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Elementary Algebra Textbook Second Edition Department of Mathematics College of the Redwoods 2012-2013 ii Copyright All parts of this prealgebra textbook are copyrighted c 2011 in the name of the Department of Mathematics, College of the Redwoods. They are not in the public domain. However, they are being made available free for use in educational institutions. This offer does not extend to any application that is made for profit. Users who have such applications in mind should contact David Arnold at [email protected] or Bruce Wagner at [email protected]. This work is licensed under the Creative Commons Attribution-Non- Commercial-NoDerivs 3.0 Unported License. To view a copy of this li- cense, visit http://creativecommons.org/licenses/by-nc-nd/3.0/ or send a letter to Creative Commons, 171 Second Street, Suite 300, San Fran- cisco, California, 94105, USA. Contents 1 The Arithmetic of Numbers 1 1.1 An Introduction to the Integers .................. 2 The Integers ........................... 3 Absolute Value .......................... 4 Integer Addition ......................... 5 Mathematical Properties of Addition .............. 6 Integer Subtraction ....................... 7 Integer Multiplication ...................... 8 Mathematical Properties of Multiplication ........... 9 Exponents ............................ 10 Graphing Calculator: Negating versus Subtracting ...... 11 Exercises ............................. 14 Answers .............................. 15 1.2 Order of Operations ........................ 16 Grouping Symbols ........................ 19 Absolute Value Bars as Grouping Symbols .......... 20 Nested Grouping Symbols .................... 20 Evaluating Algebraic Expressions ................ 21 Evaluating Fractions ....................... 22 Using the Graphing Calculator ................. 23 Exercises ............................. 27 Answers .............................. 30 1.3 The Rational Numbers ....................... 31 Reducing Fractions to Lowest Terms .............. 31 Multiplying Fractions ...................... 33 Dividing Fractions ........................ 35 Adding Fractions ......................... 36 Order of Operations ....................... 37 Fractions on the Graphing Calculator ............. 39 Exercises ............................. 43 Answers .............................. 46 iii iv CONTENTS 1.4 Decimal Notation .......................... 48 Adding and Subtracting Decimals ............... 49 Multiplication and Division of Decimals ............ 51 Order of Operations ....................... 52 Rounding Using the Graphing Calculator ........... 55 Exercises ............................. 57 Answers .............................. 59 1.5 Algebraic Expressions ....................... 60 The Distributive Property .................... 61 Speeding Things Up a Bit .................... 62 Distributing a Negative Sign .................. 63 Combining Like Terms ...................... 64 Order of Operations ....................... 66 Exercises ............................. 69 Answers .............................. 70 2 Solving Linear Equations and Inequalities 73 2.1 Solving Equations: One Step ................... 74 Equivalent Equations ...................... 74 Wrap and Unwrap, Do and Undo ................ 75 Operations that Produce Equivalent Equations ........ 76 More Operations That Produce Equivalent Equations .... 79 Writing Mathematics ...................... 82 Exercises ............................. 84 Answers .............................. 85 2.2 Solving Equations: Multiple Steps ................ 87 Variables on Both Sides of the Equation ............ 90 Simplifying Expressions When Solving Equations ....... 92 Exercises ............................. 96 Answers .............................. 97 2.3 Clearing Fractions and Decimals ................. 99 Canceling is More Efficient ................... 100 Clearing Fractions from an Equation .............. 101 Clearing Decimals from an Equation .............. 106 Exercises ............................. 109 Answers .............................. 110 2.4 Formulae .............................. 111 Clearing Fractions ........................ 113 Geometric Formulae ....................... 115 Exercises ............................. 118 Answers .............................. 120 2.5 Applications ............................. 121 Exercises ............................. 132 Answers .............................. 134 2.6 Inequalities ............................. 135 Ordering the Real Numbers ................... 136 CONTENTS v Set-Builder Notation ....................... 136 Interval Notation ......................... 138 Equivalent Inequalities ..................... 139 Reversing the Inequality Sign .................. 141 Multiple Steps .......................... 143 Summary Table of Set-Builder and Interval Notation ..... 146 Exercises ............................. 147 Answers .............................. 149 3 Introduction to Graphing 151 3.1 Graphing Equations by Hand ................... 152 The Cartesian Coordinate System ............... 152 Plotting Ordered Pairs ..................... 154 Equations in Two Variables ................... 156 Graphing Equations in Two Variables ............. 157 Guidelines and Requirements .................. 159 Using the TABLE Feature of the Graphing Calculator .... 160 Exercises ............................. 165 Answers .............................. 168 3.2 The Graphing Calculator ..................... 171 Reproducing Calculator Results on Homework Paper .... 174 Adjusting the Viewing Window ................. 176 Exercises ............................. 179 Answers .............................. 180 3.3 Rates and Slope .......................... 183 Measuring the Change in a Variable .............. 184 Slope as Rate ........................... 185 The Steepness of a Line ..................... 189 The Geometry of the Slope of a Line .............. 192 Exercises ............................. 196 Answers .............................. 199 3.4 Slope-Intercept Form of a Line .................. 201 Applications ........................... 206 Exercises ............................. 209 Answers .............................. 212 3.5 Point-Slope Form of a Line .................... 213 Parallel Lines ........................... 217 Perpendicular Lines ....................... 219 Applications ........................... 222 Exercises ............................. 224 Answers .............................. 225 3.6 Standard Form of a Line ...................... 228 Slope-Intercept to Standard Form ............... 230 Point-Slope to Standard Form ................. 232 Intercepts ............................. 234 Horizontal and Vertical Lines .................. 237 vi CONTENTS Exercises ............................. 240 Answers .............................. 242 4 Systems of Linear Equations 245 4.1 Solving Systems by Graphing ................... 246 Exceptional Cases ........................ 250 Solving Systems with the Graphing Calculator ........ 253 Exercises ............................. 259 Answers .............................. 260 4.2 Solving Systems by Substitution ................. 262 Exceptional Cases Revisited .................. 268 Exercises ............................. 270 Answers .............................. 272 4.3 Solving Systems by Elimination .................. 273 Exceptional Cases ........................ 276 Exercises ............................. 279 Answers .............................. 280 4.4 Applications of Linear Systems .................. 282 Exercises ............................. 290 Answers .............................. 291 5 Polynomial Functions 293 5.1 Functions .............................. 294 Mapping Diagrams ........................ 296 Function Definition ....................... 297 Mapping Diagram Notation ................... 299 Function Notation ........................ 301 Interchanging y and f(x) .................... 303 Exercises ............................. 305 Answers .............................. 307 5.2 Polynomials ............................. 308 Ascending and Descending Powers ............... 309 The Degree of a Polynomial ................... 311 Polynomial Functions ...................... 312 The Graph of a Polynomial Function ............. 313 Exercises ............................. 317 Answers .............................. 319 5.3 Applications of Polynomials .................... 321 Zeros and x-intercepts of a Function .............. 325 Exercises ............................. 330 Answers .............................. 333 5.4 Adding and Subtracting Polynomials ............... 334 Negating a Polynomial ..................... 335 Subtracting Polynomials ..................... 336 Some Applications ........................ 338 Exercises ............................. 340 CONTENTS vii Answers .............................. 342 5.5 Laws of Exponents ......................... 343 Multiplying With Like Bases .................. 344 Dividing With Like Bases .................... 345 Raising a Power to a Power ................... 347 Raising a Product to a Power .................. 349 Raising a Quotient to a Power ................. 350 Exercises ............................. 353 Answers .............................. 355 5.6 Multiplying Polynomials ...................... 356 The Product of Monomials ................... 356 Multiplying a Monomial and a Polynomial .......... 357 Multiplying Polynomials ..................... 359 Speeding Things Up a Bit ...................
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