Algebra I All-in-One For Dummies

Mary Jane Sterling taught algebra, business calculus, geometry, and finite mathematics at Bradley University in Peoria, Illinois, for more than 30 years. When she’s not solving for X, she’s often writing other math books, such as Trigonometry For Dummies and Finite Math For Dummies.

• Introduction 1About This Book 1Foolish Assumptions 2Icons Used in This Book 2Beyond the Book 3Where to Go from Here 3Book 1: Starting Out With Numbers and Properties 5Chapter 1: Assembling Your Tools: Number Systems 7Identifying Numbers by Name 8Realizing real numbers 8Counting on natural numbers 8Whittling out whole numbers 8Integrating integers 9Being reasonable: Rational numbers 9Restraining irrational numbers 9Picking out primes and composites 10Zero: It’s Complicated 10Imagining imaginary numbers 10Coping with complex numbers 10Placing Numbers on the Number Line 12Speaking in Algebra 13Being precise with words 13Describing the size of an expression 15Relating operations with symbols 15Taking Aim at Algebra-Speak 17Herding numbers with grouping symbols 17Defining relationships 19Taking on algebraic tasks 20Practice Questions Answers and Explanations 22Whaddya Know? Chapter 1 Quiz 23Answers to Chapter 1 Quiz 25Chapter 2: Deciphering Signs in Expressions 27Assigning Numbers Their Place 27Using the number line 28Comparing positives and negatives with symbols 29Zeroing in on Zero 31Going in for Operations 32Sorting out types of operations 32Tackling the Basic Binary Operations 35Adding signed numbers 35Making a Difference with Signed Numbers 38Multiplying Signed Numbers 40Dividing Signed Numbers 41Working with Nothing: Zero and Signed Numbers 42Practice Questions Answers and Explanations 44Whaddya Know? Chapter 2 Quiz 47Answers to Chapter 2 Quiz 48Chapter 3: Incorporating Algebraic Properties 49Getting a Grip on Grouping Symbols 49Spreading, Grouping, and Changing the Order 52Distributing the wealth 52Making Associations Work 53Computing by Commuting 55Relating Inverses and Identities 56Investigating Inverses 56Identifying Identities 58Working with Factorial 59Applying the Greatest Integer Function 60Practice Question Answers and Explanations 62Whaddya Know? Chapter 3 Quiz 66Answers to Chapter 3 Quiz 67Chapter 4: Coordinating Fractions and Decimals 69Converting Improper Fractions and Mixed Numbers 70Finding Fraction Equivalences 72Rewriting fractions 72Determining lowest terms 74Making Proportional Statements 75Finding Common Denominators 77Creating common denominators from multiples of factors 78Using the box method 80Applying Fractional Operations 81Adding and subtracting fractions 81Multiplying and dividing fractions 83Simplifying Complex Fractions 86Performing Operations with Decimals 88Changing Fractions to Decimals and Vice Versa 89Making fractions become decimals 90Rounding decimals 91Writing decimals as equivalent fractions 91Practice Question Answers and Explanations 94Whaddya Know? Chapter 4 Quiz 101Answers to Chapter 4 Quiz 103Book 2: Operating On Operations 105Chapter 5: Taming Rampaging Radicals 107Simplifying Radical Terms 108Working through Radical Expressions 109Recognizing perfect square terms 109Rewriting radical terms 110Rationalizing Fractions 113Managing Radicals as Exponential Terms 114Using Fractional Exponents 116Making the switch to fractional exponents 117Simplifying expressions with exponents 118Estimating Answers 119Practice Questions Answers and Explanations 120Whaddya Know? Chapter 5 Quiz 124Answers to Chapter 5 Quiz 125Chapter 6: Exploring Exponents 127Powering up with Exponential Notation 127Using Negative Exponents 130Multiplying and Dividing Exponentials 132Multiplying the same base 132Multiplying the same power 133Dividing with exponents 135Raising Powers to Powers 137Testing the Power of Zero 139Writing Numbers with Scientific Notation 140Practice Questions Answers and Explanations 143Whaddya Know? Chapter 6 Quiz 145Answers to Chapter 6 Quiz 146Book 3: Making Things Simple by Simplifying 149Chapter 7: Simplifying Algebraic Expressions 151Addressing the Order of Operations 152Adding and Subtracting Like Terms 154Multiplying and Dividing Algebraically 157Dealing with factors 157Diving into dividing 158Gathering Terms with Grouping Symbols 160Evaluating Expressions 163Checking Your Answers 165Seeing if it makes sense 165Plugging in values 166Practice Questions Answers and Explanations 168Whaddya Know? Chapter 7 Quiz 171Answers to Chapter 7 Quiz 172Chapter 8: Working with Numbers in Their Prime 175Beginning with the Basics 176Composing Composite Numbers 178Writing Prime Factorizations 178Dividing while standing on your head 178Getting to the root of primes with a tree 180Wrapping your head around the rules of divisibility 182Making Use of a Prime Factor 185Taking primes into account 185Pulling out factors and leaving the rest 187Practice Questions Answers and Explanations 190Whaddya Know? Chapter 8 Quiz 194Answers to Chapter 8 Quiz 195Chapter 9: Specializing in Multiplication Matters 197Distributing One Factor Over Many 198Distributing Signs 200Mixing It up with Numbers and Variables 201Negative exponents yielding fractional answers 203Working with Fractional Powers 205Distributing More Than One Term 207Distributing binomials 207Distributing trinomials 209Curses, Foiled Again — Or Not 210Squaring Binomials 211Multiplying the Sum and Difference of the Same Two Terms 212Powering Up Binomials 213Cubing binomials 213Raising Binomials to Higher Powers 215Creating the Sum and Difference of Cubes 217Multiplying Conjugates 218Practice Questions Answers and Explanations 220Whaddya Know? Chapter 9 Quiz 225Answers to Chapter 9 Quiz 226Chapter 10: Dividing the Long Way to Simplify Algebraic Expressions 229Dividing by a Monomial 229Dividing by a Binomial 231Dividing by Polynomials with More Terms 233Simplifying Division Synthetically 235Practice Questions Answers and Explanations 237Whaddya Know? Chapter 10 Quiz 240Answers to Chapter 10 Quiz 241Book 4: Factoring 243Chapter 11: Figuring on Factoring 245Factoring out the Greatest Common Factor 245Factoring out numbers 246Factoring out variables 249Unlocking combinations of numbers and variables 252Using the Box Method 255Changing Factoring into a Division Problem 257Reducing Algebraic Fractions 258Practice Questions Answers and Explanations 260Whaddya Know? Chapter 11 Quiz 262Answers to Chapter 11 Quiz 264Chapter 12: Taking the Bite out of Binomial Factoring 267Reining in Big and Tiny Numbers 268Factoring the Difference of Squares 269Factoring Differences and Sums of Cubes 271Making Factoring a Multiple Mission 274Practice Questions Answers and Explanations 277Whaddya Know? Chapter 12 Quiz 279Answers to Chapter 12 Quiz 280Chapter 13: Factoring Trinomials and Special Polynomials 281Recognizing the Standard Quadratic Expression 281Focusing First on the Greatest Common Factor 283Unwrapping the FOILing Package 284The opening to unFOIL 284Coming to the end of the FOIL roll 287Factoring Quadratic-Like Trinomials 290Factoring Trinomials Using More Than One Method 291Factoring by Grouping 293Putting All the Factoring Together and Making Factoring Choices 297Combining unFOIL and the GCF 297Grouping and unFOILing in the same package 298Incorporating the Remainder Theorem 301Synthesizing with synthetic division 302Choosing numbers for synthetic division 303Practice Questions Answers and Explanations 305Whaddya Know? Chapter 13 Quiz 310Answers to Chapter 13 Quiz 311Book 5: Solving Linear And Polynomial Equations 313Chapter 14: Establishing Ground Rules for Solving Equations 315Creating the Correct Setup for Solving Equations 316Setting up equations for further action 316Making plans for solving equations 316Keeping Equations Balanced 318Balancing with binary operations 318Squaring both sides and suffering the consequences 320Taking a root of both sides 322Solving with Reciprocals 323Making a List and Checking It Twice 324Doing a reality check 324Thinking like a car mechanic when checking your work 325Practice Problems Answers and Explanations 328Whaddya Know? Chapter 14 Quiz 330Answers to Chapter 14 Quiz 331Chapter 15: Lining Up Linear Equations 333Playing by the Rules 334Using the Addition/Subtraction Property 334Using the Multiplication/Division Property 336Devising a method using division 336Making the most of multiplication 337Reciprocating the invitation 338Putting Several Operations Together 339Solving Linear Equations with Grouping Symbols 343Nesting isn’t for the birds 343Distributing first 343Multiplying or dividing before distributing 346Working with Proportions 349Using the rules for proportions 349Transforming fractional equations into proportions 351Solving for Variables in Formulas 352Practice Questions Answers and Explanations 355Whaddya Know? Chapter 15 Quiz 363Answers to Chapter 15 Quiz 364Chapter 16: Muscling Up to Quadratic Equations 367Using the Square-Root Rule 368Factoring for a Solution 370Zeroing in on the multiplication property of zero 370Assigning the greatest common factor and multiplication property of zero to solving quadratics 372Solving Quadratics with Three Terms 373Using the Quadratic Formula 379Completing the Square 383Imagining the Worst with Imaginary Numbers 384Practice Problems Answers and Explanations 387Whaddya Know? Chapter 16 Quiz 392Answers to Chapter 16 Quiz 393Book 6: Dealing With Non-Polynomial Equations and Inequalities 395Chapter 17: Yielding to Higher Powers 397Queuing Up to Cubic Equations 397Solving perfectly cubed equations 398Working with the not-so-perfectly cubed 400Going for the greatest common factor 401Grouping cubes 404Solving cubics with integers 405Determining How Many Possible Roots 407Applying the Rational Root Theorem 408Using the Factor/Root Theorem 410Solving by Factoring 411Solving Powers That Are Quadratic-Like 412Solving Synthetically 416Practice Questions Answers and Explanations 420Whaddya Know? Chapter 17 Quiz 426Answers to Chapter 17 Quiz 427Chapter 18: Reeling in Radical and Absolute Value Equations 429Raising Both Sides to Solve Radical Equations 430Powering up by squaring both sides 430Raising to higher powers 433Doubling the Fun with Radical Equations .435Solving Absolute Value Equations 437Making linear absolute value equations absolutely wonderful 437Factoring absolute value equations for solutions 440Checking for Absolute Value Extraneous Roots 441Practice Questions Answers and Explanations 443Whaddya Know? Chapter 18 Quiz 447Answers to Chapter 18 Quiz 448Chapter 19: Getting Even with Inequalities 449Defining the Inequality Notation 450Pointing in the right direction 450Grappling with graphing inequalities 450Using the Rules to Work on Inequality Statements 451Rewriting Inequalities Using Interval Notation 453Solving Linear Inequalities 455Solving Quadratic Inequalities 457Dealing with Polynomial and Rational Inequalities 460Solving Absolute-Value Inequalities 463Solving Complex Inequalities 465Practice Questions Answers and Explanations 467Whaddya Know? Chapter 19 Quiz 471Answers to Chapter 19 Quiz 472Book 7: Evaluating Formulas and Story Problems 475Chapter 20: Facing Up to Formulas 477Working with Formulas 477Measuring Up 479Finding out how long: Units of length 479Putting the Pythagorean Theorem to work 482Deciphering Perimeter, Area, and Volume 484Using perimeter formulas to get around 484Squaring off with area formulas 486Soaring with Heron’s formula 488Working with volume formulas 490Getting Interested in Using Percent 492Compounding interest formulas 492Gauging taxes and discounts 495Working out the Combinations and Permutations 497Counting down to factorials 498Counting on combinations 498Ordering up permutations 500Practice Questions Answers and Explanations 502Whaddya Know? Chapter 20 Quiz 505Answers to Chapter 20 Quiz 507Chapter 21: Making Formulas Work in Basic Story Problems 509Setting Up to Solve Story Problems 509Applying the Pythagorean Theorem 511Using Geometry to Solve Story Problems 513Working around Perimeter, Area, and Volume 515Parading out perimeter and arranging area 515Adjusting the area 517Pumping up the volume 519Going ’Round in Circles 523Putting Distance, Rate, and Time in a Formula 525Going the distance with the distance-rate-time formula 525Figuring distance plus distance 527Equating distances 528Figuring distance and fuel 529Counting on Interest and Percent 530Practice Questions Answers and Explanations 532Whaddya Know? Chapter 21 Quiz 537Answers to Chapter 21 Quiz 539Chapter 22: Relating Values in Story Problems 541Tackling Age Problems 542Tackling Consecutive Integer Problems 543Working Together on Work Problems 545Throwing an Object into the Air 547Practice Questions Answers and Explanations 550Whaddya Know? Chapter 22 Quiz 553Answers to Chapter 22 Quiz 554Chapter 23: Measuring Up with Quality and Quantity Story Problems 555Achieving the Right Blend with Mixture Problems 556Concocting the Correct Solution 100% of the Time 559Dealing with Money Problems 561Investigating investments and interest 561Going for the green: Money 564Practice Questions Answers and Explanations 567Whaddya Know? Chapter 23 Quiz 571Answers to Chapter 23 Quiz 572Book 8: Getting a Grip On Graphing 573Chapter 24: Getting a Handle on Graphing 575Thickening the Plot with Points 575Interpreting ordered pairs 576Actually Graphing Points 577Sectioning Off by Quadrants 578Graphing Lines 579Using points to lay out lines 579Going with the horizontal and vertical 581Graphing Lines Using Intercepts 583Computing Slopes of Lines 585Sighting the slope 585Formulating slope 587Graphing with the Slope-Intercept Form 590Changing to the Slope-Intercept Form 591Writing Equations of Lines 592Given a point and a slope 593Given two points 593Picking on Parallel and Perpendicular Lines 594Finding Distances between Points 595Finding Midpoints of Segments 597Practice Questions Answers and Explanations 598Whaddya Know? Chapter 24 Quiz 602Answers to Chapter 24 Quiz 604Chapter 25: Extending the Graphing Horizon 605Finding the Intersections of Lines 605Graphing for intersections 606Substituting to find intersections 607Graphing Parabolas and Circles 608Curling Up with Parabolas 609Trying out the basic parabola 609Putting the vertex on an axis 610Going around in circles with a circular graph 612Plotting and Plugging in Polynomial Graphs 614Investigating Graphs of Inequality Functions 618Taking on Absolute-Value Function Graphs 620Graphing with Transformations 621Sliding and multiplying 622Practice Questions Answers and Explanations 625Whaddya Know? Chapter 25 Quiz 632Answers to Chapter 25 Quiz 633Chapter 26: Coordinating Systems of Equations and Graphing 639Defining Solutions of Systems of Equations 639Solving Systems of Linear Equations 641Using elimination 641Using substitution 642Introducing intersections of lines 643Solving Systems Involving Non-Linear Equations 644Taking on Systems of Three Linear Equations 647Practice Problems Answers and Explanations 648Whaddya Know? Chapter 26 Quiz 651Answers to Chapter 26 Quiz 652Index 655