Algebra I For Dummies

Mary Jane Sterling (Peoria, Illinois) is the author of Algebra I For Dummies, Algebra Workbook For Dummies, Algebra II For Dummies, Algebra II Workbook For Dummies, and five other For Dummies books. She has been at Bradley University in Peoria, Illinois for nearly 30 years, teaching algebra, business calculus, geometry, finite mathematics, and whatever interesting material comes her way.

• Introduction 1About This Book 1Conventions Used in This Book 2What You’re Not to Read 2Foolish Assumptions 3How This Book Is Organized 3Part 1: Starting Off with the Basics 3Part 2: Figuring Out Factoring 4Part 3: Working Equations 4Part 4: Applying Algebra 4Part 5: The Part of Tens 5Icons Used in This Book 5Where to Go from Here 6Part 1: Starting off with the Basics 7Chapter 1: Assembling Your Tools 9Beginning with the Basics: Numbers 10Really real numbers 10Counting on natural numbers 10Wholly whole numbers 11Integrating integers 12Being reasonable: Rational numbers 12Restraining irrational numbers 12Picking out primes and composites 13Speaking in Algebra 13Taking Aim at Algebra Operations 14Deciphering the symbols 14Grouping 15Defining relationships 16Taking on algebraic tasks 16Chapter 2: Assigning Signs: Positive and Negative Numbers 19Showing Some Signs 20Picking out positive numbers 20Making the most of negative numbers 20Comparing positives and negatives 21Zeroing in on zero 22Going In for Operations 22Breaking into binary operations 22Introducing non-binary operations 23Operating with Signed Numbers 25Adding like to like: Same-signed numbers 25Adding different signs 26Subtracting signed numbers 27Multiplying and dividing signed numbers 29Working with Nothing: Zero and Signed Numbers 31Associating and Commuting with Expressions 31Reordering operations: The commutative property 32Associating expressions: The associative property 33Chapter 3: Figuring Out Fractions and Dealing with Decimals 35Pulling Numbers Apart and Piecing Them Back Together 36Making your bow to proper fractions 36Getting to know improper fractions 37Mixing it up with mixed numbers 37Following the Sterling Low-Fraction Diet 38Inviting the loneliest number one 39Figuring out equivalent fractions 40Realizing why smaller or fewer is better 41Preparing Fractions for Interactions 43Finding common denominators 43Working with improper fractions 45Taking Fractions to Task 46Adding and subtracting fractions 46Multiplying fractions 47Dividing fractions 50Dealing with Decimals 51Changing fractions to decimals 52Changing decimals to fractions 53Chapter 4: Exploring Exponents and Raising Radicals 55Multiplying the Same Thing Over and Over and Over 55Powering up exponential notation 56Comparing with exponents 57Taking notes on scientific notation 58Exploring Exponential Expressions 60Multiplying Exponents 65Dividing and Conquering 66Testing the Power of Zero 66Working with Negative Exponents 67Powers of Powers 68Squaring Up to Square Roots 69Chapter 5: Doing Operations in Order and Checking Your Answers 73Ordering Operations 74Gathering Terms with Grouping Symbols 76Checking Your Answers 78Making sense or cents or scents 79Plugging in to get a charge of your answer 79Curbing a Variable’s Versatility 80Representing numbers with letters 81Attaching factors and coefficients 82Interpreting the operations 82Doing the Math 83Adding and subtracting variables 84Adding and subtracting with powers 85Multiplying and Dividing Variables 86Multiplying variables 86Dividing variables 87Doing it all 88Part 2: Figuring Out Factoring 91Chapter 6: Working with Numbers in Their Prime 93Beginning with the Basics 94Composing Composite Numbers 95Writing Prime Factorizations 96Dividing while standing on your head 96Getting to the root of primes with a tree 98Wrapping your head around the rules of divisibility 99Getting Down to the Prime Factor 100Taking primes into account 100Pulling out factors and leaving the rest 103Chapter 7: Sharing the Fun: Distribution 107Giving One to Each 108Distributing first 109Adding first 109Distributing Signs 110Distributing positives 110Distributing negatives 111Reversing the roles in distributing 112Mixing It Up with Numbers and Variables 113Negative exponents yielding fractional answers 115Working with fractional powers 115Distributing More Than One Term 117Distributing binomials 117Distributing trinomials 118Multiplying a polynomial times another polynomial 119Making Special Distributions 120Recognizing the perfectly squared binomial 120Spotting the sum and difference of the same two terms 121Working out the difference and sum of two cubes 123Chapter 8: Getting to First Base with Factoring 127Factoring 127Factoring out numbers 128Factoring out variables 130Unlocking combinations of numbers and variables 131Changing factoring into a division problem 133Grouping Terms 134Chapter 9: Getting the Second Degree 139The Standard Quadratic Expression 140Reining in Big and Tiny Numbers 141FOILing 142FOILing basics 142FOILed again, and again 143Applying FOIL to a special product 146UnFOILing 147Unwrapping the FOILing package 148Coming to the end of the FOIL roll 151Making Factoring Choices 152Combining unFOIL and the greatest common factor 153Grouping and unFOILing in the same package 154Chapter 10: Factoring Special Cases 157Befitting Binomials 157Factoring the difference of two perfect squares 158Factoring the difference of perfect cubes 159Factoring the sum of perfect cubes 162Tinkering with Multiple Factoring Methods 163Starting with binomials 163Ending with binomials 164Knowing When to Quit 165Incorporating the Remainder Theorem 166Synthesizing with synthetic division 166Choosing numbers for synthetic division 167Part 3: Working Equations 169Chapter 11: Establishing Ground Rules for Solving Equations 171Creating the Correct Setup for Solving Equations 172Keeping Equations Balanced 172Balancing with binary operations 173Squaring both sides and suffering the consequences 174Taking a root of both sides 175Undoing an operation with its opposite 176Solving with Reciprocals 176Making a List and Checking It Twice 179Doing a reality check 179Thinking like a car mechanic when checking your work 180Finding a Purpose 181Chapter 12: Solving Linear Equations 183Playing by the Rules 184Solving Equations with Two Terms 184Devising a method using division 185Making the most of multiplication 186Reciprocating the invitation 188Extending the Number of Terms to Three 189Eliminating the extra constant term 189Vanquishing the extra variable term 190Simplifying to Keep It Simple 191Nesting isn’t for the birds 192Distributing first 192Multiplying or dividing before distributing 194Featuring Fractions 196Promoting practical proportions 196Transforming fractional equations into proportions 198Solving for Variables in Formulas 199Chapter 13: Taking a Crack at Quadratic Equations 203Squaring Up to Quadratics 204Rooting Out Results from Quadratic Equations 206Factoring for a Solution 208Zeroing in on the multiplication property of zero 209Assigning the greatest common factor and multiplication property of zero to solving quadratics 210Solving Quadratics with Three Terms 211Applying Quadratic Solutions 217Figuring Out the Quadratic Formula 219Imagining the Worst with Imaginary Numbers 221Chapter 14: Distinguishing Equations with Distinctive Powers 223Queuing Up to Cubic Equations 224Solving perfectly cubed equations 224Working with the not-so-perfectly cubed 225Going for the greatest common factor 226Grouping cubes 228Solving cubics with integers 228Working Quadratic-Like Equations 230Rooting Out Radicals 234Powering up both sides 235Squaring both sides twice 237Solving Synthetically 239Chapter 15: Rectifying Inequalities 243Translating between Inequality and Interval Notation 244Intervening with interval notation 244Grappling with graphing inequalities 246Operating on Inequalities 247Adding and subtracting inequalities 247Multiplying and dividing inequalities 248Solving Linear Inequalities 249Working with More Than Two Expressions 250Solving Quadratic and Rational Inequalities 252Working without zeros 255Dealing with more than two factors 255Figuring out fractional inequalities 256Working with Absolute-Value Inequalities 258Working absolute-value equations 258Working absolute-value inequalities 260Part 4: Applying Algebra 263Chapter 16: Taking Measure with Formulas 265Measuring Up 265Finding out how long: Units of length 266Putting the Pythagorean theorem to work 267Working around the perimeter 269Spreading Out: Area Formulas 273Laying out rectangles and squares 273Tuning in triangles 274Going around in circles 276Pumping Up with Volume Formulas 276Prying into prisms and boxes 277Cycling cylinders 277Scaling a pyramid 278Pointing to cones 279Rolling along with spheres 279Chapter 17: Formulating for Profit and Pleasure 281Going the Distance with Distance Formulas 282Calculating Interest and Percent 283Compounding interest formulas 284Gauging taxes and discounts 286Working Out the Combinations and Permutations 287Counting down to factorials 288Counting on combinations 288Ordering up permutations 290Chapter 18: Sorting Out Story Problems 291Setting Up to Solve Story Problems 292Working around Perimeter, Area, and Volume 294Parading out perimeter and arranging area 294Adjusting the area 295Pumping up the volume 297Making Up Mixtures 300Mixing up solutions 301Tossing in some solid mixtures 302Investigating investments and interest 302Going for the green: Money 304Going the Distance 305Figuring distance plus distance 306Figuring distance and fuel 307Going ’Round in Circles 307Chapter 19: Going Visual: Graphing 311Graphing Is Good 312Grappling with Graphs 313Making a point 314Ordering pairs, or coordinating coordinates 315Actually Graphing Points 316Graphing Formulas and Equations 317Lining up a linear equation 317Going around in circles with a circular graph 318Throwing an object into the air 319Curling Up with Parabolas 321Trying out the basic parabola 321Putting the vertex on an axis 322Sliding and multiplying 324Chapter 20: Lining Up Graphs of Lines 327Graphing a Line 327Graphing the equation of a line 329Investigating Intercepts 332Sighting the Slope 333Formulating slope 335Combining slope and intercept 337Getting to the slope-intercept form 337Graphing with slope-intercept 338Marking Parallel and Perpendicular Lines 339Intersecting Lines 341Graphing for intersections 341Substituting to find intersections 342Part 5: The Part of Tens 345Chapter 21: The Ten Best Ways to Avoid Pitfalls 347Keeping Track of the Middle Term 348Distributing: One for You and One for Me 348Breaking Up Fractions (Breaking Up Is Hard to Do) 348Renovating Radicals 349Order of Operations 349Fractional Exponents 349Multiplying Bases Together 350A Power to a Power 350Reducing for a Better Fit 351Negative Exponents 351Chapter 22: The Ten Most Famous Equations 353Albert Einstein’s Theory of Relativity 353The Pythagorean Theorem 354The Value of e 354Diameter and Circumference Related with Pi 354Isaac Newton’s Formula for the Force of Gravity 355Euler’s Identity 355Fermat’s Last Theorem 356Monthly Loan Payments 356The Absolute-Value Inequality 356The Quadratic Formula 357Index 359