Algebra for the Clueless

BOB MILLER’S ALGEBRA FOR THE CLUELESS

ALGEBRA

OTHER BOOKS BY BOB MILLER

Bob Miller’s Basic Math and Pre-Algebra for the Clueless

Bob Miller’s Geometry for the Clueless, Second Edition

Bob Miller’s SAT® Math for the Clueless, Second Edition

Bob Miller’s Precalc with Trig for the Clueless, Third Edition

Bob Miller’s Calc I for the Clueless, Second Edition

Bob Miller’s Calc II for the Clueless, Second Edition

Bob Miller’s Calc III for the Clueless

BOB MILLER’S ALGEBRA FOR THE CLUELESS

ALGEBRA

Robert Miller

Formerly of the Mathematics Department

City University of New York

Second Edition

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DOI: 10.1036/0071473661

This book is written for you: not for your teacher, not

for your next-door neighbor, not for anyone but you.

Unfortunately, most math books today teach algebra

in a way that does not give you the basics you need to

succeed. Many students immediately have problems,

while some manage to succeed, only to have problems

in algebra 2 or precalculus. This book gives and

explains the topics you will need to succeed.

However, as much as I hate to admit it, I am not per￾fect. If you find something that is unclear or a topic

that should be added to the book, you can contact me

in one of two ways. You can write me c/o McGraw￾Hill, Two Penn Plaza, New York, NY 10121-2298.

Please enclose a self-addressed stamped envelope.

Be patient; I will answer. You can also see me

at www.bobmiller.com and contact me at

[email protected]. I will answer

faster than if you write, but again, please be

patient.

If you need more advanced stuff, there is Geometry

for the Clueless, Precalc with Trig for the Clueless, and

Calc I, Calc II, and Calc III for the Clueless. If you are

preparing for the SAT, SAT® Math for the Clueless will

help you.

Now enjoy the book and learn!!!

Bob Miller

v

TO THE STUDENT:

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vii

CONTENTS

To the Student v

CHAPTER 1 Natural Numbers and Introductory Terms 1

Congratulations 1

Introductory Terms 2

Order of Operations, Numerical Evaluations, and Formulas

from the Past 8

Square 10

Rectangle 10

Triangles 11

Circles 12

Some Definitions, Addition, and Subtraction 14

Products, Quotients, and the Distributive Law 18

Products 18

Quotients 20

Division 21

Distributive Law 22

CHAPTER 2 Integers Plus More 23

Addition 24

Subtraction 26

Multiplication 27

Rational Numbers 30

Division 31

Short Division 33

Distributive Law 34

For more information about this title, click here

CHAPTER 3 First-Degree Equations 35

Solving Linear Equations 37

CHAPTER 4 Problems with Words: Why So Many Students Have

Problems on the SAT 47

Basic Problems 47

Consecutive Integer Problems 51

Geometry Problems 53

Mixtures, Coins, Percentages, and Interest 55

Age Problems 58

Distance Problems 59

CHAPTER 5 Factoring 63

More About the Distributive Law 63

The Game of Factoring 67

The Largest Common Factor 67

Difference of Two Squares 71

Factoring Trinomials I 72

Factoring Trinomials II 76

Sum and Difference of Cubes 79

Grouping 80

Solving Polynomial Equations by Factoring 82

Factors Whose Product Is Zero 82

Solving Equations by Factoring 83

Word Problems 86

CHAPTER 6 Algebraic Fractions 91

Reducing 91

Multiplication and Division 93

Adding and Subtracting 94

Complex Fractions 99

Long Division 101

Fractional Equations 104

viii CONTENTS

Word Problems with Fractions 107

Age Problems Revisited 107

Work Problems 109

Distance Problems Revisited 110

CHAPTER 7 Radicals and Exponents 113

Simplifying Radicals 117

Adding and Subtracting Radicals 118

Multiplying Radicals 119

Dividing Radicals 121

Rationalize the Denominator I 121

Division 122

Rationalize the Denominator II 124

Exponents 125

Law of Exponents for Positive Integers 125

Law of Exponents for All Integers 126

Fractional Exponents 128

Number Theory 131

Divisibility 131

Rational Numbers 133

Irrational Numbers 134

Pythagorean Theorem 134

Real Numbers 138

Complex (Imaginary) Numbers 139

CHAPTER 8 Quadratics 145

Solve by Square Rooting 145

Solve by Completing the Square 146

The Quadratic Formula 149

More About the Quadratic Formula and Quadratic Equation 150

Equations with Radicals 153

CHAPTER 9 Points, Lines, and Planes 157

Points 157

Graphing Lines 159

Slope 165

C O N T E N T S ix

Equation of the Line 167

Parallel and Perpendicular Lines 172

Solving Two Equations in Two Unknowns 174

Solving by Graphing 174

Solving by Substitution 174

Solving by Elimination 176

Relating the Graphs to the Algebra 179

Word Problems 180

Old Ones Revisited 180

New Problems 183

A New Distance Problem 184

Three Equations in Three Unknowns 185

Why We Are Not Doing This by Graphing 185

Solving Three Equations in Three Unknowns 187

Word Problems in Three Dimensions 191

CHAPTER 10 Odds and Ends 195

Ratio and Proportion 195

Geometric Formulas 197

Other Geometric Facts 201

Linear Inequalities 203

Variations of Variation 208

Functions 209

Functional Notation 210

Composite Functions 215

Inverse Functions 216

Translations, Flips, Stretches, and Contractions 220

Preimages and Images Revisited 225

CHAPTER 11 Miscellaneous Miscellany 229

Absolute Value 229

Counting, Probability, and Statistics 233

Statistics 233

Counting 235

Probability 237

Matrices 239

x CONTENTS

APPENDIX 1 Fractions, Decimals, Percents, and Graphs 247

APPENDIX 2 Sets 265

Acknowledgments 269

About Bob Miller: In His Own Words 271

Index 273

C O N T E N T S xi

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CHAPTER 1

NATURAL NUMBERS

AND INTRODUCTORY

TERMS

CONGRATULATIONS

Congratulations!!!! You have reached a point that most

of the world does not even come near, believe it or not.

You are starting algebra. It is a great adventure we are

beginning.

Algebra is a new subject, even if you had a little in

the past. You may have some trouble at the beginning.

I did too!!!! Even though I was getting almost every￾thing correct, for more than two months I didn’t really

understand what was happening, really!!!! After that

things got better. Next there are new vocabulary words.

There are always some at the start of a new course. In

algebra there are less than 100. (In English you need

about 7000 new words for high school.) Since there are

so few words, every word is very important. You must

not only memorize the words but also understand

them. Many of these words occur right at the begin￾ning. This may be kind of boring, but learning these

words is super necessary. If you need to review your

fractions, decimals, percents, and graphs, look at the

appendix at the back of the book.

1

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Now relax. Read the text slowly. If you have trouble

with an example, write it out and don’t go to the next

step until you understand the previous step.

I really love this stuff. I hope after reading parts of

this book, you will too.

Okay. Let’s get started.

INTRODUCTORY TERMS

At the beginning, we will deal with two sets of num￾bers. The first is the set of natural numbers nn, which

are the numbers 1, 2, 3, 4, . . . and the second is the set

of whole numbers 0, 1, 2, 3, 4, . . . . The three dots at

the end mean the set is infinite, that it goes on forever.

The first four numbers show the pattern. Numbers like

5.678, 3/4, −7/45,
7, π, and so on are not natural

numbers and not whole numbers.

We will talk about equality statements, such as 4 + 5 =

9 and 7 − 3 = 4.

We will write 3 + 4 ≠ 10, which says 3 plus 4 does

not equal 10.

A prime natural number is a natural number with

two distinct natural number factors, itself and 1. 1 is

not a prime. The first eight prime factors are 2, 3, 5, 7,

11, 13, 17, and 19.

9 is not a prime since it has three nn prime factors,

1, 3, and 9. Numbers like 9 are called composites.

The even natural numbers are the set 2, 4, 6, 8, ....

The odd natural numbers are the set 1, 3, 5, 7, ....

We would like to graph numbers. We will do it on a

line graph or number line. Let’s give some examples.

EXAMPLE 1—

Graph the first four even natural numbers.

First, draw a straight line with a ruler.

Next, divide the line into convenient lengths.

2 B O B M I L L E R ’ S A L G E B R A F O R T H E C L U E L E S S