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Algebra I For Dummies, 2nd Edition (1119297567) cover image

Algebra I For Dummies, 2nd Edition

ISBN: 978-1-119-29756-7
384 pages
May 2016
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About This Book 1

Conventions Used in This Book 2

What You’re Not to Read 2

Foolish Assumptions 3

How This Book Is Organized 3

Part 1: Starting Off with the Basics 3

Part 2: Figuring Out Factoring 4

Part 3: Working Equations 4

Part 4: Applying Algebra 4

Part 5: The Part of Tens 5

Icons Used in This Book 5

Where to Go from Here 6


CHAPTER 1: Assembling Your Tools 9

Beginning with the Basics: Numbers 10

Really real numbers 10

Counting on natural numbers 10

Wholly whole numbers 11

Integrating integers 12

Being reasonable: Rational numbers 12

Restraining irrational numbers 12

Picking out primes and composites 13

Speaking in Algebra 13

Taking Aim at Algebra Operations 14

Deciphering the symbols 14

Grouping 15

Defining relationships 16

Taking on algebraic tasks 16

CHAPTER 2: Assigning Signs: Positive and Negative Numbers 19

Showing Some Signs 20

Picking out positive numbers 20

Making the most of negative numbers 20

Comparing positives and negatives 21

Zeroing in on zero 22

Going In for Operations 22

Breaking into binary operations 22

Introducing non-binary operations 23

Operating with Signed Numbers 25

Adding like to like: Same-signed numbers 25

Adding different signs 26

Subtracting signed numbers 27

Multiplying and dividing signed numbers 29

Working with Nothing: Zero and Signed Numbers 31

Associating and Commuting with Expressions 31

Reordering operations: The commutative property 32

Associating expressions: The associative property 33

CHAPTER 3: Figuring Out Fractions and Dealing with Decimals 35

Pulling Numbers Apart and Piecing Them Back Together 36

Making your bow to proper fractions 36

Getting to know improper fractions 37

Mixing it up with mixed numbers 37

Following the Sterling Low-Fraction Diet 38

Inviting the loneliest number one 39

Figuring out equivalent fractions 40

Realizing why smaller or fewer is better 41

Preparing Fractions for Interactions 43

Finding common denominators 43

Working with improper fractions 45

Taking Fractions to Task 46

Adding and subtracting fractions 46

Multiplying fractions 47

Dividing fractions 50

Dealing with Decimals 51

Changing fractions to decimals 52

Changing decimals to fractions 53

CHAPTER 4: Exploring Exponents and Raising Radicals 55

Multiplying the Same Thing Over and Over and Over 55

Powering up exponential notation 56

Comparing with exponents 57

Taking notes on scientific notation 58

Exploring Exponential Expressions 60

Multiplying Exponents 65

Dividing and Conquering 66

Testing the Power of Zero 66

Working with Negative Exponents 67

Powers of Powers 68

Squaring Up to Square Roots 69

CHAPTER 5: Doing Operations in Order and Checking Your Answers 73

Ordering Operations 74

Gathering Terms with Grouping Symbols 76

Checking Your Answers 78

Making sense or cents or scents 79

Plugging in to get a charge of your answer 79

Curbing a Variable’s Versatility 80

Representing numbers with letters 81

Attaching factors and coefficients 82

Interpreting the operations 82

Doing the Math 83

Adding and subtracting variables 84

Adding and subtracting with powers 85

Multiplying and Dividing Variables 86

Multiplying variables 86

Dividing variables 87

Doing it all 88


CHAPTER 6: Working with Numbers in Their Prime 93

Beginning with the Basics 94

Composing Composite Numbers 95

Writing Prime Factorizations 96

Dividing while standing on your head 96

Getting to the root of primes with a tree 98

Wrapping your head around the rules of divisibility 99

Getting Down to the Prime Factor 100

Taking primes into account 100

Pulling out factors and leaving the rest 103

CHAPTER 7: Sharing the Fun: Distribution 107

Giving One to Each 108

Distributing first 109

Adding first 109

Distributing Signs 110

Distributing positives 110

Distributing negatives 111

Reversing the roles in distributing 112

Mixing It Up with Numbers and Variables 113

Negative exponents yielding fractional answers 115

Working with fractional powers 115

Distributing More Than One Term 117

Distributing binomials 117

Distributing trinomials 118

Multiplying a polynomial times another

polynomial 119

Making Special Distributions 120

Recognizing the perfectly squared binomial 120

Spotting the sum and difference of the same two terms 121

Working out the difference and sum of two cubes 123

CHAPTER 8: Getting to First Base with Factoring 127

Factoring 127

Factoring out numbers 128

Factoring out variables 130

Unlocking combinations of numbers and variables 131

Changing factoring into a division problem 133

Grouping Terms 134

CHAPTER 9: Getting the Second Degree 139

The Standard Quadratic Expression 140

Reining in Big and Tiny Numbers 141

FOILing 142

FOILing basics 142

FOILed again, and again 143

Applying FOIL to a special product 146

UnFOILing 147

Unwrapping the FOILing package 148

Coming to the end of the FOIL roll 151

Making Factoring Choices 152

Combining unFOIL and the greatest common factor 153

Grouping and unFOILing in the same package 154

CHAPTER 10: Factoring Special Cases 157

Befitting Binomials 157

Factoring the difference of two perfect squares 158

Factoring the difference of perfect cubes 159

Factoring the sum of perfect cubes 162

Tinkering with Multiple Factoring Methods 163

Starting with binomials 163

Ending with binomials 164

Knowing When to Quit 165

Incorporating the Remainder Theorem 166

Synthesizing with synthetic division 166

Choosing numbers for synthetic division 167


CHAPTER 11: Establishing Ground Rules for Solving Equations 171

Creating the Correct Setup for Solving Equations 172

Keeping Equations Balanced 172

Balancing with binary operations 173

Squaring both sides and suffering the consequences 174

Taking a root of both sides 175

Undoing an operation with its opposite 176

Solving with Reciprocals 176

Making a List and Checking It Twice 179

Doing a reality check 179

Thinking like a car mechanic when checking your work 180

Finding a Purpose 181

CHAPTER 12: Solving Linear Equations 183

Playing by the Rules 184

Solving Equations with Two Terms 184

Devising a method using division 185

Making the most of multiplication 186

Reciprocating the invitation 188

Extending the Number of Terms to Three 189

Eliminating the extra constant term 189

Vanquishing the extra variable term 190

Simplifying to Keep It Simple 191

Nesting isn’t for the birds 192

Distributing first 192

Multiplying or dividing before distributing 194

Featuring Fractions 196

Promoting practical proportions 196

Transforming fractional equations into proportions 198

Solving for Variables in Formulas 199

CHAPTER 13: Taking a Crack at Quadratic Equations 203

Squaring Up to Quadratics 204

Rooting Out Results from Quadratic Equations 206

Factoring for a Solution 208

Zeroing in on the multiplication property of zero 209

Assigning the greatest common factor and multiplication property of zero to solving quadratics 210

Solving Quadratics with Three Terms 211

Applying Quadratic Solutions 217

Figuring Out the Quadratic Formula 219

Imagining the Worst with Imaginary Numbers 221

CHAPTER 14: Distinguishing Equations with Distinctive Powers 223

Queuing Up to Cubic Equations 224

Solving perfectly cubed equations 224

Working with the not-so-perfectly cubed 225

Going for the greatest common factor 226

Grouping cubes 228

Solving cubics with integers 228

Working Quadratic-Like Equations 230

Rooting Out Radicals 234

Powering up both sides 235

Squaring both sides twice 237

Solving Synthetically 239

CHAPTER 15: Rectifying Inequalities 243

Translating between Inequality and Interval Notation 244

Intervening with interval notation 244

Grappling with graphing inequalities 246

Operating on Inequalities 247

Adding and subtracting inequalities 247

Multiplying and dividing inequalities 248

Solving Linear Inequalities 249

Working with More Than Two Expressions 250

Solving Quadratic and Rational Inequalities 252

Working without zeros 255

Dealing with more than two factors 255

Figuring out fractional inequalities 256

Working with Absolute-Value Inequalities 258

Working absolute-value equations 258

Working absolute-value inequalities 260


CHAPTER 16: Taking Measure with Formulas 265

Measuring Up 265

Finding out how long: Units of length 266

Putting the Pythagorean theorem to work 267

Working around the perimeter 269

Spreading Out: Area Formulas 273

Laying out rectangles and squares 273

Tuning in triangles 274

Going around in circles 276

Pumping Up with Volume Formulas 276

Prying into prisms and boxes 277

Cycling cylinders 277

Scaling a pyramid 278

Pointing to cones 279

Rolling along with spheres 279

CHAPTER 17: Formulating for Profit and Pleasure 281

Going the Distance with Distance Formulas 282

Calculating Interest and Percent 283

Compounding interest formulas 284

Gauging taxes and discounts 286

Working Out the Combinations and Permutations 287

Counting down to factorials 288

Counting on combinations 288

Ordering up permutations 290

CHAPTER 18: Sorting Out Story Problems 291

Setting Up to Solve Story Problems 292

Working around Perimeter, Area, and Volume 294

Parading out perimeter and arranging area 294

Adjusting the area 295

Pumping up the volume 297

Making Up Mixtures 300

Mixing up solutions 301

Tossing in some solid mixtures 302

Investigating investments and interest 302

Going for the green: Money 304

Going the Distance 305

Figuring distance plus distance 306

Figuring distance and fuel 307

Going ’Round in Circles 307

CHAPTER 19: Going Visual: Graphing 311

Graphing Is Good 312

Grappling with Graphs 313

Making a point 314

Ordering pairs, or coordinating coordinates 315

Actually Graphing Points 316

Graphing Formulas and Equations 317

Lining up a linear equation 317

Going around in circles with a circular graph 318

Throwing an object into the air 319

Curling Up with Parabolas 321

Trying out the basic parabola 321

Putting the vertex on an axis 322

Sliding and multiplying 324

CHAPTER 20: Lining Up Graphs of Lines 327

Graphing a Line 327

Graphing the equation of a line 329

Investigating Intercepts 332

Sighting the Slope 333

Formulating slope 335

Combining slope and intercept 337

Getting to the slope-intercept form 337

Graphing with slope-intercept 338

Marking Parallel and Perpendicular Lines 339

Intersecting Lines 341

Graphing for intersections 341

Substituting to find intersections 342


CHAPTER 21: The Ten Best Ways to Avoid Pitfalls 347

Keeping Track of the Middle Term 348

Distributing: One for You and One for Me 348

Breaking Up Fractions (Breaking Up Is Hard to Do) 348

Renovating Radicals 349

Order of Operations 349

Fractional Exponents 349

Multiplying Bases Together 350

A Power to a Power 350

Reducing for a Better Fit 351

Negative Exponents 351

CHAPTER 22: The Ten Most Famous Equations 353

Albert Einstein’s Theory of Relativity 353

The Pythagorean Theorem 354

The Value of e 354

Diameter and Circumference Related with Pi 354

Isaac Newton’s Formula for the Force of Gravity 355

Euler’s Identity 355

Fermat’s Last Theorem 356

Monthly Loan Payments 356

The Absolute-Value Inequality 356

The Quadratic Formula 357


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