Beginning and Intermediate Algebra

[Pages:489]Beginning and Intermediate Algebra

An open source (CC-BY) textbook Available for free download at:

by Tyler Wallace

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ISBN #978-1-4583-7768-5 Copyright 2010, Some Rights Reserved CC-BY.

Beginning and Intermediate Algebra by Tyler Wallace is licensed under a Creative Commons Attribution 3.0 Unported License. () Based on a work at .

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Special thanks to: My beautiful wife, Nicole Wallace who spent countless hours typing problems and my two wonderful kids for their patience and support during this project

Another thanks goes to the faculty reviewers who reviewed this text: Donna Brown, Michelle Sherwood, Ron Wallace, and Barbara Whitney One last thanks to the student reviewers of the text: Eloisa Butler, Norma Cabanas, Irene Chavez, Anna Dahlke, Kelly Diguilio, Camden Eckhart, Brad Evers, Lisa Garza, Nickie Hampshire, Melissa Hanson, Adriana Hernandez, Tiffany Isaacson, Maria Martinez, Brandon Platt, Tim Ries, Lorissa Smith, Nadine Svopa, Cayleen Trautman, and Erin White

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Table of Contents

Chapter 0: Pre-Algebra 0.1 Integers.........................................7 0.2 Fractions.....................................12 0.3 Order of Operations....................18 0.4 Properties of Algebra..................22 Chapter 1: Solving Linear Equations 1.1 One-Step Equations....................28 1.2 Two-Step Equations....................33 1.3 General Linear Equations...........37 1.4 Solving with Fractions................43 1.5 Formulas.....................................47 1.6 Absolute Value Equations...........52 1.7 Variation.....................................57 1.8 Application: Number/Geometry.64 1.9 Application: Age.........................72 1.10 Application: Distance................79 Chapter 2: Graphing 2.1 Points and Lines.........................89 2.2 Slope...........................................95 2.3 Slope-Intercept Form................102 2.4 Point-Slope Form......................107 2.5 Parallel & Perpendicular Lines.112

Chapter 3: Inequalities 3.1 Solve and Graph Inequalities....118 3.2 Compound Inequalitites............124 3.3 Absolute Value Inequalities.......128 Chapter 4: Systems of Equations 4.1 Graphing...................................134 4.2 Substitution..............................139 4.3 Addition/Elimination................146 4.4 Three Variables.........................151 4.5 Application: Value Problems.....158 4.6 Application: Mixture Problems.167 Chapter 5: Polynomials 5.1 Exponent Properties.................177 5.2 Negative Exponents..................183 5.3 Scientific Notation.....................188 5.4 Introduction to Polynomials.....192 5.5 Multiply Polynomials................196 5.6 Multiply Special Products.........201 5.7 Divide Polynomials...................205

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Chapter 6: Factoring 6.1 Greatest Common Factor..........212 6.2 Grouping...................................216 6.3 Trinomials where a = 1..............221 6.4 Trinomials where a 1..............226 6.5 Factoring Special Products.......229 6.6 Factoring Strategy....................234 6.7 Solve by Factoring.....................237 Chapter 7: Rational Expressions 7.1 Reduce Rational Expressions....243 7.2 Multiply and Divide..................248 7.3 Least Common Denominator....253 7.4 Add and Subtract.....................257 7.5 Complex Fractions....................262 7.6 Proportions...............................268 7.7 Solving Rational Equations.......274 7.8 Application: Dimensional Analysis....279 Chapter 8: Radicals 8.1 Square Roots.............................288 8.2 Higher Roots.............................292 8.3 Adding Radicals........................295 8.4 Multiply and Divide Radicals...298 8.5 Rationalize Denominators.........303 8.6 Rational Exponents...................310 8.7 Radicals of Mixed Index...........314 8.8 Complex Numbers.....................318

Chapter 9: Quadratics 9.1 Solving with Radicals................326 9.2 Solving with Exponents............332 9.3 Complete the Square.................337 9.4 Quadratic Formula....................343 9.5 Build Quadratics From Roots...348 9.6 Quadratic in Form....................352 9.7 Application: Rectangles............357 9.8 Application: Teamwork.............364 9.9 Simultaneous Products.............370 9.10 Application: Revenue and Distance.373 9.11 Graphs of Quadratics..............380 Chapter 10: Functions 10.1 Function Notation...................386 10.2 Operations on Functions.........393 10.3 Inverse Functions....................401 10.4 Exponential Functions.............406 10.5 Logarithmic Functions............410 10.6 Application: Compound Interest.414 10.7 Trigonometric Functions.........420 10.8 Inverse Trigonometric Functions.428 Answers........................................438

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Chapter 0 : Pre-Algebra

0.1 Integers ............................................................................................................7 0.2 Fractions ........................................................................................................12 0.3 Order of Operations .......................................................................................18 0.4 Properties of Algebra .....................................................................................22

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0.1

Pre-Algebra - Integers

Objective: Add, Subtract, Multiply and Divide Positive and Negative Numbers. The ability to work comfortably with negative numbers is essential to success in algebra. For this reason we will do a quick review of adding, subtracting, multiplying and dividing of integers. Integers are all the positive whole numbers, zero, and their opposites (negatives). As this is intended to be a review of integers, the descriptions and examples will not be as detailed as a normal lesson. World View Note: The first set of rules for working with negative numbers was written out by the Indian mathematician Brahmagupa. When adding integers we have two cases to consider. The first is if the signs match, both positive or both negative. If the signs match we will add the numbers together and keep the sign. This is illustrated in the following examples

Example 1.

- 5 + ( - 3) Same sign, add 5 + 3, keep the negative - 8 Our Solution

Example 2.

- 7 + ( - 5) Same sign, add 7 + 5, keep the negative - 12 Our Solution

If the signs don't match, one positive and one negative number, we will subtract the numbers (as if they were all positive) and then use the sign from the larger number. This means if the larger number is positive, the answer is positive. If the larger number is negative, the answer is negative. This is shown in the following examples.

Example 3.

- 7 + 2 Different signs, subtract 7 - 2, use sign from bigger number, negative - 5 Our Solution

Example 4.

- 4 + 6 Different signs, subtract 6 - 4, use sign from bigger number, positive 2 Our Solution

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Example 5.

4 + ( - 3) Different signs, subtract 4 - 3, use sign from bigger number, positive 1 Our Solution

Example 6.

7 + ( - 10) Different signs, subtract 10 - 7, use sign from bigger number, negative - 3 Our Solution

For subtraction of negatives we will change the problem to an addition problem which we can then solve using the above methods. The way we change a subtraction to an addition is to add the opposite of the number after the subtraction sign. Often this method is refered to as "add the opposite." This is illustrated in the following examples.

Example 7.

8-3 8 + ( - 3)

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Add the opposite of 3 Different signs, subtract 8 - 3, use sign from bigger number, positive Our Solution

Example 8.

-4-6 - 4 + ( - 6)

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Add the opposite of 6 Same sign, add 4 + 6, keep the negative Our Solution

Example 9.

9 - ( - 4) 9+4 13

Add the opposite of - 4 Same sign, add 9 + 4, keep the positive Our Solution

Example 10.

- 6 - ( - 2) -6+2 -4

Add the opposite of - 2 Different sign, subtract 6 - 2, use sign from bigger number, negative Our Solution

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