts with a

short review of pre-algebra (Chapters 1 and 2) to review some necessary

background. The rest of the book (Chapters 3 through 13) focuses on the

foundations of algebra.

Why You Need This Book

Can you answer yes to any of these questions?

■Do you need to review the fundamentals of algebra?

■Do you need a course supplement to Algebra I?

■Do you need a concise, comprehensive reference for algebraic concepts?

If so, then CliffsQuickReview Algebra I is for you!

How to Use This Book

You can use this book in any way that fits your personal style for study and

review—you decide what works best with your needs. You can either read

the book from cover to cover or just look for the information you want

and put it back on the shelf for later. Here are just a few ways you can

search for topics:

■Use the Pocket Guide to find essential information, such as the prop-

erties of equalities and inequalities, the quadratic equation, and rules

for operations with powers and exponents.

■Look for areas of interest in the book’s Table of Contents, or use the

index to find specific topics.

6370-X Intro.F 4/27/01 9:34 AM Page 1

■Flip through the book looking for subject areas at the top of each

page.

■Get a glimpse of what you’ll gain from a chapter by reading through

the “Chapter Check-In” at the beginning of each chapter.

■Use the Chapter Checkout at the end of each chapter to gauge your

grasp of the important information you need to know.

■Test your knowledge more completely in the CQR Review and look

for additional sources of information in the CQR Resource Center.

■Use the glossary to find key terms fast. This book defines new terms

and concepts where they first appear in the chapter. If a word is bold-

faced, you can find a more complete definition in the book’s glossary.

■Or flip through the book until you find what you’re looking for—we

organized this book to gradually build on key concepts.

Visit Our Web Site

A great resource, www.cliffsnotes.com, features review materials, valu-

able Internet links, quizzes, and more to enhance your learning. The site

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When you stop by our site, don’t hesitate to share your thoughts about this

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ton. We welcome your feedback!

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