Life of FredBeginning 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 merryscience. We go hunting for a little animal whose name we don’t know, so

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we 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, howfar have you gone? In algebra, you may have started your trip at 9 a.m.and have traveled at 45 miles per hour and then, after you’ve traveled halfway to your destination, you suddenly speed up to 60 miles per hour andarrive at 5 p.m. Algebra can answer: At what time did you change speed? That question would “blow away” most arithmetic students, but it is aroutine 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.

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A Note to Students

Hi! This is going to be fun.

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.

Just before the Index is the A.R.T. section, which very brieflysummarizes much of beginning algebra. If you have to review for a finalexam or you want to quickly look up some topic eleven years after you’veread this book, the A.R.T. section is the place to go.

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Contents

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

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

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

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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

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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); Grand Masterpiece(bright red); Mikado (cherry red). On his right were yellow roses: GoldMedal (golden yellow); Lemon Spice (soft yellow). Yellow roses were hisfavorite.

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 the onlything you can’t do in a dream is shut your eyes and fall asleep. So Fredknew 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 to fly. 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.

Chapter One Numbers & Sets

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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 in detail 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.

Chapter One Numbers & Sets

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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

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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, . . .} iscalled 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.

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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 number

system will be written {. . . –3, –2, –1, 0, +1, +2, +3, +4, . . . }. We’ll callthis new set the integers.

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A.R.T.All Reorganized Together

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

Topics:Absolute valueArithmetic of the IntegersExponentsFractional EquationsFractionsGeometryGraphingInequalitiesLawsMultiplying and Factoring BinomialsNumbersQuadratic EquationsRadicalsSetsTwo Equations and Two UnknownsWord ProblemsWords/Expressions

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

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

Arithmetic of the IntegersGoing from –7 to +8 means 8 – (–7) = 15 (p. 20)4 – (–6) becomes 4 + (+6) (p. 21)

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

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

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

A.R.T.

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

When the bases are the same (that’s the number under the exponent), then you add the exponents(x²)³ = x An exponent-on-an-exponent multiply (p. 155, 307)6

x n = x (p. 158)m – n x m

x equals 1/x³ (p. 158)–3

x always equals 1. (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 factorsOne tricky simplification:

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

Adding, subtracting, multiplying & dividing—see p. 202Long Division by a binomial—see p. 253

For example: 2x + 5)& &6x&³& +& 1&9&x&²&& +&& 2&&2x& +& &30&

FunctionsA function is any rule which associates to each element of the first set (called the domain) exactly one element of the second set (called the codomain).

Each element in the domain has an image in the codomain.The set of images is called the range.Examples of functions start on p. 260The identity function maps each element onto itself. (p. 281)

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Index

! (factorial) . . . . . . . . . . . . . . 148< . . . . . . . . . . . . . . . . . . . . . . . 34> . . . . . . . . . . . . . . . . . . . . . . . 37$ . . . . . . . . . . . . . . . . . . . . . . 285. . . . . . . . . . . . . . . . . . . . . . . 143B . . . . . . . . . . . . . . . . . . . . . . . . . 47abscissa . . . . . . . . . . . . . . . . . . 122absolute value . . . . . . . . . . . . . 296approximately equal to . . . . . . . 143atoms in the observable universe

. . . . . . . . . . . . . . . . . . . 16averages

mean, median, mode . . . . . . 126a² + b² = c²

Pythagorean theorem . . . . . 218base

exponents . . . . . . . . . . . . . . 154Bernard of Morlas . . . . . . . . . . . 89binomial . . . . . . . . . . . . . . . . . . 169braces . . . . . . . . . . . . . . . . . . . . . 17brackets . . . . . . . . . . . . 17, 18, 313C = (5/9)(F – 32) . . . . . . . . . . . 285C = Bd . . . . . . . . . . . . . . . . . . . . 47cancel crazy . . . . . . . . . . . . . . . 209changing two of the three signs of a

fraction . . . . . . . . . . . . 196Christina Rossetti . . . . . . . . . . . 200circumference . . . . . . . . . . . . . . . 43Clint Eastwood . . . . . . . . . . . . . 191codomain . . . . . . . . . . . . . . . . . 261coefficient . . . . . . . . . . . . . . . . . 44commutative law of addition . . 207,

208commutative law of multiplication

. . . . . . . . . . . . . . . . . . 155completing the square . . . . . . . 241complex fractions . . . . . . . . . . . 209

conjugate . . . . . . . . . . . . . . . . . 231consecutive numbers . . . . . . . . . 68

even integers . . . . . . . . . . . . . 71odd integer . . . . . . . . . . . . . . 71

continuous . . . . . . . . . . . . . . . . 132coordinates . . . . . . . . . . . . . . . . 122cube root . . . . . . . . . . . . . . . . . 226Der Rosenkavalier . . . . . . . . . . 213diameter . . . . . . . . . . . . . . . . . . . 43

of the earth . . . . . . . . . . . . . . 45distributive property . . . . . . . . . . 76

the proof . . . . . . . . . . . . . 88, 89division by zero . . . . 250, 293-295,

303domain . . . . . . . . . . . . . . . . . . . 261empty set . . . . . . . . . . . . . . 27, 117Erasmus . . . . . . . . . . . . . . . . . . . 57extraneous answers . . . . . . . . . 230factorial . . . . . . . . . . . . . . . . . . 148factoring

ax² + bx + c where a � 1 . . 181common factor . . . . . . . . . . 174difference of squares . . . . . 178grouping . . . . . . . . . . . . . . . 179x² + bx + c . . . . . . . . . . . . . 175x² – y² . . . . . . . . . . . . . . . . . 178

Fadiman’s Lifetime Reading Plan. . . . . . . . . . . . . . . . . . 156

Fahrenheit . . . . . . . . . . . . . . . . . 19finite . . . . . . . . . . . . . . . . . . . . . . 16fractions

add . . . . . . . . . . . . . . . . . . . 202adding using interior decorating

. . . . . . . . . . . . . . 203-205divide . . . . . . . . . . . . . . . . . 203multiply . . . . . . . . . . . . . . . 202simplify . . . . . . . . . . . . . . . 202subtract . . . . . . . . . . . . . . . . 202

Frank Lloyd Wright . . . . . . . . . 151Fred teaching style . . . . . . . . . . . 63

317

function . . . . . . . . . . . . . . . . . . 260codomain . . . . . . . . . . . . . . 261domain . . . . . . . . . . . . . . . . 261range . . . . . . . . . . . . . . . . . . 277

graphing any equation . . . . . . . 130Guess the Function . . . . . . . . . . 262hebdomadal . . . . . . . . . . . . . . . 150Heron’s formula . . . . . . . . . . . . 234hexagon . . . . . . . . . . . . . . . . . . . 36hyperbola . . . . . . . . . . . . . 132, 299hypotenuse . . . . . . . . . . . . . . . . 222i before e . . . . . . . . . . . . . . . . . 216identity mapping . . . . . . . . . . . 281image . . . . . . . . . . . . . . . . . . . . 261imaginary number . . . . . . . . . . 157index

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

graphing . . . . . . . . . . . 287, 289infinite . . . . . . . . . . . . . . . . . . . . 16infinite geometric progression

. . . . . . . . . . . . . . . . . . 302integers . . . . . . . . . . . . . . . . . . . . 18interior decorating

adding fractions . . . . . 203-205Invent a Function . . . . . . . . . . . 276irony . . . . . . . . . . . . . . . . . . . . . 110irrational numbers . . . . . . . . . . 221Joan of Arc . . . . . . . . . . . . . . . . 157KITTENS University . . . . . . . . . 22legs . . . . . . . . . . . . . . . . . . . . . . 222Leibnitz . . . . . . . . . . . . . . . . . . . 37like terms . . . . . . . . . . . . . . . . . 154limit of a function . . . . . . . . . . 297linear equations . . . . . . . . . . . . 129long division of polynomials . 253-

255Maggie A Lady . . . . . . . . . . . . 200mean average . . . . . . . . . . . . . . 126median average . . . . . . . . . . . . 127mnemonics . . . . . . . . . . . . . . . . 205mode average . . . . . . . . . . . . . . 126monomial . . . . . . . . . . . . . . . . . 169multiplying signed numbers . . . . 36,

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multiplying two binomials together. . . . . . . . . . . . . . . . . . 168

natural numbers . . . . . . . . . . . . . 17negative times a negative equals a

positiveshorter proof . . . . . . . . . . . . . 82the proof . . . . . . . . . . . . . . . . 80

Newton . . . . . . . . . . . . . . . . . . . . 37null set . . . . . . . . . . . . . . . . . . . . 27number line . . . . . . . . . . . . . . . 220order of operations . . . . . . . 67, 161ordered pair . . . . . . . . . . . . . . . 121ordinate . . . . . . . . . . . . . . . . . . 122origin . . . . . . . . . . . . . . . . . . . . 134parabola . . . . . . . . . . . . . . . . . . 133pentadactyl appendage . . . . . . . . 52pentagon . . . . . . . . . . . . . . . . . . . 32perfect square . . . . . . . . . . . . . . 221perimeter . . . . . . . . . . . . . . . . . . 65

of a triangle . . . . . . . . . . . . 233pi . . . . . . . . . . . . . . . . . . . . . . . . 47

2000 digits . . . . . . . . . . . 45, 46point-plotting . . . . . . . . . . . . . . 289polynomial . . . . . . . . . . . . . . . . 169Priestly, Joseph . . . . . . . . . . . . . 24proportion . . . . . . . . . . . . . . . . . . 42pure quadratic . . . . . . . . . . . . . 214Pythagorean theorem . . . . . . . . 218quadrants . . . . . . . . . . . . . . . . . 122quadratic equations

solved by completing the square. . . . . . . . . . . . . . 239-241

solved by factoring . . . 172, 173quadratic formula

derived . . . . . . . . . . . . . . . . 246using . . . . . . . . . . . . . . 247, 248

quadratic terms . . . . . . . . . . . . . 172radical equation . . . . . . . . . . . . 228

solving . . . . . . . . . . . . 230, 231radical sign . . . . . . . . . . . . . . . . 215radicand . . . . . . . . . . . . . . . . . . 222raised dot . . . . . . . . . . . . . . . . . . 33range . . . . . . . . . . . . . . . . . . . . . 277ratio . . . . . . . . . . . . . . . . . . . 22, 23

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

318

rational expressions . . . . . . . . . 201rational numbers . . . . . . . . . . . . 73rationalizing the denominator . 229real numbers . . . . . . . . . . . . . . . 220rectangular coordinate system

. . . . . . . . . . . . . . . . . . 149rectangular parallelepiped . . . . 214reflexive property of equality . . 80right triangle . . . . . . . . . . . . . . . 218

hypotenuse . . . . . . . . . . . . . 222legs . . . . . . . . . . . . . . . . . . . 222

Santa Clausapproach to solving fractional

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

sector . . . . . . . . . . . . . . . . . . . . . 65perimeter . . . . . . . . . . . . . . . 83

set . . . . . . . . . . . . . . . . . . . . . . . . 17set builder notation . . . . . . . . . . 73slope . . . . . . . . . . . . . . . . . . . . . 267

equal to zero . . . . . . . . . . . . 291slope-intercept y = mx + b . . . . 274square root . . . . . . . . . . . . . . . . 215

principal . . . . . . . . . . . . . . . 215simplifying . . . . . . . . . 222, 223

subset . . . . . . . . . . . . . . . . . . . . 277symmetric law of equality . 66, 219theorem . . . . . . . . . . . . . . . . . . 218transpose . . . . . . . . . . . . . . . . . 110trapezoid . . . . . . . . . . . . . . . . . . . 65

area . . . . . . . . . . . . . . . . . . . . 66trinomial . . . . . . . . . . . . . . . . . . 169two equations and two unknowns

dependent equations . . . . . . 147inconsistent equations . . . . 147solved by addition . . . . . . . 113solved by graphing . . . . . . . 141solved by substitution . . . . . 141

unionof sets . . . . . . . . . . . . . . . . . 115

volume of a cylinderV = Br²h . . . . . . . . . . . . . . . 142

volume of a sphereV = (4/3)Br³ . . . . . . . . . . . . 152

Weimar Constitution of Germany. . . . . . . . . . . . . . . . . . . 73

whole numbers . . . . . . . . . . . . . . 17y = mx + b . . . . . . . . . . . . . . . . 274y-intercept . . . . . . . . . . . . . . . . 274Zeno . . . . . . . . . . . . . . . . . . . . . . 36zero-sum game . . . . . . . . . . . . . . 39