Beginning and Intermediate Algebr a An open source (CC-BY) textbook Available for free download at: http://wallace.ccfaculty.org/book/book.html by Tyler Wallace 1 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. (http://creativecommons.org/licenses/by/3.0/) Based on a work at http://wallace.ccfaculty.org/b ook/book.html. You are free: • to Share: to copy, distribute and transmit the work • to Remix: to adapt the work Under the following conditions: • Attribution: You must attribute the work in the manner specified by the author or licensor (but not in any way that suggests that they endorse you or your use of the work). 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The best way to do this is with a link to the following web page: http://creativecommons.org/licenses/by/3.0/ This is a human readable summary of the full legal co de which can be read at the following URL: http://creativecommons.org/licenses/by/3.0/legalcode 2 Special thanks to: My beautiful wife, Nicole Wallace who spe nt 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 Hamp- shire, Melissa Hanson, Adriana Hernandez, Tiffany Isaacson, Maria Martinez, Brandon Platt, Tim Ries, Lorissa Smith, Nadine Svopa, Cayleen Trautman, and Erin White 3 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: G raphing 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.11 2 Chapter 3: In equalities 3.1 Solve and G raph Inequalities 118 3.2 Compound Inequalitites 124 3.3 Absolute Value Inequalities 128 Chapter 4: Systems of Equations 4.1 Graphing 134 4.2 Su bstitution 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 17 7 5.2 Negative Exponents 183 5.3 Scientific Notation 188 5.4 Introduction to Polynomials 192 5.5 M ultiply Polynomials 196 5.6 M ultiply Special Products 201 5.7 Divide Polynomials 205 4 Chapter 6: Fact oring 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: Ra tional Expressions 7.1 Reduce Rational E xpressions 243 7.2 M ultiply 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 Ap plication: Dimensional Analysis 279 Chapter 8: Ra dicals 8.1 Square Roots 288 8.2 Higher Roots 292 8.3 Adding Radical s 295 8.4 M ultiply and Divide Radicals 298 8.5 Rat ionalize Denominators 303 8.6 Rat ional 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 R oots 348 9.6 Quadratic in Form 35 2 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 Fu nctions 406 10.5 Lo garithmic Functions 410 10.6 Application: Compound Interest.414 10.7 Trigonometric Functions 420 10.8 Inverse Trigonometric Functions.428 Answers 438 5 Chapte r 0 : Pre-Algebra 0.1 Integers 7 0.2 Fractions 12 0.3 Order of Operations 18 0.4 Properties of Algebra 2 2 6 0.1 Pre-Algebra - Integers Objective: Add, Subtract, Multiply and Divide Positive and Negative Numbers. The ab ility to work comfortably with negative numbers is essential to success in algebra. For t his reason we will do a quick review of adding, subtracting, multi- plying 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 fo r working with nega tive numbers was written out by the Indian mathema tician Brahmagupa. When adding integers we have two case s to consider. The first is if the signs match, both positive or both negative. If the signs match we will add t he num- bers together and keep the sign. This is illustra ted 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 subtra ct 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 i s 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 7 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 1 0 −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 subtrac- tion to an a ddition is to add the opposite of the number after the subtraction sign. Often this metho d is refered to as “add the opposite.” This is illustrated in the following examples. Example 7. 8 −3 Add the opposite of 3 8 + ( −3) Different signs, subtract 8 −3, use sign from bigger number, positive 5 Our Solution Example 8. −4 −6 Add the opposite of 6 −4 + ( −6) Same sign, add 4 + 6, keep the negative −10 Our Solution Example 9. 9 −( −4) Add the opposite of −4 9 + 4 Same sign, add 9 + 4, keep the positive 13 Our Solution Example 10. −6 −( −2) Add the opposite of −2 −6 + 2 Different sign, subtract 6 −2, use sign from bigger number, negative −4 Our Solution 8 Multiplication and division of integers both work in a ve r y similar pattern. The short description of the process is we multiply and divide like normal, if the signs match (both positive or both negative) the answer is positive. If the signs don’t match (one positive and one negative) then the answer is negative. This is shown in the following examples Example 11. (4)( −6) Signs do not match, answer is negat ive −24 Our Soluti on Example 12. −36 −9 Signs match, answer is positive 4 Our Solution Example 13. −2( −6) Signs match, answer is positive 12 Our So lution Example 14. 15 −3 Signs do not match, answer is negative −5 Our Solution A few things to be careful of when working with integers. First be sure not to confuse a problem like − 3 − 8 with − 3( − 8). The second problem is a multipli- cation problem because there is nothin g between the 3 and the parenthesis. If there is no operation written in between the parts, then we assume that means we are multiplying. The −3 −8 problem, is subtraction because the subtraction sep - arates the 3 from what comes after it. Another item to watch out for is to be careful not to mix up the pattern for adding and subtracting integers with the pattern for multiplying and dividing integers. They can look very similar, for example if the signs ma t ch on addition, the we keep t he negative, −3 + ( −7) = − 10, but if the signs match on multiplicati on, the answer is positive, ( − 3)( − 7) = 21. 9 0.1 Practice - Integers Evaluate each expression. 1) 1 −3 3) ( −6) −( −8) 5) ( −3) −3 7) 3 −( −5) 9) ( −7) −( −5) 11) 3 −( −1) 13) 6 −3 15) ( −5) + 3 17) 2 −3 19) ( −8) −( −5) 21) ( −2) + ( −5) 23) 5 −( −6) 25) ( −6) + 3 27) 4 −7 29) ( −7) + 7 2) 4 −( −1) 4) ( −6) + 8 6) ( −8) −( −3) 8) 7 −7 10) ( −4) + ( −1) 12) ( −1) + ( −6) 14) ( −8) + ( −1) 16) ( −1) −8 18) 5 −7 20) ( −5) + 7 22) 1 + ( −1) 24) 8 −( −1) 26) ( −3) + ( −1) 28) 7 −3 30) ( −3) + ( −5) Find each product. 31) (4)( −1) 33) (10)( −8) 35) ( −4)( −2) 37) ( −7)(8) 39) (9)( −4) 41) ( −5)(2) 43) ( −5)(4) 32) (7)( −5) 34) ( −7)( −2) 36) ( −6)( −1) 38) (6)( −1) 40) ( −9)( −7) 42) ( −2)( −2) 44) ( −3)( −9) 10 [...]... 2 [5 + 3(22 − 5)] + 22 − 5|2 21 0.4 Pre -Algebra - Properties of Algebra Objective: Simplify algebraic expressions by substituting given values, distributing, and combining like terms In algebra we will often need to simplify an expression to make it easier to use There are three basic forms of simplifying which we will review here World View Note: The term Algebra comes from the Arabic word al-jabr... Reduce 21 and 28 by dividing by 7 Reduce 6 and 16 by dividing by 2 Multiply numerators across and denominators across Our Soultion To add and subtract fractions we will first have to find the least common denominator (LCD) There are several ways to find an LCD One way is to find the smallest multiple of the largest denominator that you can also divide the small denomiator by Example 19 Find the LCD of 8 and. .. numerator and denominator by the same number This is shown in the following example Example 15 36 84 36 ÷ 4 9 = 84 ÷ 4 21 Both numerator and denominator are divisible by 4 Both numerator and denominator are still divisible by 3 12 3 9÷3 = 21 ÷ 3 7 Our Soultion The previous example could have been done in one step by dividing both numerator and denominator by 12 We also could have divided by 2 twice and then... Variation 57 1.8 Application: Number and Geometry 64 1.9 Application: Age 72 1.10 Application: Distance, Rate and Time 79 27 1.1 Solving Linear Equations - One Step Equations Objective: Solve one step linear equations by balancing using inverse operations Solving linear equations is an important and fundamental skill in algebra In algebra, we are often presented with a problem... multiplication or exponents When we want to do something out of order and make it come first we will put it in parenthesis (or grouping symbols) This list then is our order of operations we will use to simplify expressions Order of Operations: Parenthesis (Grouping) Exponents Multiply and Divide (Left to Right) Add and Subtract (Left to Right) Multiply and Divide are on the same level because they are the same... denominator Example 17 25 32 · 24 55 5 4 · 3 11 20 33 Reduce 25 and 55 by dividing by 5 Reduce 32 and 24 by dividing by 8 Multiply numerators across and denominators across Our Solution Dividing fractions is very similar to multiplying with one extra step Dividing fractions requires us to first take the reciprocal of the second fraction and multiply Once we do this, the multiplication problem solves... replace each variable with the equivalent number and simplify what remains using order of operations Example 30 p(q + 6) when p = 3 and q = 5 (3)((5) + 6) (3)(11) 33 Replace p with 3 and q with 5 Evaluate parenthesis Multiply Our Solution Whenever a variable is replaced with something, we will put the new number inside a set of parenthesis Notice the 3 and 5 in the previous example are in parenthesis... value of the variables In this case, we will have to simplify what we can and leave the variables in our final solution One way we can simplify expressions is to combine like terms Like terms are terms where the variables match exactly (exponents included) Examples of like terms would be 3xy and − 7xy or 3a2b and 8a2b or − 3 and 5 If we have like terms we are allowed to add (or subtract) the numbers... Distribute 4 into first parenthesis, − 1 into second Combine like terms 12x − 2x and − 32 + 7 Our Solution 24 0.4 Practice - Properties of Algebra Evaluate each using the values given 2) y 2 + y − z; use y = 5, z = 1 1) p + 1 + q − m; use m = 1, p = 3, q = 4 3) p − pq ; use 6 p = 6 and q = 5 4) 5) c2 − (a − 1); use a = 3 and c = 5 7) 5j + 9) kh ; use h = 5, 2 4 − (p − m) 2 6+z −y ; use 3 y = 1, z = 4... − 35 −5 54) 80 −8 55) −8 −2 56) 50 5 57) − 16 2 58) 48 8 59) 60) 54 −6 60 − 10 11 0.2 Pre -Algebra - Fractions Objective: Reduce, add, subtract, multiply, and divide with fractions Working with fractions is a very important foundation to algebra Here we will briefly review reducing, multiplying, dividing, adding, and subtracting fractions As this is a review, concepts will not be explained in detail as . : Pre -Algebra 0.1 Integers 7 0.2 Fractions 12 0.3 Order of Operations 18 0.4 Properties of Algebra 2 2 6 0.1 Pre -Algebra - Integers Objective: Add, Subtract, Multiply and Divide Positive and Negative Numbers. The. Hanson, Adriana Hernandez, Tiffany Isaacson, Maria Martinez, Brandon Platt, Tim Ries, Lorissa Smith, Nadine Svopa, Cayleen Trautman, and Erin White 3 Table of Contents Chapter 0: Pre -Algebra 0.1 Integers. Tyler Wallace 1 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