The Complete Idiot''s Guide To Algebra - IGCSE STUDY BANK
Algebra
Algebraby W. Michael KelleyA member of Penguin Group (USA) Inc.
For my wife, Lisa, who makes my life worth living, and my son, Nicholas, who taught me thatwaking up in the morning with the people you love is just the best thing in the world.ALPHA BOOKSPublished by the Penguin GroupPenguin Group (USA) Inc., 375 Hudson Street, New York, New York 10014, U.S.A.Penguin Group (Canada), 10 Alcorn Avenue, Toronto, Ontario, Canada M4V 3B2 (a division of PearsonPenguin Canada Inc.)Penguin Books Ltd, 80 Strand, London WC2R 0RL, EnglandPenguin Ireland, 25 St Stephen’s Green, Dublin 2, Ireland (a division of Penguin Books Ltd)Penguin Group (Australia), 250 Camberwell Road, Camberwell, Victoria 3124, Australia (a division ofPearson Australia Group Pty Ltd)Penguin Books India Pvt Ltd, 11 Community Centre, Panchsheel Park, New Delhi—10 017, IndiaPenguin Group (NZ), cnr Airborne and Rosedale Roads, Albany, Auckland 1310, New Zealand (a division ofPearson New Zealand Ltd)Penguin Books (South Africa) (Pty) Ltd, 24 Sturdee Avenue, Rosebank, Johannesburg 2196, South AfricaPenguin Books Ltd, Registered Offices: 80 Strand, London WC2R 0RL, EnglandCopyright 2004 by W. Michael KelleyAll rights reserved. No part of this book shall be reproduced, stored in a retrieval system, or transmitted byany means, electronic, mechanical, photocopying, recording, or otherwise, without written permission from thepublisher. No patent liability is assumed with respect to the use of the information contained herein. Althoughevery precaution has been taken in the preparation of this book, the publisher and author assume no responsibility for errors or omissions. Neither is any liability assumed for damages resulting from the use of information contained herein. For information, address Alpha Books, 800 East 96th Street, Indianapolis, IN 46240.THE COMPLETE IDIOT’S GUIDE TO and Design are registered trademarks of Penguin Group (USA) Inc.International Standard Book Number: 1-4295-1385-3Library of Congress Catalog Card Number: 2004103222Note: This publication contains the opinions and ideas of its author. It is intended to provide helpful andinformative material on the subject matter covered. It is sold with the understanding that the author and publisher are not engaged in rendering professional services in the book. If the reader requires personal assistanceor advice, a competent professional should be consulted.The author and publisher specifically disclaim any responsibility for any liability, loss, or risk, personal or otherwise, which is incurred as a consequence, directly or indirectly, of the use and application of any of the contents of this book.Publisher: Marie Butler-KnightProduct Manager: Phil KitchelSenior Managing Editor: Jennifer ChisholmSenior Acquisitions Editor: Mike SandersDevelopment Editor: Nancy D. LewisSenior Production Editor: Billy FieldsCopy Editor: Amy BorrelliIllustrator: Richard KingCover/Book Designer: Trina WurstIndexer: Brad HerrimanLayout/Proofreading: Becky Harmon, Donna Martin
Contents at a GlancePart 1:A Final Farewell to Numbers1 Getting Cozy with NumbersFind out a thing or two you may not have known about numbers.Part 2:Part 3:32 Making Friends with FractionsNobody really likes fractions, but ignoring them won’t makethem go away.173 Encountering ExpressionsWitness numbers and variables living together in perfectharmony.29Equations and Inequalities414 Solving Basic EquationsFinding the value of x is a delicate balancing act.435 Graphing Linear EquationsYou put your slope in, you take your intercept out, you plug apoint in, and you shake it all about.556 Cooking Up Linear EquationsLearn the recipe for lines, passed down from generation to generation. (The secret ingredient is love.)717 Linear InequalitiesAnd you thought that oak tree in your yard gave off a lot of shade.83Systems of Equations and Matrix Algebra8 Systems of Linear Equations and InequalitiesThe action-packed sequel to solving one equation: Solving twoequations at once. This time, it’s personal.9 The Basics of the MatrixWhat do Neo, Morpheus, Trinity, and slow-motion bullets haveto do with algebra?Part 4:1Now You’re Playing with (Exponential) Power!979911112510 Introducing PolynomialsPolynomial: Yet another fancy algebra term that basically means“more clumps of numbers and variables.”12711 Factoring PolynomialsReverse time and undo polynomial multiplication.141
12 Wrestling with RadicalsEver wondered how square roots really work?15113 Quadratic Equations and InequalitiesEquations that have more than one correct answer? Are youpulling my leg?16514 Solving High-Powered EquationsOkay, so maybe equations can have two answers, but how canthey possibly have more than that?179Part 5:The Function Junction18915 Introducing the FunctionLearn how functions are really just like vending machinesthat dispense cheese crackers.19116 Graphing FunctionsThey say a picture is worth a thousand words. Wouldn’t it begreat if you could draw a graph without plotting a thousandpoints? You can!203Part 6:Please, Be Rational!21917 Rational ExpressionsIt’s time to look at bigger, stronger, and meatier fractionsthan you’ve ever seen before.22118 Rational Equations and InequalitiesFractions make equations tastier, just like little stale marshmallow bits make breakfast cereal more delicious.233Part 7:Wrapping Things Up24719 Whipping Word ProblemsHow to handle your panic when you look at an algebra problem that’s as long as a short story.24920 A Plethora of PracticeThe practice problems are jammed in this chapter so denselythat not even light can escape their gravitational pull.261AppendixesA Solutions to “You’ve Got Problems”275B Glossary295Index303
ContentsPart 1: A Final Farewell to Numbers1 Getting Cozy with Numbers13Classifying Number Sets .4Familiar Classifications .4Intensely Mathematical Classifications .5Persnickety Signs .6Addition and Subtraction .7Multiplication and Division .8Opposites and Absolute Values .9Come Together with Grouping Symbols .9Important Assumptions .11Associative Property .12Commutative Property .13Identity Properties .14Inverse Properties .142 Making Friends with Fractions17What Is a Fraction? .18Ways to Write Fractions .19Simplifying Fractions .20Locating the Least Common Denominator .21Operations with Fractions .24Adding and Subtracting Fractions .24Multiplying Fractions .25Dividing Fractions .263 Encountering Expressions29Introducing Variables .30Translating Words into Math .31Behold the Power of Exponents .32Big Things Come in Small Packages .32Exponential Rules .33Living Large with Scientific Notation .35Dastardly Distribution .36Get Your Operations in Order .37Evaluating Expressions .39
viii The Complete Idiot’s Guide to AlgebraPart 2: Equations and Inequalities414 Solving Basic Equations43Maintaining a Balance .44Adding and Subtracting .45Multiplying and Dividing .46Equations with Multiple Steps .47Absolute Value Equations .51Equations with Multiple Variables .535 Graphing Linear Equations55Climb Aboard the Coordinate Plane .56Sketching Line Graphs .60Graphing with Tables .61Graphing with Intercepts .62It’s a Slippery Slope .64Calculating Slope of a Line .64Interpreting Slope Values .66Kinky Absolute Value Graphs .676 Cooking Up Linear Equations71Point-Slope Form .72Slope-Intercept Form .73Graphing with Slope-Intercept Form .74Standard Form of a Line .76Tricky Linear Equations .78How to Get from Point A to Point B .79Parallel and Perpendicular Lines .817 Linear Inequalities83Equations vs. Inequalities .84Solving Basic Inequalities .85The Inequality Mood Swing .85Graphing Solutions .86Compound Inequalities .89Solving Compound Inequalities .89Graphing Compound Inequalities .90Inequalities with Absolute Values .91Inequalities Involving “Less Than” .91Inequalities Involving “Greater Than” .92Graphing Linear Inequalities .94
ContentsPart 3: Systems of Equations and Matrix Algebra8 Systems of Linear Equations and Inequalities9799Solving a System by Graphing .100The Substitution Method .102The Elimination Method .104Systems That Are Out of Whack .105Systems of Inequalities .1079 The Basics of the Matrix111What Is a Matrix? .112Matrix Operations .112Multiplying by a Scalar .113Adding and Subtracting Matrices .113Multiplying Matrices .115When Can You Multiply Matrices? .115Use Your Fingers to Calculate Matrix Products .116Determining Determinants .1182 2 Determinants .1193 3 Determinants .120Cracking Cramer’s Rule .121Part 4: Now You’re Playing with (Exponential) Power!10 Introducing Polynomials125127Classifying Polynomials .128Adding and Subtracting Polynomials .130Multiplying Polynomials .131Products of Monomials .131Binomials, Trinomials, and Beyond .132Dividing Polynomials .134Long Division .134Synthetic Division .13611 Factoring Polynomials141Greatest Common Factors .142Factoring by Grouping .143Special Factoring Patterns .144Factoring Trinomials Using Their Coefficients .147Factoring with the Bomb Method .149ix
xThe Complete Idiot’s Guide to Algebra12 Wrestling with Radicals151Introducing the Radical Sign .152Simplifying Radical Expressions .152Unleashing Radical Powers .154Radical Operations .155Addition and Subtraction .155Multiplication .156Division .157Solving Radical Equations .159When Things Get Complex .160There’s Something in Your i .161Simplifying Complex Expressions .16213 Quadratic Equations and Inequalities165Solving Quadratics by Factoring .166Completing the Square .168Solving Basic Exponential Equations .169Bugging Out with Squares .170The Quadratic Formula .172All Signs Point to the Discriminant .173Solving One-Variable Quadratic Inequalities .17514 Solving High-Powered Equations179There’s No Escaping Your Roots! .180Finding Factors .181Baby Steps to Solving Cubic Equations .182Calculating Rational Roots .183What About Imaginary and Irrational Roots? .186Part 5: The Function Junction15 Introducing the Function189191Getting to Know Your Relations .191Operating on Functions .194Composition of Functions .196Inverse Functions .197Defining Inverse Functions .198Calculating Inverse Functions .199Bet You Can’t Solve Just One: Piecewise-Defined Functions .200
Contents16 Graphing Functions203Second Verse, Same as the First .204Two Important Line Tests .206The Vertical Line Test .206The Horizontal Line Test .207Determining Domain and Range .208Important Function Graphs .211Graphing Function Transformations .213Making Functions Flip Out .213Stretching Functions .214Moving Functions Around .215Multiple Transformations .216Part 6: Please, Be Rational!21917 Rational Expressions221Simplifying Rational Expressions .222Combining Rational Expressions .224Multiplying and Dividing Rationally .227Encountering Complex Fractions .22918 Rational Equations and Inequalities233Solving Rational Equations .233Proportions and Cross Multiplying .236Investigating Variation .237Direct Variation .238Indirect Variation .240Solving Rational Inequalities .242Part 7: Wrapping Things Up19 Whipping Word Problems247249Interest Problems .250Simple Interest .251Compound Interest .252Area and Volume Problems .254Speed and Distance Problems .256Mixture and Combination Problems .25820 A Plethora of Practice261xi
AppendixesA Solutions to “You’ve Got Problems”275B Glossary295
ForewordJust when you were getting used to dealing with numbers, along comes algebra!“Suddenly all these x’s and y’s start sprouting up all over”—as Mike Kelley so eloquently puts it—“like pimples on prom night.” But have no fear, The Complete Idiot’sGuide to Algebra is here!As a naive junior in Mike Kelley’s high school classroom, I daily looked forward tohearing his relevant and entertaining spin on math. I learned without even realizingit. His laid-back and humorous attitude—which is evident throughout the book—gently eases students into new and complex material.Now that I am an English major at the University of Maryland, Baltimore County, Iam no longer required to study math. So, why am I writing about this book? Becauseit wor
relevant to any high school or college student’s life. The Complete Idiot’s Guide to Algebra. The Complete Idiot’s Guide to Algebra. The Complete Idiot’s Guide to Algebra
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