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 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