Algebra I For Dummies

INTRODUCTION 1 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 PART 1: STARTING OFF WITH THE BASICS 7 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 PART 2: FIGURING OUT FACTORING 91 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 PART 3: WORKING EQUATIONS 169 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 PART 4: APPLYING ALGEBRA 263 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 PART 5: THE PART OF TENS 345 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 INDEX 359