Algebra I All-in-One For Dummies

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