2.6 Absolute Value Equations

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1 96 CHAPTER 2 Equations, Inequalities, and Problem Solving The formula for onverting Fahrenheit temperatures to Celsius temperatures is C = 5 1F Use this formula for Eerises 9 91 and During a reent year, the temperatures in Chiago ranged from -29 C to 35 C. Use a ompound inequality to onvert these temperatures to Fahrenheit temperatures. 92. In Oslo, the average temperature ranges from -10 to 18 Celsius. Use a ompound inequality to onvert these temperatures to the Fahrenheit sale. Solve. 93. Christian D Angelo has sores of 68, 65, 75, and 78 on his algebra tests. Use a ompound inequality to find the sores he an make on his final eam to reeive a C in the ourse. The final eam ounts as two tests, and a C is reeived if the final ourse average is from 70 to Wendy Wood has sores of 80, 90, 82, and 75 on her hemistry tests. Use a ompound inequality to find the range of sores she an make on her final eam to reeive a B in the ourse. The final eam ounts as two tests, and a B is reeived if the final ourse average is from 80 to Absolute Value Equations 1 Solve Absolute Value Equations. 1 Solving Absolute Value Equations In Chapter 1, we defined the absolute value of a number as its distane from 0 on a number line. 2 units 3 units = 2 and 3 = In this setion, we onentrate on solving equations ontaining the absolute value of a variable or a variable epression. Eamples of absolute value equations are 0 0 = 3-5 = 0 2y z = 0 3z Sine distane and absolute value are so losely related, absolute value equations and inequalities (see Setion 2.7) are etremely useful in solving distane-type problems suh as alulating the possible error in a measurement. For the absolute value equation 0 0 = 3, its solution set will ontain all numbers whose distane from 0 is 3 units. Two numbers are 3 units away from 0 on the number line: 3 and units 3 units Thus, the solution set of the equation 0 0 = 3 is 53, -36. This suggests the following: Solving Equations of the Form 0 X 0 a If a is a positive number, then 0 X 0 = a is equivalent to X = a or X = -a. EXAMPLE 1 Solve: 0 p 0 = Solution Sine 2 is positive, p = 2 is equivalent to p = 2 or p = -2. To hek, let p = 2 and then p = - 2 in the original equation. p = 2 Original equation p = 2 Original equation 2 = 2 Let p = = 2 Let p = = 2 True 2 = 2 True The solutions are 2 and -2 or the solution set is 52, Solve: q = 3.

2 Setion 2.6 Absolute Value Equations 97 If the epression inside the absolute value bars is more ompliated than a single variable, we an still apply the absolute value property. Helpful Hint For the equation 0 X 0 = a in the bo on the previous page, X an be a single variable or a variable epression. EXAMPLE 2 Solve: 0 5w = 7. Solution Here the epression inside the absolute value bars is 5w + 3. If we think of the epression 5w + 3 as X in the absolute value property, we see that 0 X 0 = 7 is equivalent to X = 7 or X = -7 Then substitute 5w + 3 for X, and we have 5w + 3 = 7 or 5w + 3 = -7 Solve these two equations for w. 5w + 3 = 7 or 5w + 3 = -7 5w = 4 or 5w = -10 w = 4 5 or w = -2 Chek: To hek, let w = -2 and then w = 4 in the original equation. 5 Let w = -2 Let w = = 7 ` 5a 4 5 b + 3 ` = = = 7-7 = 7 7 = 7 7 = 7 True 7 = 7 True Both solutions hek, and the solutions are -2 and 4 5 or the solution set is e -2, 4 5 f. 2 Solve: 2-3 = 5. EXAMPLE 3 Solution ` 2 Solve: ` 2-1 ` = ` = 11 is equivalent to 2-1 = 11 or 2-1 = -11 2a 2-1b = or 2a 2-1b = Clear frations. - 2 = 22 or - 2 = -22 Apply the distributive property. = 24 or = -20 The solutions are 24 and Solve: ` ` = 15.

3 98 CHAPTER 2 Equations, Inequalities, and Problem Solving To apply the absolute value property, first make sure that the absolute value epression is isolated. Helpful Hint If the equation has a single absolute value epression ontaining variables, isolate the absolute value epression first. EXAMPLE 4 Solve: = 7. Solution We want the absolute value epression alone on one side of the equation, so begin by subtrating 5 from both sides. Then apply the absolute value property = = 2 Subtrat 5 from both sides. 2 = 2 or 2 = -2 = 1 or = -1 The solutions are -1 and 1. 4 Solve: = 14. EXAMPLE 5 Solve: 0 y 0 = 0. Solution We are looking for all numbers whose distane from 0 is zero units. The only number is 0. The solution is 0. 5 Solve: z = 0. The net two eamples illustrate a speial ase for absolute value equations. This speial ase ours when an isolated absolute value is equal to a negative number. EXAMPLE 6 Solve: = 23. Solution First, isolate the absolute value = = -2 Subtrat 25 from both sides. 0 0 = -1 Divide both sides by 2. The absolute value of a number is never negative, so this equation has no solution. The solution set is 5 6 or. 6 Solve: 3 z + 9 = 7. EXAMPLE 7 Solve: ` ` = Solution Again, the absolute value of any epression is never negative, so no solution eists. The solution set is 5 6 or. 7 Solve: ` ` = -8. 4

4 Setion 2.6 Absolute Value Equations 99 Given two absolute value epressions, we might ask, when are the absolute values of two epressions equal? To see the answer, notie that = 0 2 0, = 0-2 0, = 0 2 0, and = same same opposites opposites Two absolute value epressions are equal when the epressions inside the absolute value bars are equal to or are opposites of eah other. EXAMPLE 8 Solve: = Solution This equation is true if the epressions inside the absolute value bars are equal to or are opposites of eah other = 5-8 or = Net, solve eah equation = 5-8 or = = -8 or = 8-2 = -10 or 8 = 6 The solutions are 3 and 5. 4 = 5 or = Solve: = 3-1. EXAMPLE 9 Solve: = Solution - 3 = 5 - or - 3 = = 5 or - 3 = = 8 or = = 4 or -3 = -5 False Reall from Setion 2.1 that when an equation simplifies to a false statement, the equation has no solution. Thus, the only solution for the original absolute value equation is 4. 9 Solve: - 2 = 8 -. CONCEPT CHECK True or false? Absolute value equations always have two solutions. Eplain your answer. The following bo summarizes the methods shown for solving absolute value equations. Answer to Conept Chek: false; answers may vary Absolute Value Equations If a is positive, then solve X = a or X = -a. 0 X 0 = a If a is 0, solve X = 0. If a is negative, the equation 0 X 0 = a has no solution. 0 X 0 = 0 Y 0 Solve X = Y or X = -Y.

5 100 CHAPTER 2 Equations, Inequalities, and Problem Solving Voabulary, Readiness & Video Chek Math eah absolute value equation with an equivalent statement = = = = = -5 A. - 2 = 0 B. - 2 = + 3 or - 2 = C. - 2 = 5 or - 2 = -5 D. E. + 3 = 5 or + 3 = -5 Martin-Gay Interative Videos Wath the setion leture videos and answer the following question As eplained in Eample 3, why is a positive in the rule 0 X 0 = a is equivalent to X = a or X = -a? See Video Eerise Set Solve eah absolute value equation. See Eamples 1 through = 7 2. y = = n = = n 0 = 4 7. ` 2-3 ` = 1 8. ` n ` = z = = = = = z 0 = n = z = = y = 0 Solve. See Eamples 8 and = y = 0 6y z = 0 z = MIXED Solve eah absolute value equation. See Eamples 1 through = = y = y = z 0 = y 0 = = m = = z = p 0 = m 0 = = = 1 z 37. ` ` = ` 5-1 ` = v - 3 = b = n + 1 = = = m - 9 = = = = = = n = 4n = y 0 = = n = 0 4n ` 2-5 ` = ` n 0 = m 0 = ` 2-1 ` = ` y = 0 9-4y ` 3n + 2 ` = n ` = ` = = y 0 = 0 y ` 8-7 ` = ` z = z ` 2r - 6 ` = d + 1 ` =

6 Setion 2.7 Absolute Value Inequalities 101 REVIEW AND PREVIEW The irle graph shows the types of heese produed in the United States in Use this graph to answer Eerises 69 through 72. See Setion 2.2. Swiss 3.2% Hispani 2.1% Cream Cheese 7.1% Muenster 1.1% Brik 0.1% U.S. Cheese 1 Prodution by Variety, 2010 All Others 3.0% Other Italian 9.0% 1 Eludes Cottage Cheese Soure: USDA, Dairy Produts Annual Survey Cheddar 31.0% Mozzarella 33.4% Other Amerian 10.0% 69. In 2010, heddar heese made up what perent of U.S. heese prodution? 70. Whih heese had the highest U.S. prodution in 2010? 71. A irle ontains 360. Find the number of degrees found in the 9% setor for Other Italian Cheese. 72. In 2010, the total prodution of heese in the United States was 10,109,293,000 pounds. Find the amount of ream heese produed during that year. List five integer solutions of eah inequality. See Setions 1.2 through Ú y y CONCEPT EXTENSIONS Without going through a solution proedure, determine the solution of eah absolute value equation or inequality = Write an absolute value equation representing all numbers whose distane from 0 is 5 units. 80. Write an absolute value equation representing all numbers whose distane from 0 is 2 units. 81. Eplain why some absolute value equations have two solutions. 82. Eplain why some absolute value equations have one solution. 83. Write an absolute value equation representing all numbers whose distane from 1 is 5 units. 84. Write an absolute value equation representing all numbers whose distane from 7 is 2 units. 85. Desribe how solving an absolute value equation suh as = 3 is similar to solving an absolute value equation suh as = Desribe how solving an absolute value equation suh as = 3 is different from solving an absolute value equation suh as = Write eah as an equivalent absolute value equation. 87. = 6 or = = 4 or 2-1 = = 3-4 or - 2 = For what value(s) of will an absolute value equation of the form 0 a + b 0 = have a. one solution? b. no solution?. two solutions? 2.7 Absolute Value Inequalities S 1 Solve Absolute Value Inequalities of the Form 0 X 0 6 a. 2 Solve Absolute Value Inequalities of the Form 0 X 0 7 a. 1 Solving Absolute Value Inequalities of the Form 0 X 0 * a The solution set of an absolute value inequality suh as ontains all numbers whose distane from 0 is less than 2 units, as shown below. Distane from 0: less than 2 units Distane from 0: less than 2 units The solution set is , or 1-2, 22 in interval notation. EXAMPLE 1 Solve: and graph the solution set. Solution The solution set of this inequality ontains all numbers whose distane from 0 is less than or equal to 3. Thus 3, -3, and all numbers between 3 and -3 are in the solution set.