A. B. A. B. ( ( ) ) Beginning and Intermediate Algebra 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/book/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). With the understanding that: Waiver: Any of the above conditions can be waived if you get permission from the copyright holder. 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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 code 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 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 3 Table of Contents Chapter 0: Pre-Algebra Chapter 3: Inequalities 0.1 Integers.........................................7 3.1 Solve and Graph Inequalities....118 0.2 Fractions.....................................12 3.2 Compound Inequalitites............124 0.3 Order of Operations....................18 3.3 Absolute Value Inequalities.......128 0.4 Properties of Algebra..................22 Chapter 4: Systems of Equations Chapter 1: Solving Linear Equations 4.1 Graphing...................................134 1.1 One-Step Equations....................28 4.2 Substitution..............................139 1.2 Two-Step Equations....................33 4.3 Addition/Elimination................146 1.3 General Linear Equations...........37 4.4 Three Variables.........................151 1.4 Solving with Fractions................43 4.5 Application: Value Problems.....158 1.5 Formulas.....................................47 4.6 Application: Mixture Problems.167 1.6 Absolute Value Equations...........52 Chapter 5: Polynomials 1.7 Variation.....................................57 5.1 Exponent Properties.................177 1.8 Application: Number/Geometry.64 5.2 Negative Exponents..................183 1.9 Application: Age.........................72 5.3 Scientific Notation.....................188 1.10 Application: Distance................79 5.4 Introduction to Polynomials.....192 Chapter 2: Graphing 5.5 Multiply Polynomials................196 2.1 Points and Lines.........................89 5.6 Multiply Special Products.........201 2.2 Slope...........................................95 5.7 Divide Polynomials...................205 2.3 Slope-Intercept Form................102 2.4 Point-Slope Form......................107 2.5 Parallel & Perpendicular Lines.112 4 Chapter 6: Factoring Chapter 9: Quadratics 6.1 Greatest Common Factor..........212 9.1 Solving with Radicals................326 6.2 Grouping...................................216 9.2 Solving with Exponents............332 6.3 Trinomials where a = 1..............221 9.3 Complete the Square.................337 6.4 Trinomials where a \u0002 1..............226 9.4 Quadratic Formula....................343 6.5 Factoring Special Products.......229 9.5 Build Quadratics From Roots...348 6.6 Factoring Strategy....................234 9.6 Quadratic in Form....................352 6.7 Solve by Factoring.....................237 9.7 Application: Rectangles............357 Chapter 7: Rational Expressions 9.8 Application: Teamwork.............364 7.1 Reduce Rational Expressions....243 9.9 Simultaneous Products.............370 7.2 Multiply and Divide..................248 9.10 Application: Revenue and Distance.373 7.3 Least Common Denominator....253 9.11 Graphs of Quadratics..............380 7.4 Add and Subtract.....................257 Chapter 10: Functions 7.5 Complex Fractions....................262 10.1 Function Notation...................386 7.6 Proportions...............................268 10.2 Operations on Functions.........393 7.7 Solving Rational Equations.......274 10.3 Inverse Functions....................401 7.8 Application: Dimensional Analysis....279 10.4 Exponential Functions.............406 Chapter 8: Radicals 10.5 Logarithmic Functions............410 8.1 Square Roots.............................288 10.6 Application: Compound Interest.414 8.2 Higher Roots.............................292 10.7 Trigonometric Functions.........420 8.3 Adding Radicals........................295 10.8 Inverse Trigonometric Functions.428 8.4 Multiply and Divide Radicals...298 Answers........................................438 8.5 Rationalize Denominators.........303 8.6 Rational Exponents...................310 8.7 Radicals of Mixed Index...........314 8.8 Complex Numbers.....................318 5 Chapter 0 : Pre-Algebra 0.1 Integers ............................................................................................................7 0.2 Fractions ........................................................................................................12 0.3 Order of Operations .......................................................................................18 0.4 Properties of Algebra .....................................................................................22 6 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) 8 Same sign, add 5 + 3, keep the negative Our Solution 7 + ( 5) 12 Same sign, add 7 + 5, keep the negative Our Solution Example 2. 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 5 Different signs, subtract 7 2, use sign from bigger number, negative Our Solution Example 4. 4+6 2 Different signs, subtract 6 4, use sign from bigger number, positive Our Solution 7 Example 5. 4 + ( 3) 1 Different signs, subtract 4 3, use sign from bigger number, positive Our Solution Example 6. 7 + ( 10) 3 Different signs, subtract 10 7, use sign from bigger number, negative 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. 83 8 + ( 3) 5 Add the opposite of 3 Different signs, subtract 8 3, use sign from bigger number, positive Our Solution Example 8. 46 4 + ( 6) 10 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 8 Multiplication and division of integers both work in a very 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. Signs do not match, answer is negative Our Solution (4)( 6) 24 Example 12. 36 9 4 Signs match, answer is positive Our Solution Example 13. 2( 6) 12 Signs match, answer is positive Our Solution 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 multiplication problem because there is nothing 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 separates 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 match on addition, the we keep the negative, 3 + ( 7) = 10, but if the signs match on multiplication, the answer is positive, ( 3)( 7) = 21. 9 0.1 Practice - Integers Evaluate each expression. 1) 1 3 2) 4 ( 1) 3) ( 6) ( 8) 4) ( 6) + 8 5) ( 3) 3 6) ( 8) ( 3) 7) 3 ( 5) 8) 7 7 9) ( 7) ( 5) 10) ( 4) + ( 1) 11) 3 ( 1) 12) ( 1) + ( 6) 13) 6 3 14) ( 8) + ( 1) 15) ( 5) + 3 16) ( 1) 8 17) 2 3 18) 5 7 19) ( 8) ( 5) 20) ( 5) + 7 21) ( 2) + ( 5) 22) 1 + ( 1) 23) 5 ( 6) 24) 8 ( 1) 25) ( 6) + 3 26) ( 3) + ( 1) 27) 4 7 28) 7 3 29) ( 7) + 7 30) ( 3) + ( 5) Find each product. 31) (4)( 1) 32) (7)( 5) 33) (10)( 8) 34) ( 7)( 2) 35) ( 4)( 2) 36) ( 6)( 1) 37) ( 7)(8) 38) (6)( 1) 39) (9)( 4) 40) ( 9)( 7) 41) ( 5)(2) 42) ( 2)( 2) 43) ( 5)(4) 44) ( 3)( 9) 10 45) (4)( 6) Find each quotient. 46) 30 10 47) 49 7 48) 12 4 49) 2 1 50) 30 6 51) 20 10 52) 27 3 53) 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 other lessons are. World View Note: The earliest known use of fraction comes from the Middle Kingdom of Egypt around 2000 BC! We always like our final answers when working with fractions to be reduced. Reducing fractions is simply done by dividing both the 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 93 = 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 divided by 3 once (in any order). It is not important which method we use as long as we continue reducing our fraction until it cannot be reduced any further. The easiest operation with fractions is multiplication. We can multiply fractions by multiplying straight across, multiplying numerators together and denominators together. Example 16. 6 3 7 5 18 35 Multiply numerators across and denominators across Our Solution When multiplying we can reduce our fractions before we multiply. We can either reduce vertically with a single fraction, or diagonally with several fractions, as long as we use one number from the numerator and one number from the 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 just as the previous problem. 13 Example 18. 21 28 6 16 21 6 16 28 3 3 8 4 9 32 Multiply by the reciprocal 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 12 12 12? 8 24 24? =3 8 24 Test multiples of 12 Can t divide 12 by 8 Yes! We can divide 24 by 8! Our Soultion Adding and subtracting fractions is identical in process. If both fractions already have a common denominator we just add or subtract the numerators and keep the denominator. Example 20. 7 3 + 8 8 10 8 5 4 Same denominator, add numerators 7 + 3 Reduce answer, dividing by 2 Our Solution 1 5 While 4 can be written as the mixed number 1 4 , in algebra we will almost never use mixed numbers. For this reason we will always use the improper fraction, not the mixed number. 14 Example 21. 13 9 6 6 Same denominator, subtract numerators 13 9 4 6 Reduce answer, dividing by 2 2 3 Our Solution If the denominators do not match we will first have to identify the LCD and build up each fraction by multiplying the numerators and denominators by the same number so the denominator is built up to the LCD. Example 22. 5 4 + 6 9 35 42 + 36 92 15 8 + 18 18 23 18 LCD is 18. Multiply first fraction by 3 and the second by 2 Same denominator, add numerators, 15 + 8 Our Solution Example 23. 2 1 3 6 22 1 23 6 4 1 6 6 LCD is 6 Multiply first fraction by 2, the second already has a denominator of 6 Same denominator, subtract numerators, 4 1 3 6 Reduce answer, dividing by 3 1 2 Our Solution 15 0.2 Practice - Fractions Simplify each. Leave your answer as an improper fraction. 1) 42 12 2) 25 20 3) 35 25 4) 24 9 5) 54 36 6) 30 24 7) 45 36 8) 36 27 9) 27 18 10) 48 18 11) 40 16 12) 48 42 13) 63 18 14) 16 12 15) 80 60 16) 72 48 17) 72 60 18) 126 108 19) 36 24 20) 160 140 Find each product. 5 8 22) ( 2)( 6 ) 21) (9)( 9 ) 2 24) ( 2)( 3 ) 1 13 26) ( 2 )( 2 ) 23) (2)( 9 ) 25) ( 2)( 8 ) 6 27) ( 5 )( 3 28) ( 7 )( 11 ) 8 30) ( 2)( 7 ) 9 3 32) ( 31) ( 3 )( 4 ) 3 3 17 )( 5 ) 9 3 17 33 (2)( 2 ) 1 1 1 29) (8)( 2 ) 2 11 ) 8 3 34) ( 9 )( 5 ) 7 35) ( 2 )( 5 ) 1 5 36) ( 2 )( 7 ) 16 Find each quotient. 7 37) 2 4 38) 12 7 9 5 3 2 39) 1 9 1 2 40) 2 41) 3 2 13 7 42) 5 3 2 44) 10 9 6 46) 1 6 5 3 1 48) 13 8 3 2 50) 4 5 52) 5 3 3 54) 1 7 +( 56) 1 3 +3 43) 1 3 45) 8 9 1 47) 9 7 5 49) 2 9 51) 1 10 5 3 2 7 5 15 8 13 8 5 Evaluate each expression. 53) 1 3 + ( 3) 55) 3 7 7 57) 11 6 +6 59) 3 5 +4 5 60) ( 1) 3 61) 2 5 +4 5 62) 63) 9 8 + ( 7) 4 1 7 5 1 66) 1 2 68) 11 8 2 70) 6 5 8 3 3 +4 5 3 2 77) ( 11 6 1 5 8 72) ( 3 ) + ( 5 ) 5 8 7 74) ( 6) + ( 3 ) 15 8 76) ( 1) ( 3 ) 15 5 )+ 3 8 78) 1 1 5 3 1 15 8 79) ( 1) ( 6 ) 81) 9 7 64) ( 2) + 6 71) ( 7 ) 75) 12 7 2 1 73) 6 15 ) 8 2 67) ( 2 ) + 2 1 5 5 58) ( 2) + ( 65) 1 + ( 3 ) 69) 11 ) 7 1 ( 3) 3 2 9 +7 1 3 80) ( 2 ) ( 5 ) 82) 17 9 7 5 ( 3) 0.3 Pre-Algebra - Order of Operations Objective: Evaluate expressions using the order of operations, including the use of absolute value. When simplifying expressions it is important that we simplify them in the correct order. Consider the following problem done two different ways: Example 24. 2 + 5 3 Add First 73 Multiply 21 Solution 2 + 5 3 Multiply 2 + 15 Add 17 Solution The previous example illustrates that if the same problem is done two different ways we will arrive at two different solutions. However, only one method can be correct. It turns out the second method, 17, is the correct method. The order of operations ends with the most basic of operations, addition (or subtraction). Before addition is completed we must do repeated addition or multiplication (or division). Before multiplication is completed we must do repeated 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 operation (division is just multiplying by the reciprocal). This means they must be done left to right, so some problems we will divide first, others we will multiply first. The same is true for adding and subtracting (subtracting is just adding the opposite). Often students use the word PEMDAS to remember the order of operations, as the first letter of each operation creates the word PEMDAS. However, it is the P E author's suggestion to think about PEMDAS as a vertical word written as: MD AS so we don't forget that multiplication and division are done left to right (same with addition and subtraction). Another way students remember the order of operations is to think of a phrase such as \"Please Excuse My Dear Aunt Sally\" where each word starts with the same letters as the order of operations start with. World View Note: The first use of grouping symbols are found in 1646 in the Dutch mathematician, Franciscus van Schooten's text, Vieta. He used a bar over 18 the expression that is to be evaluated first. So problems like 2(3 + 5) were written as 2 3 + 5. Example 25. 2 + 3(9 4)2 2 + 3(5)2 2 + 3(25) 2 + 75 77 Parenthesis first Exponents Multiply Add Our Solution It is very important to remember to multiply and divide from from left to right! Example 26. 30 3 2 10 2 20 Divide first (left to right!) Multiply Our Solution In the previous example, if we had multiplied first, five would have been the answer which is incorrect. If there are several parenthesis in a problem we will start with the inner most parenthesis and work our way out. Inside each parenthesis we simplify using the order of operations as well. To make it easier to know which parenthesis goes with which parenthesis, different types of parenthesis will be used such as { } and [ ] and ( ), these parenthesis all mean the same thing, they are parenthesis and must be evaluated first. Example 27. 2{82 7[32 4(32 + 1)]( 1)} 2{82 7[32 4(9 + 1)]( 1)} 2{82 7[32 4(10)]( 1)} 2{82 7[32 40]( 1)} 2{82 7[ 8]( 1)} 2{64 7[ 8]( 1)} 2{64 + 56( 1)} 2{64 56} 2{8} 16 Inner most parenthesis, exponents first Add inside those parenthesis Multiply inside inner most parenthesis Subtract inside those parenthesis Exponents next Multiply left to right, sign with the number Finish multiplying Subtract inside parenthesis Multiply Our Solution As the above example illustrates, it can take several steps to complete a problem. The key to successfully solve order of operations problems is to take the time to show your work and do one step at a time. This will reduce the chance of making a mistake along the way. 19 There are several types of grouping symbols that can be used besides parenthesis. One type is a fraction bar. If we have a fraction, the entire numerator and the entire denominator must be evaluated before we reduce the fraction. In these cases we can simplify in both the numerator and denominator at the same time. Example 28. 24 ( 8) 3 15 5 1 Exponent in the numerator, divide in denominator 16 ( 8) 3 31 Multiply in the numerator, subtract in denominator 16 ( 24) 2 Add the opposite to simplify numerator, denominator is done. 40 2 Reduce, divide 20 Our Solution Another type of grouping symbol that also has an operation with it, absolute value. When we have absolute value we will evaluate everything inside the absolute value, just as if it were a normal parenthesis. Then once the inside is completed we will take the absolute value, or distance from zero, to make the number positive. Example 29. 1 + 3| 42 ( 8)| + 2|3 + ( 5)2| 1 + 3| 16 ( 8)| + 2|3 + 25| 1 + 3| 8| + 2|28| 1 + 3(8) + 2(28) 1 + 24 + 2(28) 1 + 24 + 56 25 + 56 81 Evaluate absolute values first, exponents Add inside absolute values Evaluate absolute values Multiply left to right Finish multiplying Add left to right Add Our Solution The above example also illustrates an important point about exponents. Exponents only are considered to be on the number they are attached to. This means when we see 42, only the 4 is squared, giving us (42) or 16. But when the negative is in parentheses, such as ( 5)2 the negative is part of the number and is also squared giving us a positive solution, 25. 20 0.3 Practice - Order of Operation Solve. 1) 6 4( 1) 2) ( 6 6)3 3) 3 + (8) |4| 4) 5( 5 + 6) 62 5) 8 4 2 6) 7 5 + 6 7) [ 9 (2 5)] ( 6) 9) 6 + ( 3 3)2 3 11) 4 2 32 16 8) ( 2 23 2) ( 4) 13) [ 1 ( 5)]|3 + 2| 15) 2 + 4 7 + 22 42+53 18 17) [6 2 + 2 ( 6)]( 5 + 6 ) 19) 25) 12) 10 6 ( 2)2 5 14) 3 {3 [ 3(2 + 4) ( 2)]} 16) 4 [2 + 4( 6) 4 22 5 2 ] 18) 2 ( 3) + 3 6[ 2 ( 1 3)] 13 2 2 ( 1)3 + ( 6) [ 1 ( 3)] 20) 8 4 + ( 4) [ 4 ( 3)] (42 + 32) 5 52 + ( 5)2 |42 25| 2 3 22) 23 + 4 18 6 + ( 4) [ 5( 1)( 5)] 9 2 (3 6) 1 ( 2 + 1) ( 3) 24) 13 + ( 3)2 + 4( 3) + 1 [ 10 ( 6)] {[4 + 5] [42 32(4 3) 8]} + 12 21) 6 23) 10) ( 7 5) [ 2 2 ( 6)] 5 + 32 24 6 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 which means \"reunion\". It was first used in Iraq in 830 AD by Mohammad ibnMusa al-Khwarizmi. The first form of simplifying expressions is used when we know what number each variable in the expression represents. If we know what they represent we can 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. This is to preserve operations that are sometimes lost in a simple replacement. Sometimes the parenthesis won't make a difference, but it is a good habbit to always use them to prevent problems later. Example 31. \u0010x\u0011 x + zx(3 z) when x = 6 and z = 2 3 \u0012 \u0013 ( 6) ( 6) + ( 2)( 6)(3 ( 2)) 3 6 + ( 2)( 6)(5)( 2) 6 + 12(5)( 2) 6 + 60( 2) 6 120 126 22 Replace all x s with 6 and z s with 2 Evaluate parenthesis Multiply left to right Multiply left to right Multiply Subtract Our Solution It will be more common in our study of algebra that we do not know the 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 in front of the variables, then keep the variables the same. This is shown in the following examples Example 32. 5x 2y 8x + 7y 3x + 5y Combine like terms 5x 8x and 2y + 7y Our Solution Example 33. 8x2 3x + 7 2x2 + 4x 3 6x2 + x + 4 Combine like terms 8x2 2x2 and 3x + 4x and 7 3 Our Solution As we combine like terms we need to interpret subtraction signs as part of the following term. This means if we see a subtraction sign, we treat the following term like a negative term, the sign always stays with the term. A final method to simplify is known as distributing. Often as we work with problems there will be a set of parenthesis that make solving a problem difficult, if not impossible. To get rid of these unwanted parenthesis we have the distributive property. Using this property we multiply the number in front of the parenthesis by each term inside of the parenthesis. Distributive Property: a(b + c) = ab + ac Several examples of using the distributive property are given below. Example 34. 4(2x 7) 8x 28 Multiply each term by 4 Our Solution Example 35. 7(5x 6) 35 + 42 Multiply each term by 7 Our Solution In the previous example we again use the fact that the sign goes with the number, this means we treat the 6 as a negative number, this gives ( 7)( 6) = 42, a positive number. The most common error in distributing is a sign error, be very careful with your signs! 23 It is possible to distribute just a negative through parenthesis. If we have a negative in front of parenthesis we can think of it like a 1 in front and distribute the 1 through. This is shown in the following example. Example 36. (4x 5y + 6) 1(4x 5y + 6) 4x + 5y 6 Negative can be thought of as 1 Multiply each term by 1 Our Solution Distributing through parenthesis and combining like terms can be combined into one problem. Order of operations tells us to multiply (distribute) first then add or subtract last (combine like terms). Thus we do each problem in two steps, distribute then combine. Example 37. 5 + 3(2x 4) 5 + 6x 12 7 + 6x Distribute 3, multipling each term Combine like terms 5 12 Our Solution Example 38. 3x 2(4x 5) 3x 8x + 10 5x + 10 Distribute 2, multilpying each term Combine like terms 3x 8x Our Solution In the previous example we distributed 2, not just 2. This is because we will always treat subtraction like a negative sign that goes with the number after it. This makes a big difference when we multiply by the 5 inside the parenthesis, we now have a positive answer. Following are more involved examples of distributing and combining like terms. Example 39. 2(5x 8) 6(4x + 3) 10x 16 24x 18 14x 34 Distribute 2 into first parenthesis and 6 into second Combine like terms 10x 24x and 16 18 Our Solution Example 40. 4(3x 8) (2x 7) 4(3x 8) 1(2x 7) 12x 32 2x + 7 10x 25 Negative (subtract) in middle can be thought of as 1 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 6) x + 6z 4y; use x = 6, y = 4, z = 4 j = 4, k = 2 8) 5(b + a) + 1 + c; use a = 2, b = 6, c = 5 10) z + x (12)3; use x = 5, z = 4 + q; use m = 4, p = 6, q = 6 n 12) 3 + z 1 + y 1; use y = 5, z = 4 11) m + n + m + 2 ; use m = 1 and n = 2 13) q p (q 1 3); use p = 3, q = 6 14) p + (q r)(6 p); use p = 6, q = 5, r = 5 15) y [4 y (z x)]; use x = 3, y = 1, z = 6 16) 4z (x + x (z z)); use x = 3, z = 2 17) k 32 (j + k) 5; use j = 4, k = 5 19) zx (z 4+x ); use x = 2, z 6 =6 18) a3(c2 c); use a = 3, c = 2 20) 5 + qp + pq q; use p = 6, q = 3 Combine Like Terms 21) r 9 + 10 22) 4x + 2 4 23) n + n 24) 4b + 6 + 1 + 7b 25) 8v + 7v 26) x + 8x 27) 7x 2x 28) 7a 6 + 5 29) k 2 + 7 30) 8p + 5p 31) x 10 6x + 1 32) 1 10n 10 33) m 2m 34) 1 r 6 35) 9n 1 + n + 4 36) 4b + 9b 25 Distribute 37) 8(x 4) 38) 3(8v + 9) 39) 8n(n + 9) 40) ( 5 + 9a) 41) 7k( k + 6) 42) 10x(1 + 2x) 43) 6(1 + 6x) 44) 2(n + 1) 45) 8m(5 m) 46) 2p(9p 1) 47) 9x(4 x) 48) 4(8n 2) 49) 9b(b 10) 50) 4(1 + 7r) 51) 8n(5 + 10n) 52) 2x(8x 10) Simplify. 53) 9(b + 10) + 5b 54) 4v 7(1 8v) 55) 3x(1 4x) 4x2 56) 8x + 9( 9x + 9) 57) 4k 2 8k (8k + 1) 58) 9 10(1 + 9a) 59) 1 7(5 + 7p) 60) 10(x 2) 3 61) 10 4(n 5) 62) 6(5 m) + 3m 63) 4(x + 7) + 8(x + 4) 64) 2r(1 + 4r) + 8r( r + 4) 65) 8(n + 6) 8n(n + 8) 66) 9(6b + 5) 4b(b + 3) 67) 7(7 + 3v) + 10(3 10v) 68) 7(4x 6) + 2(10x 10) 69) 2n( 10n + 5) 7(6 10n) 70) 3(4 + a) + 6a(9a + 10) 71) 5(1 6k) + 10(k 8) 72) 7(4x + 3) 10(10x + 10) 73) (8n2 3n) (5 + 4n2) 74) (7x2 3) (5x2 + 6x) 75) (5p 6) + (1 p) 76) (3x2 x) (7 8x) 77) (2 4v 2) + (3v 2 + 2v) 78) (2b 8) + (b 7b2) 79) (4 2k 2) + (8 2k 2) 80) (7a2 + 7a) (6a2 + 4a) 81) (x2 8) + (2x2 7) 82) (3 7n2) + (6n2 + 3) 26 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 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 where the answer is known, but part of the problem is missing. The missing part of the problem is what we seek to find. An example of such a problem is shown below. Example 41. 4x + 16 = 4 Notice the above problem has a missing part, or unknown, that is marked by x. If we are given that the solution to this equation is 5, it could be plugged into the equation, replacing the x with 5. This is shown in Example 2. Example 42. 4( 5) + 16 = 4 20 + 16 = 4 4=4 Multiply 4( 5) Add 20 + 16 True! Now the equation comes out to a true statement! Notice also that if another number, for example, 3, was plugged in, we would not get a true statement as seen in Example 3. Example 43. 4(3) + 16 = 4 12 + 16 = 4 28 \u0002 4 Multiply 4(3) Add 12 + 16 False! Due to the fact that this is not a true statement, this demonstates that 3 is not the solution. However, depending on the complexity of the problem, this \"guess and check\" method is not very efficient. Thus, we take a more algebraic approach to solving equations. Here we will focus on what are called \"one-step equations\" or equations that only require one step to solve. While these equations often seem very fundamental, it is important to master the pattern for solving these problems so we can solve more complex problems. 28 Addition Problems To solve equations, the general rule is to do the opposite. For example, consider the following example. Example 44. x+7=5 7 7 x = 12 The 7 is added to the x Subtract 7 from both sides to get rid of it Our solution! Then we get our solution, x = 12. The same process is used in each of the following examples. Example 45. 4+x=8 4 4 x=4 7=x+9 9 9 2=x 5=8+x 8 8 3=x Table 1. Addition Examples Subtraction Problems In a subtraction problem, we get rid of negative numbers by adding them to both sides of the equation. For example, consider the following example. Example 46. x5=4 +5 +5 x=9 The 5 is negative, or subtracted from x Add 5 to both sides Our Solution! Then we get our solution x = 9. The same process is used in each of the following examples. Notice that each time we are getting rid of a negative number by adding. 29 Example 47. 10 = x 7 +7 +7 3=x 6+x=2 +6 +6 x=4 5=8+x +8 +8 13 = x Table 2. Subtraction Examples Multiplication Problems With a multiplication problem, we get rid of the number by dividing on both sides. For example consider the following example. Example 48. 4x = 20 4 4 x=5 Variable is multiplied by 4 Divide both sides by 4 Our solution! Then we get our solution x = 5 With multiplication problems it is very important that care is taken with signs. If x is multiplied by a negative then we will divide by a negative. This is shown in example 9. Example 49. 5x = 30 5 5 x=6 Variable is multiplied by 5 Divide both sides by 5 Our Solution! The same process is used in each of the following examples. Notice how negative and positive numbers are handled as each problem is solved. Example 50. 30 8x = 24 8 8 x=3 4x = 20 4 4 x=5 42 = 7x 7 7 6=x Table 3. Multiplication Examples Division Problems: In division problems, we get rid of the denominator by multiplying on both sides. For example consider our next example. Example 51. x =3 5 x (5) = 3(5) 5 x = 15 Variable is divided by 5 Multiply both sides by 5 Our Solution! Then we get our solution x = 15. The same process is used in each of the following examples. Example 52. x =2 7 x ( 7) 7 = 2( x = 14 7) x =5 8 x (8) 8 = 5(8) x = 40 x =9 4 x ( 4) 4 = 9( 4) x = 36 Table 4. Division Examples The process described above is fundamental to solving equations. once this process is mastered, the problems we will see have several more steps. These problems may seem more complex, but the process and patterns used will remain the same. World View Note: The study of algebra originally was called the \"Cossic Art\" from the Latin, the study of \"things\" (which we now call variables). 31 1.1 Practice - One Step Equations Solve each equation. 1) v + 9 = 16 2) 14 = b + 3 3) x 11 = 16 4) 14 = x 18 5) 30 = a + 20 6) 1 + k = 5 7) x 7 = 26 8) 13 + p = 19 9) 13 = n 5 10) 22 = 16 + m 11) 340 = 17x 12) 4r = 28 n 5 9 13) 9 = 12 14) 15) 20v = 160 16) 20x = 80 17) 340 = 20n 19) 16x = 320 21) 16 + n = 13 23) p 8 = 21 25) 180 = 12x b =9 18) 1 2 20) k 13 a =8 = 16 22) 21 = x + 5 24) m 4 = 13 26) 3n = 24 27) 20b = 200 29) r 14 = 5 14 31) 7 = a + 4 33) 10 = x 4 35) 13a = 143 37) p 20 = 12 39) 9 + m = 7 x 28) 17 = 12 30) n + 8 = 10 32) v 16 = 30 34) 15 = x 16 36) 8k = 120 x 38) 15 = 9 n 40) 19 = 20 32 1.2 Linear Equations - Two-Step Equations Objective: Solve two-step equations by balancing and using inverse opperations. After mastering the technique for solving equations that are simple one-step equations, we are ready to consider two-step equations. As we solve two-step equations, the important thing to remember is that everything works backwards! When working with one-step equations, we learned that in order to clear a \"plus five\" in the equation, we would subtract five from both sides. We learned that to clear \"divided by seven\" we multiply by seven on both sides. The same pattern applies to the order of operations. When solving for our variable x, we use order of operations backwards as well. This means we will add or subtract first, then multiply or divide second (then exponents, and finally any parentheses or grouping symbols, but that's another lesson). So to solve the equation in the first example, Example 53. 4x 20 = 8 We have two numbers on the same side as the x. We need to move the 4 and the 20 to the other side. We know to move the four we need to divide, and to move the twenty we will add twenty to both sides. If order of operations is done backwards, we will add or subtract first. Therefore we will add 20 to both sides first. Once we are done with that, we will divide both sides by 4. The steps are shown below. 4x 20 = 8 + 20 + 20 4x = 12 4 4 x=3 Start by focusing on the subtract 20 Add 20 to both sides Now we focus on the 4 multiplied by x Divide both sides by 4 Our Solution! Notice in our next example when we replace the x with 3 we get a true statement. 4(3) 20 = 8 12 20 = 8 8=8 Multiply 4(3) Subtract 12 20 True! 33 The same process is used to solve any two-step equations. Add or subtract first, then multiply or divide. Consider our next example and notice how the same process is applied. Example 54. 5x + 7 = 7 7 7 5x = 0 5 5 x= 0 Start by focusing on the plus 7 Subtract 7 from both sides Now focus on the multiplication by 5 Divide both sides by 5 Our Solution! Notice the seven subtracted out completely! Many students get stuck on this point, do not forget that we have a number for \"nothing left\" and that number is zero. With this in mind the process is almost identical to our first example. A common error students make with two-step equations is with negative signs. Remember the sign always stays with the number. Consider the following example. Example 55. 4 2x = 10 4 4 2x = 6 2 2 x=3 Start by focusing on the positive 4 Subtract 4 from both sides Negative (subtraction) stays on the 2x Divide by 2 Our Solution! The same is true even if there is no coefficient in front of the variable. Consider the next example. Example 56. 8x=2 8 8 x=6 1x = 6 Start by focusing on the positive 8 Subtract 8 from both sides Negative (subtraction) stays on the x Remember, no number in front of variable means 1 34 1 1 x=6 Divide both sides by 1 Our Solution! Solving two-step equations is a very important skill to master, as we study algebra. The first step is to add or subtract, the second is to multiply or divide. This pattern is seen in each of the following examples. Example 57. 3x + 7 = 8 7 7 3x = 15 3 3 x=5 7 5x = 17 7 7 5x = 10 5 5 x=2 2 + 9x = 7 +2 +2 9x = 9 9 9 x=1 5 3x = 5 +5 +5 3x = 0 3 3 x=0 8 = 2x + 10 10 10 2 = 2x 2 2 1=x x 3= 5 4 +4 +4 x (5)(1) = 5 (5) 5=x Table 5. Two-Step Equation Examples As problems in algebra become more complex the process covered here will remain the same. In fact, as we solve problems like those in the next example, each one of them will have several steps to solve, but the last two steps are a twostep equation like we are solving here. This is why it is very important to master two-step equations now! Example 58. 3x2 + 4 x + 6 1 1 1 + = x8 x 3 5x 5 + 1 = x log5(2x 4) = 1 World View Note: Persian mathematician Omar Khayyam would solve algebraic problems geometrically by intersecting graphs rather than solving them algebraically. 35 1.2 Practice - Two-Step Problems Solve each equation. n 1) 5 + 4 = 4 2) 2 = 2m + 12 3) 102 = 7r + 4 4) 27 = 21 3x 5) 8n + 3 = 77 6) 4 b = 8 7) 0 = 6v 8) 2 + 2 = 4 x x 9) 8 = 5 6 k 11) 0 = 7 + 2 13) 12 + 3x = 0 15) 24 = 2n 8 17) 2 = 12 + 2r 19) b 3 + 7 = 10 23) 16 = 8a + 64 25) 56 + 8k = 64 27) 2x + 4 = 22 29) 20 = 4p + 4 33) r 8 12) 6 = 15 + 3p 14) 5m + 2 = 27 16) 37 = 8 + 3x n 18) 8 + 12 = 7 20) 21) 152 = 8n + 64 31) 5 = 3 + a 10) 5 = 4 1 n 2 6=5 35) 40 = 4n 32 37) 87 = 3 7v 39) x + 1 = 11 x 1 8=8 v 22) 11 = 8 + 2 24) 2x 3 = 29 26) 4 3n = 16 28) 67 = 5m 8 x 30) 9 = 8 + 6 32) m 4 1=2 34) 80 = 4x 28 36) 33 = 3b + 3 38) 3x 3 = 3 a 40) 4 + 3 = 1 36 1.3 Solving Linear Equations - General Equations Objective: Solve general linear equations with variables on both sides. Often as we are solving linear equations we will need to do some work to set them up into a form we are familiar with solving. This section will focus on manipulating an equation we are asked to solve in such a way that we can use our pattern for solving two-step equations to ultimately arrive at the solution. One such issue that needs to be addressed is parenthesis. Often the parenthesis can get in the way of solving an otherwise easy problem. As you might expect we can get rid of the unwanted parenthesis by using the distributive property. This is shown in the following example. Notice the first step is distributing, then it is solved like any other two-step equation. Example 59. 4(2x 6) = 16 8x 24 = 16 + 24 + 24 8x = 40 8 8 x=5 Distribute 4 through parenthesis Focus on the subtraction first Add 24 to both sides Now focus on the multiply by 8 Divide both sides by 8 Our Solution! Often after we distribute there will be some like terms on one side of the equation. Example 2 shows distributing to clear the parenthesis and then combining like terms next. Notice we only combine like terms on the same side of the equation. Once we have done this, our next example solves just like any other two-step equation. Example 60. 3(2x 4) + 9 = 15 6x 12 + 9 = 15 6x 3 = 15 +3 +3 6x = 18 Distribute the 3 through the parenthesis Combine like terms, 12 + 9 Focus on the subtraction first Add 3 to both sides Now focus on multiply by 6 37 6 6 x=3 Divide both sides by 6 Our Solution A second type of problem that becomes a two-step equation after a bit of work is one where we see the variable on both sides. This is shown in the following example. Example 61. 4x 6 = 2x + 10 Notice here the x is on both the left and right sides of the equation. This can make it difficult to decide which side to work with. We fix this by moving one of the terms with x to the other side, much like we moved a constant term. It doesn't matter which term gets moved, 4x or 2x, however, it would be the author's suggestion to move the smaller term (to avoid negative coefficients). For this reason we begin this problem by clearing the positive 2x by subtracting 2x from both sides. 4x 6 = 2x + 10 2x 2x 2x 6 = 10 +6 +6 2x = 16 2 2 x=8 Notice the variable on both sides Subtract 2x from both sides Focus on the subtraction first Add 6 to both sides Focus on the multiplication by 2 Divide both sides by 2 Our Solution! The previous example shows the check on this solution. Here the solution is plugged into the x on both the left and right sides before simplifying. Example 62. 4(8) 6 = 2(8) + 10 32 6 = 16 + 10 26 = 26 Multiply 4(8) and 2(8) first Add and Subtract True! The next example illustrates the same process with negative coefficients. Notice first the smaller term with the variable is moved to the other side, this time by adding because the coefficient is negative. 38 Example 63. 3x + 9 = 6x 27 + 3x + 3x 9 = 9x 27 + 27 + 27 36 = 9x 9 9 4=x Notice the variable on both sides, 3x is smaller Add 3x to both sides Focus on the subtraction by 27 Add 27 to both sides Focus on the mutiplication by 9 Divide both sides by 9 Our Solution Linear equations can become particularly intersting when the two processes are combined. In the following problems we have parenthesis and the variable on both sides. Notice in each of the following examples we distribute, then combine like terms, then move the variable to one side of the equation. Example 64. 2(x 5) + 3x = x + 18 2x 10 + 3x = x + 18 5x 10 = x + 18 x x 4x 10 = 18 + 10 + 10 4x = 28 4 4 x=7 Distribute the 2 through parenthesis Combine like terms 2x + 3x Notice the variable is on both sides Subtract x from both sides Focus on the subtraction of 10 Add 10 to both sides Focus on multiplication by 4 Divide both sides by 4 Our Solution Sometimes we may have to distribute more than once to clear several parenthesis. Remember to combine like terms after you distribute! Example 65. 3(4x 5) 4(2x + 1) = 5 12x 15 8x 4 = 5 4x 19 = 5 + 19 + 19 4x = 24 Distribute 3 and 4 through parenthesis Combine like terms 12x 8x and 15 4 Focus on subtraction of 19 Add 19 to both sides Focus on multiplication by 4 39 4 4 x=6 Divide both sides by 4 Our Solution This leads to a 5-step process to solve any linear equation. While all five steps aren't always needed, this can serve as a guide to solving equations. 1. Distribute through any parentheses. 2. Combine like terms on each side of the equation. 3. Get the variables on one side by adding or subtracting 4. Solve the remaining 2-step equation (add or subtract then multiply or divide) 5. Check your answer by plugging it back in for x to find a true statement. The order of these steps is very important. World View Note: The Chinese developed a method for solving equations that involved finding each digit one at a time about 2000 years ago! We can see each of the above five steps worked through our next example. Example 66. 4(2x 6) + 9 = 3(x 7) + 8x 8x 24 + 9 = 3x 21 + 8x 8x 15 = 11x 21 8x 8x 15 = 3x 21 + 21 + 21 6 = 3x 3 3 2=x Distribute 4 and 3 through parenthesis Combine like terms 24 + 9 and 3x + 8x Notice the variable is on both sides Subtract 8x from both sides Focus on subtraction of 21 Add 21 to both sides Focus on multiplication by 3 Divide both sides by 3 Our Solution Check: 4[2(2) 6] + 9 = 3[(2) 7] + 8(2) 4[4 6] + 9 = 3[ 5] + 8(2) Plug 2 in for each x. Multiply inside parenthesis Finish parentesis on left, multiply on right 40 4[ 2] + 9 = 15 + 8(2) 8 + 9 = 15 + 16 1=1 Finish multiplication on both sides Add True! When we check our solution of x = 2 we found a true statement, 1 = 1. Therefore, we know our solution x = 2 is the correct solution for the problem. There are two special cases that can come up as we are solving these linear equations. The first is illustrated in the next two examples. Notice we start by distributing and moving the variables all to the same side. Example 67. 3(2x 5) = 6x 15 6x 15 = 6x 15 6x 6x 15 = 15 Distribute 3 through parenthesis Notice the variable on both sides Subtract 6x from both sides Variable is gone! True! Here the variable subtracted out completely! We are left with a true statement, 15 = 15. If the variables subtract out completely and we are left with a true statement, this indicates that the equation is always true, no matter what x is. Thus, for our solution we say all real numbers or R. Example 68. 2(3x 5) 4x = 2x + 7 6x 10 4x = 2x + 7 2x 10 = 2x + 7 2x 2x 10 \u0002 7 Distribute 2 through parenthesis Combine like terms 6x 4x Notice the variable is on both sides Subtract 2x from both sides Variable is gone! False! Again, the variable subtracted out completely! However, this time we are left with a false statement, this indicates that the equation is never true, no matter what x is. Thus, for our solution we say no solution or . 41 1.3 Practice - General Linear Equations Solve each equation. 1) 2 ( 3a 8) = 1 2) 2( 3n + 8) = 20 5) 66 = 6(6 + 5x) 6) 32 = 2 5( 4n + 6) 3) 5( 4 + 2v) = 50 7) 0 = 8(p 5) 4) 2 8( 4 + 3x) = 34 8) 55 = 8 + 7(k 5) 9) 2 + 2(8x 7) = 16 10) (3 5n) = 12 13) 1 7m = 8m + 7 14) 56p 48 = 6p + 2 11) 21x + 12 = 6 3x 12) 3n 27 = 27 3n 15) 1 12r = 29 8r 16) 4 + 3x = 12x + 4 19) 32 24v = 34 2v 20) 17 2x = 35 8x 17) 20 7b = 12b + 30 18) 16n + 12 = 39 7n 21) 2 5(2 4m) = 33 + 5m 22) 25 7x = 6(2x 1) 25) 6v 29 = 4v 5(v + 1) 26) 8(8r 2) = 3r + 16 23) 4n + 11 = 2(1 8n) + 3n 27) 2(4x 4) = 20 4x 29) a 5(8a 1) = 39 7a 24) 7(1 + b) = 5 5b 28) 8n 19 = 2(8n 3) + 3n 30) 4 + 4k = 4(8k 8) 31) 57 = ( p + 1) + 2(6 + 8p) 32) 16 = 5(1 6x) + 3(6x + 7) 35) 50 = 8 (7 + 7r) (4r + 6) 36) 8(6 + 6x) + 4( 3 + 6x) = 12 33) 2(m 2) + 7(m 8) = 67 37) 8(n 7) + 3(3n 3) = 41 39) 61 = 5(5r 4) + 4(3r 4) 34) 7 = 4(n 7) + 5(7n + 7) 38) 76 = 5(1 + 3b) + 3(3b 3) 40) 6(x 8) 4(x 2) = 4 41) 2(8n 4) = 8(1 n) 42) 4(1 + a) = 2a 8(5 + 3a) 45) 7(x 2) = 4 6(x 1) 46) (n + 8) + n = 8n + 2(4n 4) 49) 2(1 7p) = 8(p 7) 50) 8( 8n + 4) = 4( 7n + 8) 43) 3( 7v + 3) + 8v = 5v 4(1 6v) 47) 6(8k + 4) = 8(6k + 3) 2 44) 6(x 3) + 5 = 2 5(x 5) 48) 5(x + 7) = 4( 8x 2) 42 1.4 Solving Linear Equations - Fractions Objective: Solve linear equations with rational coefficients by multiplying by the least common denominator to clear the fractions. Often when solving linear equations we will need to work with an equation with fraction coefficients. We can solve these problems as we have in the past. This is demonstrated in our next example. Example 69. 3 7 5 x = 4 2 6 + 7 7 + 2 2 Focus on subtraction Add 7 to both sides 2 Notice we will need to get a common denominator to add 5 6 7 + 2 . Notice we have a \u0010 \u0011 7 3 21 common denominator of 6. So we build up the denominator, 2 3 = 6 , and we can now add the fractions: 3 21 5 x = 4 6 6 + 21 21 + 6 6 Same problem, with common denominator 6 Add 21 to both sides 6 26 3 x= 4 6 Reduce 26 13 to 6 3 3 13 x= 4 3 Focus on multiplication by 3 3 4 3 We can get rid of 4 by dividing both sides by 4 . Dividing by a fraction is the 4 same as multiplying by the reciprocal, so we will multiply both sides by 3 . \u0012 \u0013 \u0012 \u0013 13 4 4 3 x= 3 3 3 4 52 x= 9 Multiply by reciprocal Our solution! While this process does help us arrive at the correct solution, the fractions can make the process quite difficult. This is why we have an alternate method for dealing with fractions - clearing fractions. Clearing fractions is nice as it gets rid of the fractions for the majority of the problem. We can easily clear the fractions 43 by finding the LCD and multiplying each term by the LCD. This is shown in the next example, the same problem as our first example, but this time we will solve by clearing fractions. Example 70. 3 7 5 x = 4 2 6 (12)3 (12)7 (12)5 x = 4 2 6 (3)3x (6)7 = (2)5 9x 42 = 10 + 42 + 42 9x = 52 9 9 52 x= 9 LCD = 12, multiply each term by 12 Reduce each 12 with denominators Multiply out each term Focus on subtraction by 42 Add 42 to both sides Focus on multiplication by 9 Divide both sides by 9 Our Solution The next example illustrates this as well. Notice the 2 isn't a fraction in the origional equation, but to solve it we put the 2 over 1 to make it a fraction. Example 71. 2 3 1 x2= x+ 3 2 6 (6)2 (6)2 (6)3 (6)1 x = x+ 3 1 2 6 (2)2x (6)2 = (3)3x + (1)1 4x 12 = 9x + 1 4x 4x 12 = 5x + 1 1 1 13 = 5x 5 5 13 =x 5 LCD = 6, multiply each term by 6 Reduce 6 with each denominator Multiply out each term Notice variable on both sides Subtract 4x from both sides Focus on addition of 1 Subtract 1 from both sides Focus on multiplication of 5 Divide both sides by 5 Our Solution We can use this same process if there are parenthesis in the problem. We will first distribute the coefficient in front of the parenthesis, then clear the fractions. This is seen in the following example. 44 Example 72. \u0012 \u0013 3 5 4 =3 x+ 2 9 27 2 5 x+ =3 9 6 (18)2 (18)3 (18)5 x+ = 9 9 6 (3)5x + (2)2 = (18)3 15x + 4 = 54 4 4 15x = 50 . 15 15 10 x= 3 Distribute 3 through parenthesis, reducing if possible 2 LCD = 18, multiply each term by 18 Reduce 18 with each denominator Multiply out each term Focus on addition of 4 Subtract 4 from both sides Focus on multiplication by 15 Divide both sides by 15. Reduce on right side. Our Solution While the problem can take many different forms, the pattern to clear the fraction is the same, after distributing through any parentheses we multiply each term by the LCD and reduce. This will give us a problem with no fractions that is much easier to solve. The following example again illustrates this process. Example 73. 3 1 1 3 7 x = ( x + 6) 4 2 3 4 2 1 1 7 3 x = x+2 2 4 2 4 (4)1 (4)1 (4)2 (4)7 (4)3 x = x+ 2 4 1 2 4 (1)3x (2)1 = (1)1x + (4)2 (2)7 3x 2 = x + 8 14 3x 2 = x 6 x x 2x 2 = 6 +2 +2 2x = 4 2 2 x=2 1 Distribute , reduce if possible 3 LCD = 4, multiply each term by 4. Reduce 4 with each denominator Multiply out each term Combine like terms 8 14 Notice variable on both sides Subtract x from both sides Focus on subtraction by 2 Add 2 to both sides Focus on multiplication by 2 Divide both sides by 2 Our Solution World View Note: The Egyptians were among the first to study fractions and linear equations. The most famous mathematical document from Ancient Egypt is the Rhind Papyrus where the unknown variable was called \"heap\" 45 1.4 Practice - Fractions Solve each equation. 3 1 21 1) 5 (1 + p) = 20 5 6 3 4 7) 635 72 5 4m = 5 9 9) 2b + 5 = 11) 6) 113 24 = 2( 11 4 3 8 11 4 + 4r = 163 32 16 9 4 5 8) + x) 11 5 3 7 3 ( n + 1) = 2 2 3 15) 55 6 17) 16 9 = 5 3 5 ( p 3) 2 2 = 4 4 4 ( 3n 3) 3 10) 3 2 12) 41 9 = 3 ( 3 + n) 7 5 5 11 3 3 3 5 5 3 3 23) ( 2 x 2 ) = 2 + x 25) 45 16 3 7 19 + 2 n = 4 n 16 3 3 7 27) 2 (v + 2 ) = 4 v 29) 47 9 3 1 7 1 2 3 10 k 3 = 7 13 8 83 16) 2 ( 3 x 4 ) 2 x = 24 9 18) 3 (m + 4 ) 20) 1 12 22) 7 6 4 10 3 53 = 18 5 7 = 3 x + 3 (x 4 ) 4 3 3 3 n = 2 n + 2(n + 2 ) 24) 149 16 7 5 11 7 r= 4 r 3 1 26) 2 ( 3 a + 3 ) = 8 19 6 2 = 2 (x + 3 ) 3 x 14) 3 ( 4 k + 1) 5 + 2 b = 2 (b 3 ) 5 9 4v = 8 2 19) 8 = 4 (r 2 ) 21) 29 3 1 5 8 19 ( 3 a + 1) = 4 4 13) a 3 4) 2 n 3 = 12 3) 0 = 4 (x 5 ) 5) 3 2) 2 = 2 k + 2 1 5 11 25 a+ 8 4 4 2 28) 3 2 x = 3 x 3 ( 1 30) 3 n + 5 5 + 2 x = 3 ( 2 x + 1) 46 29 6 4 4 4 ( 3 r + 1) 2 = 2( 3 n + 3 ) 13 x + 1) 4 1.5 Solving Linear Equations - Formulas Objective: Solve linear formulas for a given variable. Solving formulas is much like solving general linear equations. The only difference is we will have several varaibles in the problem and we will be attempting to solve for one specific variable. For example, we may have a formula such as A = r 2 + rs (formula for surface area of a right circular cone) and we may be interested in solving for the varaible s. This means we want to isolate the s so the equation has s on one side, and everything else on the other. So a solution might look like s = A r 2 . This second equation gives the same information as the first, they are s algebraically equivalent, however, one is solved for the area, while the other is solved for s (slant height of the cone). In this section we will discuss how we can move from the first equation to the second. When solving formulas for a variable we need to focus on the one varaible we are trying to solve for, all the others are treated just like numbers. This is shown in the following example. Two parallel problems are shown, the first is a normal onestep equation, the second is a formula that we are solving for x Example 74. 3x = 12 3 3 x=4 wx = z In both problems, x is multiplied by something w w To isolate the x we divide by 3 or w. z x= Our Solution w We use the same process to solve 3x = 12 for x as we use to solve w x = z for x. Because we are solving for x we treat all the other variables the same way we would treat numbers. Thus, to get rid of the multiplication we divided by w. This same idea is seen in the following example. Example 75. m + n = p for n Solving for n, treat all other variables like numbers m m Subtract m from both sides n= pm Our Solution As p and m are not like terms, they cannot be combined. For this reason we leave the expression as p m. This same one-step process can be used with grouping symbols. 47 Example 76. a(x y) = b for a Solving for a, treat (x y) like a number (x y) (x y) Divide both sides by (x y) a= b xy Our Solution Because (x y) is in parenthesis, if we are not searching for what is inside the parenthesis, we can keep them together as a group and divide by that group. However, if we are searching for what is inside the parenthesis, we will have to break up the parenthesis by distributing. The following example is the same formula, but this time we will solve for x. Example 77. a(x y) = b for x ax ay = b + ay + ay ax = b + ay a a b + ay x= a Solving for x, we need to distribute to clear parenthesis This is a two step equation, ay is subtracted from our x term Add ay to both sides The x is multipied by a Divide both sides by a Our Solution Be very careful as we isolate x that we do not try and cancel the a on top and bottom of the fraction. This is not allowed if there is any adding or subtracting in the fraction. There is no reducing possible in this problem, so our final reduced b + ay answer remains x = a . The next example is another two-step problem Example 78. y = mx + b for m b b y b = mx x x y b =m x Solving for m, focus on addition first Subtract b from both sides m is multipied by x. Divide both sides by x Our Solution It is important to note that we know we are done with the problem when the variable we are solving for is isolated or alone on one side of the equation and it does not appear anywhere on the other side of the equation. The next example is also a two-step equation, it is the problem we started with at the beginning of the lesson. 48 Example 79. A = r 2 + rs for s r 2 r 2 A r 2 = rs r r 2 A r =s r Solving for s, focus on what is added to the term with s Subtract r2 from both sides s is multipied by r Divide both sides by r Our Solution Again, we cannot reduce the r in the numerator and denominator because of the subtraction in the problem. Formulas often have fractions in them and can be solved in much the same way we solved with fractions before. First identify the LCD and then multiply each term by the LCD. After we reduce there will be no more fractions in the problem so we can solve like any general equation from there. Example 80. h= (n)h = 2m for m To clear the fraction we use LCD = n n (n)2m n nh = 2m 2 2 nh =m 2 Multiply each term by n Reduce n with denominators Divide both sides by 2 Our Solution The same pattern can be seen when we have several fractions in our problem. Example 81. a c + = e for a b b (b)a (b)c + = e (b) b b a + c = eb c c a = eb c To clear the fraction we use LCD = b Multiply each term by b Reduce b with denominators Subtract c from both sides Our Solution Depending on the context of the problem we may find a formula that uses the same letter, one capital, one lowercase. These represent different values and we must be careful not to combine a capital variable with a lower case variable. Example 82. a= A for b Use LCD (2 b) as a group 2b 49 (2 b)a = (2 b)A 2b (2 b)a = A 2a ab = A 2a 2a ab = A 2a a a A 2a b= a Multiply each term by (2 b) reduce (2 b) with denominator Distribute through parenthesis Subtract 2a from both sides The b is multipied by a Divide both sides by a Our Solution Notice the A and a were not combined as like terms. This is because a formula will often use a capital letter and lower case letter to represent different variables. Often with formulas there is more than one way to solve for a variable. The next example solves the same problem in a slightly different manner. After clearing the denominator, we divide by a to move it to the other side, rather than distributing. Example 83. a= A for b Use LCD = (2 b) as a group 2b (2 b)A 2b (2 b)a = A a a A 2b= a 2 2 A b= 2 a A ( 1)( b) = ( 1) 2( 1) a A b= +2 a (2 b)a = Multiply each term by (2 b) Reduce (2 b) with denominator Divide both sides by a Focus on the positive 2 Subtract 2 from both sides Still need to clear the negative Multiply (or divide) each term by 1 Our Solution Both answers to the last two examples are correct, they are just written in a different form because we solved them in different ways. This is very common with formulas, there may be more than one way to solve for a varaible, yet both are equivalent and correct. World View Note: The father of algebra, Persian mathematician Muhammad ibn Musa Khwarizmi, introduced the fundamental idea of blancing by subtracting the same term to the other side of the equation. He called this process al-jabr which later became the world algebra. 50 1.5 Practice - Formulas Solve each of the following equations for the indicated variable. 1) ab = c for b f 3) g x = b for x 2) g = h i for h 4) p = 3y q for y a 5) 3x = b for x 6) 7) E = mc2 for m 8) DS = ds for D 9) V = 4 r3 3 for 13) c = 4y m+n for y 15) V = Dn 12 for D 17) P = n(p c) for n Dd L c = d for y 10) E = 11) a + c = b for c 19) T = ym b for D mv2 2 for m 12) x f = g for x 14) rs a3 = k for r 16) F = k(R L) for k 18) S = L + 2B for L 20) I = Ea E q R for Ea 21) L = Lo(1 + at) for Lo 22) ax + b = c for x 23) 2m + p = 4m + q for m 24) q = 6(L p) for L 25) km r 26) R = aT + b for T = q for k 27) h = vt 16t2 for v 29) Q1 = P (Q2 Q1) for Q2 31) R = kA(T1 + T2) d for T1 28) S = rh + r 2 for h 30) L = (r1 + r2) + 2d for r1 32) P = V1(V2 V1) g 33) ax + b = c for a 34) rt = d for r 35) lwh = V for w 36) V = 37) 1 a c + b = a for a 39) at bw = s for t 38) 1 a r 2h 3 for V2 for h c + b = a for b 40) at bw = s for w 41) ax + bx = c for a 42) x + 5y = 3 for x 43) x + 5y = 3 for y 44) 3x + 2y = 7 for x 45) 3x + 2y = 7 for y 46) 5a 7b = 4 for a 47) 5a 7b = 4 for b 48) 4x 5y = 8 for x 49) 4x 5y = 8 for y 50) C = 9 (F 32) for F 5 51 1.6 Solving Linear Equations - Absolute Value Objective: Solve linear absolute value equations. When solving equations with absolute value we can end up with more than one possible answer. This is because what is in the absolute value can be either negative or positive and we must account for both possibilities when solving equations. This is illustrated in the following example. Example 84. |x| = 7 x = 7 or x = 7 Absolute value can be positive or negative Our Solution Notice that we have considered two possibilities, both the positive and negative. Either way, the absolute value of our number will be positive 7. World View Note: The first set of rules for working with negatives came from 7th century India. However, in 1758, almost a thousand years later, British mathematician Francis Maseres claimed that negatives \"Darken the very whole doctrines of the equations and make dark of the things which are in their nature excessively obvious and simple.\" When we have absolute values in our problem it is important to first isolate the absolute value, then remove the absolute value by considering both the positive and negative solutions. Notice in the next two examples, all the numbers outside of the absolute value are moved to the other side first before we remove the absolute value bars and consider both positive and negative solutions. Example 85. 5 + |x| = 8 5 5 |x| = 3 x = 3 or x = 3 Notice absolute value is not alone Subtract 5 from both sides Absolute value can be positive or negative Our Solution Example 86. 4|x| = 20 4 4 Notice absolute value is not alone Divide both sides by 4 52 |x| = 5 x = 5 or x = 5 Absolute value can be positive or negative Our Solution Notice we never combine what is inside the absolute value with what is outside the absolute value. This is very important as it will often change the final result to an incorrect solution. The next example requires two steps to isolate the absolute value. The idea is the same as a two-step equation, add or subtract, then multiply or divide. Example 87. 5|x| 4 = 26 +4 +4 5|x| = 30 5 5 |x| = 6 x = 6 or x = 6 Notice the absolute value is not alone Add 4 to both sides Absolute value still not alone Divide both sides by 5 Absolute value can be positive or negative Our Solution Again we see the same process, get the absolute value alone first, then consider the positive and negative solutions. Often the absolute value will have more than just a variable in it. In this case we will have to solve the resulting equations when we consider the positive and negative possibilities. This is shown in the next example. Example 88. |2x 1| = 7 2x 1 = 7 or 2x 1 = 7 Absolute value can be positive or negative Two equations to solve Now notice we have two equations to solve, each equation will give us a different solution. Both equations solve like any other two-step equation. 2x 1 = 7 +1+1 2x = 8 2 2 x=4 2x 1 = 7 +1 +1 2x = 6 2 2 x=3 or 53 Thus, from our previous example we have two solutions, x = 4 or x = 3. Again, it is important to remember that the absolute value must be alone first before we consider the positive and negative possibilities. This is illustrated in below. Example 89. 2 4|2x + 3| = 18 To get the absolute value alone we first need to get rid of the 2 by subtracting, then divide by 4. Notice we cannot combine the 2 and 4 becuase they are not like terms, the 4 has the absolute value connected to it. Also notice we do not distribute the 4 into the absolute value. This is because the numbers outside cannot be combined with the numbers inside the absolute value. Thus we get the absolute value alone in the following way: 2 4|2x + 3| = 18 2 2 4|2x + 3| = 20 4 4 |2x +