Life of Fred Beginning Algebra

Stanley F. Schmidt, Ph.D.

Polka Dot Publishing

What Algebra Is All About

W hen I first started studying algebra, there was no one in myfamily who could explain to me what it was all about. My Dadhad gone through the eighth grade in South Dakota, and my

Mom never mentioned to me that she had ever studied any algebra in heryears before she took a job at Planter’s Peanuts in San Francisco.

My school counselor enrolled me in beginning algebra, and Ishowed up to class on the first day not knowing what to expect. On thatday, I couldn’t have told you a thing about algebra except that it was somekind of math.

In the first month or so, I found I liked algebra better than . . .

T physical education, because there were never any fist-fights in the algebra class.

T English, because the teacher couldn’t mark me downbecause he or she didn’t like the way I expressed myself ordidn’t like my handwriting or didn’t like my face. Inalgebra, all I had to do was get the right answer and theteacher had to give me an A.

T German, because there were a million vocabulary wordsto learn. I was okay with der Finger which means finger,but besetzen, which means to occupy (a seat or a post) andbesichtigen, which means to look around, and besiegen,which means to defeat, and the zillion other words we hadto memorize by heart were just too much. In algebra, I hadto learn how to do stuff rather than just memorize a bunchof words. (I got C’s in German.)

T biology, because it was too much like German:memorize a bunch of words like mitosis and meiosis. I didenjoy the movies though. It was fun to see the little cellssplitting apart—whether it was mitosis or meiosis, I can’tremember.

So what’s algebra about? Albert Einstein said, “Algebra is a merry

7

science. We go hunting for a little animal whose name we don’t know, sowe call it x. When we bag our game, we pounce on it and give it its rightname.”

What I think Einstein was talking about was solving somethinglike 3x – 7 = 11 and getting an answer of x = 6.

But algebra is much more than just solving equations. One way tothink of it is to consider all the stuff you learned in six or eight years ofstudying arithmetic: adding, multiplying, fractions, decimals, etc. Takeall of that and stir in one new concept—the idea of an “unknown,” whichwe like to call “x.” It’s all of arithmetic taken one step higher.

Adding that little “x” makes a big difference. In arithmetic, youcould answer questions like: If you go 45 miles per hour for six hours,how far have you gone? In algebra, you may have started your trip at 9a.m. and have traveled at 45 miles per hour and then, after you’ve traveledhalf way to your destination, you suddenly speed up to 60 miles per hourand arrive at 5 p.m. Algebra can answer: At what time did you changespeed? That question would “blow away” most arithmetic students, but itis a routine algebra problem (which we solve in chapter four).

Many, many jobs require the use of algebra. Its use is so wide-spread that virtually every university requires that you have learnedalgebra before you get there. Even English majors, like my daughterMargaret, had to learn algebra before going to a university.

I also liked algebra because there were no term papers to have towrite. After I finished my algebra problems I was free to go outside andplay. Margaret had to stay inside and type all night. A lot of Englishmajors seem to have short fingers (der Finger?) because they type so much.

8

A Note to Students

When I studied algebra, my teacher told the class that we couldreasonably expect to spend 30 minutes per page to master the material inthe old algebra book we used. With the book you are holding in yourhands, you will need two reading speeds: 30 minutes per page when you'relearning algebra and whatever speed feels good when you're enjoying thelife adventures of Fred.

Our story begins on the day before Fred’s sixth birthday. Startwith chapter one, and things will explain themselves nicely.

After 12 chapters, you will have mastered all of beginning algebra.

Fred’s Home Companion: Beginning Algebra is intended to beused with this book. It divides the book you are now holding into dailylessons. It supplies the answers for all questions that are not answered inLife of Fred: Beginning Algebra. It supplies additional practice. Turn topage 319 for more details.

One Important Rule

When you come to the questions in the Your

Turn to Play or in the Cities, you must take out a piece

of paper and write out the answers to the questionsBEFORE you look at the answers. If the answers aregiven on the same page as the questions, use a pieceof paper to cover those answers so that you don’t“accidently” see them.

At first, the questions will seem easy, andyou will be tempted to ignore this rule. But bychapter four (word problems), students who havehabitually just looked at the questions and read theanswers will often find that they are “lost.”

9

Contents

Chapter 1 Numbers and Sets. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15finite and infinite setsnatural numbers, whole numbers, integersset notationnegative numbersratiosthe empty set

Chapter 2 The Integers. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32less than (<) and the number linemultiplicationproportionðcoefficients

Chapter 3 Equations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57solving equations with ratiosformulas from geometryorder of operationsconsecutive numbersrational numbersset builder notationdistance = (rate)(time) problemsdistributive propertyproof that (negative) × (negative) = positive

Chapter 4 Motion and Mixture. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87proof of the distributive propertyprice and quantity problemsmixture problemsage problems

Chapter 5 Two Unknowns. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109solving two equations, two unknowns by eliminationunion of setsgraphing of pointsmean, mode, and median averagesgraphing linear equationsgraphing any equation

Chapter 6 Exponents.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140solving two equations, two unknowns by graphingsolving two equations, two unknowns by substitutionx x , (x ) and x ÷ xm n m n m n

inconsistent and dependent equationsfactorialscommutative lawsnegative exponents

13

Chapter 7 Factoring. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 165multiplying binomialssolving quadratic equations by factoringcommon factorsfactoring x² + bx + cfactoring a difference of squaresfactoring by groupingfactoring ax² + bx + c

Chapter 8 Fractions.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 190solving equations containing fractionssimplifying fractionsadding and subtracting fractionsmultiplying and dividing fractionscomplex fractions

Chapter 9 Square Roots. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 213solving pure quadratic equationsprincipal square rootsPythagorean theoremthe real numbersthe irrational numberscube roots and indexessolving radical equationsrationalizing the denominatorextraneous roots

Chapter 10 Quadratic Equations.. . . . . . . . . . . . . . . . . . . . . . . . . . . . 236solving quadratic equations by completing the squarethe quadratic formulalong division of a polynomial by a binomial

Chapter 11 Functions and Slope. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 259definition of a functiondomain, codomain, imagesix definitions of slopeslope-intercept (y = mx + b) form of the linerange of a function

Chapter 12 Inequalities and Absolute Value. . . . . . . . . . . . . . . . . . . 285graphing inequalities in two dimensionsdivision by zeroalgebraically solving linear inequalities with one unknown

A.R.T. section (quick summary of all of beginning algebra). . . . . . . . . . 306

Index. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 316

14

Chapter OneNumbers & Sets

He stood in the middle of the largest rose garden he’d ever seen. Thesun was warm and the smell of the roses made his head spin alittle. Roses of every kind surrounded him. On his left was a

patch of red roses: Chrysler Imperial (a dark crimson); GrandMasterpiece (bright red); Mikado (cherry red). On his right were yellowroses: Gold Medal (golden yellow); Lemon Spice (soft yellow). Yellowroses were his favorite.

Up ahead on the path in front of him werewhite roses, lavender roses, orange roses and therewas even a blue rose.

Fred ran down the path. In the sheer joy ofbeing alive, he ran as any healthy five-year-oldmight do. He ran and ran and ran.

At the edge of a large green lawn, he laydown in the shade of some tall roses. He rolled hiscoat up in a ball to make a pillow.

Listening to the robins singing, he figured it was time for a littlesnooze. He tried to shut his eyes.

They wouldn’t shut. Hey! Anybody can shut their eyes. But Fred couldn’t. What was

going on? He saw the roses, the birds, the lawn, but couldn’t close hiseyes and make them disappear. And if he couldn’t shut his eyes, hecouldn’t fall asleep.

You see, Fred was dreaming. He had read somewhere that theonly thing you can’t do in a dream is shut your eyes and fall asleep. SoFred knew that he was dreaming and that gave him a lot of power.

He got to his feet and waved his hand at the sky. It turned purplewith orange polka dots. He giggled. He flapped his arms and began tofly. He settled on the lawn again and made a pepperoni pizza appear.

In short, he did all the things that five-year-olds might do whenthey find themselves King or Queen of the Universe.

15

Chapter One Numbers & Sets

And soon he was bored. He had done all the silly stuff and waslooking around for something constructive to do. So he lined up all theroses in one long row.

They stretched out in a line in both directions going on forever. Since this was a dream, he could have an unlimited (infinite) number ofroses to play with.

When Fred was three years old, he had spent some time studyingphysics and astronomy. He had learned that nothing in the physicaluniverse was infinite. Everything was finite (limited). Every object couldtravel only at a finite speed. Even the number of atoms was finite. Onebook estimated that there are only 10 atoms in the observable universe. 79

10 means 10 times 10 times 10 . . . a total of 79 times, which79

is 10,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000, which is a lot of atoms. (The “79” is an exponent—something we’ll deal with later.)

Now that he had all the roses magically lined up in a row, hedecided to count them. Math was one of Fred’s favorite activities.

Now, normally when you’ve got a bunch of stuff in a pile to count,

(² These are Fred’s dolls that he used to

play with when he was a baby.)

you line them up ..............................

and start on the left and count them 1 2 3

But Fred couldn’t do that with the roses he wanted to count. Therewere too many of them. He couldn’t start on the left as he did with hisdolls. Dolls are easy. Roses are hard.

16

Chapter One Numbers & Sets

1

So how do you count them? There wasn’t even an obvious“middle” rose to start at. In some sense, every rose is in the middle sincethere is an infinite number of roses on each side of every rose. So Fred

1

just selected a rose and called it “1.” From there it was easy to start

counting . . . . . . . . . . . . . . . . . . . . . 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15. . . .

This set (collection, group, bunch) of numbers {1, 2, 3, 4, 5, . . .}is called the natural numbers. At least, Fred figured, with the naturalnumbers he could count half of all the roses.

What to do? How would he count all the roses to the left of “1”?Then Fred remembered the movies he’d seen where rockets were ready forblastoff. The guy in the tower would count the seconds to blastoff: “Five,four, three, two, one, zero!” So he could label the rose just to the left ofthe rose marked “1” as “0.”

This new set, {0, 1, 2, 3, . . . } is called the whole numbers. It’seasy to remember the name since it’s just the natural numbers with a“hole” added. The numeral zero does look like a hole.

A set doesn’t just have to have numbers in it. Fred could gatherthe things from his dream and make a set: {roses, lawn, birds, pizza}. Thefunny looking parentheses are called braces. Braces are used to enclosesets.

Left brace: { Right brace: }

On a computer keyboard, there are actually three types of groupingsymbols: Parentheses: ( )

Braces: { } andBrackets: [ ]

In algebra, braces are used to list the members of a set, while bothparentheses and brackets are used around numbers. For example, youmight write (3 + 4) + 9 or [35 – 6] + 3.

17

Chapter One Numbers & Sets

Braces and brackets both begin with the letter “b” and to rememberwhich one is braces, think of braces on teeth. Those braces are all curlyand twisty.

In English classes, parentheses and brackets are not treated alike. If you want to make a remark in the middle of a sentence (as this sentenceillustrates), then you use parentheses (as I just did).

Brackets are used when you’re quoting someone and you want toadd your own remarks in the middle of their quote: “Four score and seven[87] years ago. . . .”

But brackets and parentheses weren’t going to help Fred withcounting all those roses. The whole numbers only got him this far:

0 1 2 3 4 5 6 7 8 9 10 11 . . .

What he needed were some new numbers. These new numberswould be numbers that would go to the left of zero. So years ago,someone invented negative numbers: minus one, minus two, minus three,minus four. . . .

Some notes on negative numbers:*#1: It would be a drag to have to write this new set as

{. . . minus 3, minus 2, minus 1, 0, 1, 2, . . .}, or even worse, to write{. . . negative 3, negative 2, negative 1, 0, 1, 2, . . .}, so we’ll invent anabbreviation for “minus.” What might we use?

How about screws? The two most common kinds look like r (Phillips screws) and s (slotted screws). Okay. Our new numbersystem will be written {. . . –3, –2, –1, 0, +1, +2, +3, +4, . . . }. We’ll callthis new set the integers.

*#2: All these integers sit on the number line.

Of course, there are more numbers on the number line than just theintegers. There is ð and fractions like ½ and decimals like 2.7.

18

A.R.T.All Reorganized Together

A Super-condensed and Reorganized-by-Topic Overview of Beginning Algebra(Highly abbreviated)

Topics:Absolute valueAdding like termsArithmetic of the IntegersExponentsFractional EquationsFractionsFunctionsGeometryGraphingInequalitiesLawsMultiplying and Factoring PolynomialsNumbersOrder of OperationsQuadratic EquationsRadicalsSetsStatisticsTwo Equations and Two UnknownsWord ProblemsWords/Expressions

Absolute valueThe absolute value of a number means take away the negative sign

if there is one. | –5 | = 5 | 0 | = 0 | 4 | = 4 (p. 296)

Adding like terms 6wz³ + 2wz³ = 8wz³ (p. 60)3 apples plus 3 apples plus 4 apples plus 6 apples plus 2 apples is 18 apples.

306

A.R.T.

Arithmetic of the IntegersGoing from –7 to +8 means 8 – (–7) = 15 (p. 20)8 – (–7) becomes 8 + (+7) (p. 21)

To subtract a negative is the same as adding the positive.For multiplication:

Signs alike ¸ Answer positiveSigns different ¸ Answer negative (p. 36)

Exponentsx²x³ = (xx)(xxx) = xxxxx = x (p. 146)5

When the bases are the same, you add the exponents.(x²)³ = x When you have an exponent-on-an-exponent, 6

you multiply. (pp. 155, 307)

x n = x (p. 158)m – n x m

x equals 1/x³ (p. 158)–3

x always equals 1 (except that 0 is undefined.)0 0

Fractional Equations

12 16 24 x + + = (p. 193) 1 1 1 1

12 16 24 x becomes + + = 1@ 48x 1 @ 48x 1 @ 48x 1 @ 48x

which simplifies to 4x + 3x + 2x = 48Memory aid: Santa Claus delivering packages (p. 198)

FractionsSimplifying: (p. 202)

x² + 6x + 8 (x + 4)(x + 2) (x + 4)(x + 2 ) x + 4 = = = x² + 5x + 6 (x + 3)(x + 2) (x + 3)(x + 2) x + 3

Memory aid: factor top; factor bottom; cancel like factors.One tricky simplification:

x(4 – x) –x(x – 4) –x = = (x – 3)(x – 4) (x – 3)(x – 4) x – 3

Adding, subtracting, multiplying and dividing—see page 202 Long Division by a binomial—see page 253

& 6&x&³ &+ &19&&x&²& &+& &2&2&x& +& 3&0&For example: 2x + 5 )

307

Index

! (factorial). . . . . . . . . . . . . . . . . 148

<. . . . . . . . . . . . . . . . . . . . . . . . . . 34

>. . . . . . . . . . . . . . . . . . . . . . . . . . 37

ð. . . . . . . . . . . . . . . . . . . . . . . . . . 47

abscissa.. . . . . . . . . . . . . . . . . . . 122

absolute value. . . . . . . . . . . . . . . 296

approximately equal to. . . . . . . . 143

atoms in the observable universe

. . . . . . . . . . . . . . . . . . . . 16

averages

mean, median, mode. . . . . . . 126

a² + b² = c²

Pythagorean theorem. . . . . . 218

base

exponents. . . . . . . . . . . . . . . 154

Bernard of Morlas. . . . . . . . . . . . 89

binomial. . . . . . . . . . . . . . . . . . . 169

braces. . . . . . . . . . . . . . . . . . . . . . 17

brackets. . . . . . . . . . . . . 17, 18, 313

C = (5/9)(F – 32). . . . . . . . . . . . 285

C = ðd.. . . . . . . . . . . . . . . . . . . . . 48

cancel crazy. . . . . . . . . . . . . . . . 209

changing two of the three signs of a

fraction. . . . . . . . . . . . . 196

Christina Rossetti. . . . . . . . . . . . 200

circumference. . . . . . . . . . . . . . . . 43

Clint Eastwood. . . . . . . . . . . . . . 191

codomain. . . . . . . . . . . . . . . . . . 261

coefficient.. . . . . . . . . . . . . . . . . . 44

commutative law of addition. . 207,

208

commutative law of multiplication

. . . . . . . . . . . . . . . . . . . 155

completing the square.. . . . . . . . 240

complex fractions. . . . . . . . . . . . 209

conjugate. . . . . . . . . . . . . . . . . . 231

consecutive numbers. . . . . . . . . . 68

even integers. . . . . . . . . . . . . . 71

odd integer. . . . . . . . . . . . . . . 71

continuous. . . . . . . . . . . . . . . . . 132

coordinates. . . . . . . . . . . . . . . . . 122

cube root. . . . . . . . . . . . . . . . . . . 226

Der Rosenkavalier. . . . . . . . . . . 213

diameter. . . . . . . . . . . . . . . . . . . . 43

of the earth. . . . . . . . . . . . . . . 45

distributive property. . . . . . . . . . . 76

the proof. . . . . . . . . . . . . . 88, 89

division by zero. . . . . 250, 293-295,

303

domain. . . . . . . . . . . . . . . . . . . . 261

empty set. . . . . . . . . . . . . . . 27, 117

Erasmus. . . . . . . . . . . . . . . . . . . . 57

extraneous answers. . . . . . . . . . . 230

factorial. . . . . . . . . . . . . . . . . . . 148

factoring

ax² + bx + c where a � 1. . . . 181

common factor. . . . . . . . . . . 174

difference of squares. . . . . . . 178

grouping. . . . . . . . . . . . . . . . 179

x² + bx + c. . . . . . . . . . . . . . 175

x² – y². . . . . . . . . . . . . . . . . . 178

Fadiman’s Lifetime Reading Plan

. . . . . . . . . . . . . . . . . . . 156

Fahrenheit.. . . . . . . . . . . . . . . . . . 19

finite. . . . . . . . . . . . . . . . . . . . . . . 16

fractions

add. . . . . . . . . . . . . . . . . . . . 202

adding using interior decorating

. . . . . . . . . . . . . . . 203-205

divide. . . . . . . . . . . . . . . . . . 203

multiply.. . . . . . . . . . . . . . . . 202

simplify. . . . . . . . . . . . . . . . . 202

subtract. . . . . . . . . . . . . . . . . 202

Frank Lloyd Wright. . . . . . . . . . 151

Fred teaching style. . . . . . . . . . . . 63

function. . . . . . . . . . . . . . . . . . . 260

codomain. . . . . . . . . . . . . . . 261

domain. . . . . . . . . . . . . . . . . 261

range. . . . . . . . . . . . . . . . . . . 277

graphing any equation. . . . . . . . 130

Guess the Function. . . . . . . . . . . 262

hebdomadal. . . . . . . . . . . . . . . . 150

Heron’s formula. . . . . . . . . . . . . 234

hexagon. . . . . . . . . . . . . . . . . . . . 36

hyperbola. . . . . . . . . . . . . . 132, 299

hypotenuse. . . . . . . . . . . . . . . . . 222

316

Index

i before e.. . . . . . . . . . . . . . . . . . 216

identity mapping.. . . . . . . . . . . . 281

image. . . . . . . . . . . . . . . . . . . . . 261

imaginary number.. . . . . . . . . . . 157

index

on a radical sign. . . . . . . . . . 226

inequality

graphing. . . . . . . . . . . . 287, 289

infinite. . . . . . . . . . . . . . . . . . . . . 16

infinite geometric progression. . 302

integers. . . . . . . . . . . . . . . . . . . . . 18

interior decorating

adding fractions. . . . . . 203-205

Invent a Function. . . . . . . . . . . . 276

irony. . . . . . . . . . . . . . . . . . . . . . 110

irrational numbers.. . . . . . . . . . . 221

Joan of Arc. . . . . . . . . . . . . . . . . 157

KITTENS University. . . . . . . . . . 22

legs. . . . . . . . . . . . . . . . . . . . . . . 222

Leibnitz. . . . . . . . . . . . . . . . . . . . 37

like terms. . . . . . . . . . . . . . . . . . 154

limit of a function. . . . . . . . . . . . 297

linear equations. . . . . . . . . . . . . 129

long division of polynomials. . 253-

255

“Maggie A Lady”. . . . . . . . . . . . 200

mean average. . . . . . . . . . . . . . . 126

median average.. . . . . . . . . . . . . 127

mnemonics. . . . . . . . . . . . . . . . . 205

mode average. . . . . . . . . . . . . . . 126

monomial. . . . . . . . . . . . . . . . . . 169

multiplying signed numbers. . . . 36,

307

multiplying two binomials together

. . . . . . . . . . . . . . . . . . . 168

natural numbers. . . . . . . . . . . . . . 17

negative times a negative equals a

positive

shorter proof. . . . . . . . . . . . . . 82

the proof. . . . . . . . . . . . . . . . . 80

Newton. . . . . . . . . . . . . . . . . . . . . 37

null set. . . . . . . . . . . . . . . . . . . . . 27

number line.. . . . . . . . . . . . . . . . 220

order of operations. . . . . . . . 67, 161

ordered pair. . . . . . . . . . . . . . . . 121

ordinate.. . . . . . . . . . . . . . . . . . . 122

origin. . . . . . . . . . . . . . . . . . . . . 134

parabola. . . . . . . . . . . . . . . . . . . 133

pentadactyl appendage. . . . . . . . . 52

pentagon. . . . . . . . . . . . . . . . . . . . 32

perfect square. . . . . . . . . . . . . . . 221

perimeter. . . . . . . . . . . . . . . . . . . 65

of a triangle. . . . . . . . . . . . . . 233

pi ð . . . . . . . . . . . . . . . . . . . . . 47

2000 digits. . . . . . . . . . . . 45, 46

point-plotting. . . . . . . . . . . . . . . 289

polynomial. . . . . . . . . . . . . . . . . 169

Priestly, Joseph. . . . . . . . . . . . . . . 24

proportion. . . . . . . . . . . . . . . . . . . 42

pure quadratic. . . . . . . . . . . . . . . 214

Pythagorean theorem. . . . . . . . . 218

quadrants. . . . . . . . . . . . . . . . . . 122

quadratic equations

solved by completing the square

. . . . . . . . . . . . . . . 239-241

solved by factoring. . . . 172, 173

quadratic formula

derived. . . . . . . . . . . . . . . . . 246

using. . . . . . . . . . . . . . . 247, 248

quadratic terms. . . . . . . . . . . . . . 172

radical equation. . . . . . . . . . . . . 228

solving. . . . . . . . . . . . . 230, 231

radical sign. . . . . . . . . . . . . . . . . 215

radicand. . . . . . . . . . . . . . . . . . . 222

raised dot. . . . . . . . . . . . . . . . . . . 33

range. . . . . . . . . . . . . . . . . . . . . . 277

ratio. . . . . . . . . . . . . . . . . . . . 22, 23

continued. . . . . . . . . . . . . . . . 58

rational expressions. . . . . . . . . . 201

rational numbers. . . . . . . . . . . . . . 73

rationalizing the denominator. . . 229

real numbers. . . . . . . . . . . . . . . . 220

rectangular coordinate system. . 149

rectangular parallelepiped. . . . . 214

reflexive property of equality. . . . 80

right triangle. . . . . . . . . . . . . . . . 218

hypotenuse. . . . . . . . . . . . . . 222

legs. . . . . . . . . . . . . . . . . . . . 222

317

Index

Santa Claus

approach to solving fractional

equations. . . . . . . . 198, 205

sector. . . . . . . . . . . . . . . . . . . . . . 65

perimeter. . . . . . . . . . . . . . . . . 84

set. . . . . . . . . . . . . . . . . . . . . . . . . 17

set builder notation. . . . . . . . . . . . 73

slope. . . . . . . . . . . . . . . . . . . . . . 267

equal to zero. . . . . . . . . . . . . 291

slope-intercept y = mx + b. . . . . 274

square root. . . . . . . . . . . . . . . . . 215

principal. . . . . . . . . . . . . . . . 215

simplifying. . . . . . . . . . 222, 223

subset. . . . . . . . . . . . . . . . . . . . . 277

symmetric law of equality.. . 66, 219

theorem.. . . . . . . . . . . . . . . . . . . 218

transpose.. . . . . . . . . . . . . . . . . . 110

trapezoid. . . . . . . . . . . . . . . . . . . . 65

area. . . . . . . . . . . . . . . . . . . . . 66

trinomial. . . . . . . . . . . . . . . . . . . 169

two equations and two unknowns

dependent equations. . . . . . . 147

inconsistent equations. . . . . . 147

solved by addition.. . . . . . . . 113

solved by graphing. . . . . . . . 141

solved by substitution. . . . . . 141

union

of sets. . . . . . . . . . . . . . . . . . 115

volume of a cylinder

V = ðr²h. . . . . . . . . . . . . . . . 142

volume of a sphere

V = (4/3)ðr³.. . . . . . . . . . . . . 152

Weimar Constitution of Germany

. . . . . . . . . . . . . . . . . . . . 73

whole numbers. . . . . . . . . . . . . . . 17

y = mx + b. . . . . . . . . . . . . . . . . 274

y-intercept.. . . . . . . . . . . . . . . . . 274

Zeno. . . . . . . . . . . . . . . . . . . . . . . 36

zero-sum game. . . . . . . . . . . . . . . 39

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