Solve Absolute Value Equations and Inequalities

1.7 Solve Absolute Value Equations and Inequalities Before You solved linear equations and inequalities. Now You will solve absolute value equations and inequalities. Why? So you can describe hearing ranges of animals, as in Ex. 81. Key Vocabulary absolute value extraneous solution Recall that the absolute value of a number x, written x, is the distance the number is from 0 on a number line. This understanding of absolute value can be extended to apply to simple absolute value equations. KEY CONCEPT Interpreting Absolute Value Equations x, if x is positive x 5 0, if x 5 0 x, if x is negative For Your Notebook Equation x 5 x 0 5 k x b 5 k Meaning The distance between x and 0 is k. The distance between x and b is k. Graph k k k k k 0 k Solutions x 0 5 k or x 0 5 k x 5 k or x 5 k b k b b 1 k x b 5 k or x b 5 k x 5 b k or x 5 b 1 k E XAMPLE 1 Solve a simple absolute value equation Solve x 5 5 7. Graph the solution. Solution x 5 5 7 Write original equation. x 5 5 7 or x 5 5 7 Write equivalent equations. x 5 5 7 or x 5 5 1 7 Solve for x. x 5 or x 5 1 Simplify. c The solutions are and 1. These are the values of x that are 7 units away from 5 on a number line. The graph is shown below. 7 7 0 4 5 6 8 10 1 1.7 Solve Absolute Value Equations and Inequalities 51

KEY CONCEPT For Your Notebook Solving an Absolute Value Equation Use these steps to solve an absolute value equation ax 1 b 5 c where c > 0. STEP 1 Write two equations: ax 1 b 5 c or ax 1 b 5 c. STEP Solve each equation. STEP 3 Check each solution in the original absolute value equation. E XAMPLE Solve an absolute value equation Solve 5x 10 5 45. 5x 10 5 45 Write original equation. 5x 10 5 45 or 5x 10 5 45 Expression can equal 45 or 45. 5x 5 55 or 5x 5 35 Add 10 to each side. x 5 11 or x 5 7 Divide each side by 5. c The solutions are 11 and 7. Check these in the original equation. CHECK 5x 10 5 45 5x 10 5 45 5(11) 10 0 45 5(7) 10 0 45 45 0 45 45 0 45 45 5 45 45 5 45 EXTRANEOUS SOLUTIONS When you solve an absolute value equation, it is possible for a solution to be extraneous. An extraneous sol ution is an apparent solution that must be rejected because it doe s not satisfy the original equation. E XAMPLE 3 Check for extraneous solutions Solve x 1 1 5 4x. Check for extraneous solutions. x 1 1 5 4x Write original equation. x 1 1 5 4x or x 1 1 5 4x Expression can equal 4x or 4x. 1 5 x or 1 5 6x Subtract x from each side. 6 5 x or 5 x Solve for x. AVOID ERRORS Always check your solutions in the original equation to make sure that they are not extraneous. Check the apparent solutions to see if either is extraneous. CHECK x 1 1 5 4x x 1 1 5 4x (6) 1 1 0 4(6) () 1 1 0 4() 4 0 4 8 0 8 4 5 4 8? 8 c The solution is 6. Reject because it is an extraneous solution. 5 Chapter 1 Equations and Inequalities

GUIDED PRACTICE for Examples 1,, and 3 Solve the equation. Check for extraneous solutions. 1. x 5 5. x 3 5 10 3. x 1 5 7 4. 3x 5 13 5. x 1 5 5 3x 6. 4x 1 5 x 1 9 INEQUALITIES You can solve an absolute value inequality by rewriting it as a compound inequality and then solving each part. KEY CONCEPT For Your Notebook Absolute Value Inequalities Inequality Equivalent form Graph of solution ax 1 b < c ax 1 b c ax 1 b > c ax 1 b c c < ax 1 b < c c ax 1 b c ax 1 b < c or ax 1 b > c ax 1 b c or ax 1 b c E XAMPLE 4 Solve an inequality of the form ax 1 b > c Solve 4x 1 5 > 13. Then graph the solution. Solution The absolute value inequality is equivalent to 4x 1 5 < 13 or 4x 1 5 > 13. First Inequality Second Inequality 4 x 1 5 < 13 Write inequalities. 4x 1 5 > 13 4x < 18 Subtract 5 from each side. 4x > 8 x < 9 } Divide each side by 4. x > c The solutions are all real numbers less than } 9 or greater than. The graph is shown below. 6 5 4 3 1 0 1 3 4 at classzone.com GUIDED PRACTICE for Example 4 Solve the inequality. Then graph the solution. 7. x 1 4 6 8. x 7 > 1 9. 3x 1 5 10 1.7 Solve Absolute Value Equations and Inequalities 53

E XAMPLE 5 Solve an inequality of the form ax 1 b c READING Tolerance is the maximum acceptable deviation of an item from some ideal or mean measurement. BASEBALL A professional baseball should weigh 5.15 ounces, with a tolerance of 0.15 ounce. Write and solve an absolute value inequality that describes the acceptable weights for a baseball. Solution STEP 1 Write a verbal model. Then write an inequality. Actual weight (ounces) Ideal weight (ounces) Tolerance (ounces) STEP w 5.15 0.15 Solve the inequality. w 5.15 0.15 Write inequality. 0.15 w 5.15 0.15 Write equivalent compound inequality. 5 w 5.5 Add 5.15 to each expression. c So, a baseball should weigh between 5 ounces and 5.5 ounces, inclusive. The graph is shown below. 0.15 0.15 4.875 5.000 5.15 5.50 5.375 E XAMPLE 6 Write a range as an absolute value inequality GYMNASTICS The thickness of the mats used in the rings, parallel bars, and vault events must be between 7.5 inches and 8.5 inches, inclusive. Write an absolute value inequality describing the acceptable mat thicknesses. REVIEW MEAN For help with finding a mean, see p. 1005. Solution STEP 1 STEP STEP 3 Calculate the mean of the extreme mat thicknesses. Mean of extremes 5 7.5 1 8.5 } 5 7.875 Find the tolerance by subtracting the mean from the upper extreme. Tolerance 5 8.5 7.875 5 0.375 Write a verbal model. Then write an inequality. Actual thickness (inches) Mean of extremes (inches) Tolerance (inches) t 7.875 0.375 c A mat is acceptable if its thickness t satisfies t 7.875 0.375. 54 Chapter 1 Equations and Inequalities

GUIDED PRACTICE for Examples 5 and 6 Solve the inequality. Then graph the solution. 10. x 1 < 6 11. x 1 1 9 1. 7 x 4 13. GYMNASTICS For Example 6, write an absolute value inequality describing the unacceptable mat thicknesses. 1.7 EXERCISES SKILL PRACTICE HOMEWORK KEY 5 WORKED-OUT SOLUTIONS on p. WS1 for Exs. 1, 47, and 77 5 STANDARDIZED TEST PRACTICE Exs., 33, 40, 63, and 64 5 MULTIPLE REPRESENTATIONS Ex. 78 1. VOCABULARY What is an extraneous solution of an equation?. WRITING The absolute value of a number cannot be negative. How, then, can the absolute value of x be x for certain values of x? CHECKING SOLUTIONS Decide whether the given number is a solution of the equation. 3. b 1 5 14; 13 4. d 1 6 5 10; 4 5. 3 6f 5 0; 6. m 1 6 5 10; 8 7. 3n 7 5 4; 1 8. 17 8r 5 15; 4 EXAMPLE 1 on p. 51 for Exs. 9 0 SOLVING EQUATIONS Solve the equation. Graph the solution. 9. x 5 9 10. y 5 5 11. z 5 0 1. f 5 5 3 13. g 5 7 14. h 4 5 4 15. k 1 3 5 6 16. m 1 5 5 1 17. n 1 9 5 10 18. 6 p 5 4 19. 5 q 5 7 0. 4 r 5 4 EXAMPLE on p. 5 for Exs. 1 3 SOLVING EQUATIONS Solve the equation. 1. d 5 5 13. 3g 1 14 5 7 3. 7h 10 5 4 4. 3p 6 5 1 5. q 1 3 5 11 6. 4r 1 7 5 43 7. 5 1 j 5 9 8. 6 3k 5 1 9. 0 9m 5 7 30. 1 } 4 x 3 5 10 31. 1 } y 1 4 5 6 3. } 3 z 6 5 1 33. SHORT RESPONSE The equation 5x 10 5 45 in Example has two solutions. Does the equation 5x 10 5 45 also have two solutions? Explain. EXAMPLE 3 on p. 5 for Exs. 34 4 EXTRANEOUS SOLUTIONS Solve the equation. Check for extraneous solutions. 34. 3x 4 5 x 35. x 1 4 5 7x 36. 8x 1 5 6x 37. 4x 1 5 5 x 1 4 38. 9 x 5 10 1 3x 39. 8 1 5x 5 7 x 1.7 Solve Absolute Value Equations and Inequalities 55

40. MULTIPLE CHOICE What is (are) the solution(s) of 3x 1 7 5 5x? A 4, } 3 B } 7 8, 7 } C 7 } 8, 7 } D 7 } ERROR ANALYSIS Describe and correct the error in solving the equation. 41. 5x 9 5 x 1 3 4. n 7 5 3n 1 5x 9 5 x 1 3 or 5x 9 5 x 1 3 n 7 5 3n 1 or n 7 5 3n 1 1 4x 9 5 3 or 6x 9 5 3 7 5 n 1 or 4n 7 5 1 4x 5 1 or 6x 5 1 6 5 n or 4n 5 8 x 5 3 or x 5 3 5 n or n 5 The solutions are 3 and. The solutions are 3 and. EXAMPLES 4 and 5 on pp. 53 54 for Exs. 43 63 SOLVING INEQUALITIES Solve the inequality. Then graph the solution. 43. j 5 44. k > 4 45. m < 7 46. n 11 1 47. d 1 4 3 48. f 1 6 < 49. g 1 > 0 50. h 1 10 10 51. 3w 15 < 30 5. x 1 6 10 53. 4y 9 7 54. 5z 1 1 > 14 55. 16 p > 3 56. 4 q 11 57. 7 r < 19 58. 19 5t > 7 59. 1 } x 10 4 60. 1 } 3 m 15 < 6 61. 1 } 7 y 1 5 > 3 6. } 5 n 8 1 4 1 at classzone.com 63. MULTIPLE CHOICE What is the solution of 6x 9 33? A 4 x 7 B 7 x 4 C x 4 or x 7 D x 7 or x 4 64. MULTIPLE CHOICE Which absolute value inequality represents the graph shown below? 1 0 1 3 4 5 6 A 1 < x < 5 B x 1 < 3 C x < 3 D x < 5 65. REASONING For the equation ax 1 b 5 c (where a, b, and c are real numbers and a Þ 0), describe the value(s) of c that yield two solutions, one solution, and no solution. SOLVING INEQUALITIES Solve the inequality. Then graph the solution. 66. x 1 1 16 67. x 1 < 5 68. 7x 1 3 0 69. x 9 > 0 CHALLENGE Solve the inequality for x in terms of a, b, and c. Assume a, b, and c are real numbers. 70. ax 1 b < c where a > 0 71. ax 1 b c where a > 0 7. ax 1 b c where a < 0 73. ax 1 b > c where a < 0 5 WORKED-OUT SOLUTIONS 56 Chapter 1 Equations p. WS1and Inequalities 5 STANDARDIZED TEST PRACTICE 5 MULTIPLE REPRESENTATIONS

PROBLEM SOLVING EXAMPLE 5 on p. 54 for Exs. 74 78 74. GYMNASTICS The horizontal bar used in gymnastics events should be placed 110.5 inches above the ground, with a tolerance of 0.4 inch. Write an absolute value inequality for the acceptable bar heights. 75. SOIL PH LEVELS Cucumbers grow in soil having a ph level of 6.5, with a tolerance of 1 point on the ph scale. Write an absolute value inequality that describes the ph levels of soil in which cucumbers can grow. 76. MULTI-STEP PROBLEM A baseball has a cushioned cork center called the pill. The pill must weigh 0.85 ounce, with a tolerance of 0.05 ounce. a. Write an absolute value inequality that describes the acceptable weights for the pill of a baseball. b. Solve the inequality to find the acceptable weights for the pill. c. Look back at Example 5 on page 54. Find the minimum and maximum percentages of a baseball s total weight that the pill can make up. 77. MANUFACTURING A regulation basketball should weigh 1 ounces, with a tolerance of 1 ounce. Write an absolute value inequality describing the weights of basketballs that should be rejected. 78. MULTIPLE REPRESENTATIONS The strength of eyeglass lenses is measured in units called diopters. The diopter number x is negative for nearsighted vision and positive for farsighted vision. Nearsightedness (focus is in front of retina) Farsightedness (focus is behind retina) Mild x 1 1.5 < 1.5 Mild x 1 < 1 Moderate x 1 4.5 < 1.5 Moderate x 3 < 1 Severe x 1 7.5 < 1.5 Severe x 5 < 1 a. Writing Inequalities Write an equivalent compound inequality for each vision category shown above. Solve the inequalities. b. Making a Graph Illustrate the six vision categories by graphing their ranges of diopter numbers on the same number line. Label each range with the corresponding category name. EXAMPLE 6 on p. 54 for Exs. 79 81 79. SLEEPING BAGS A manufacturer of sleeping bags suggests that one model is best suited for temperatures between 308F and 608F, inclusive. Write an absolute value inequality for this temperature range. 80. TEMPERATURE The recommended oven setting for cooking a pizza in a professional brick-lined oven is between 5508F and 6508F, inclusive. Write an absolute value inequality for this temperature range. 1.7 Solve Absolute Value Equations and Inequalities 57

81. AUDIBLE FREQUENCIES An elephant can hear sounds with frequencies from 16 hertz to 1,000 hertz. A mouse can hear sounds with frequencies from 1000 hertz to 91,000 hertz. Write an absolute value inequality for the hearing range of each animal. 8. CHALLENGE The depth finder on a fishing boat gives readings that are within 5% of the actual water depth. When the depth finder reading is 50 feet, the actual water depth x lies within a range given by the following inequality: x 50 0.05x a. Write the absolute value inequality as a compound inequality. b. Solve each part of the compound inequality for x. What are the possible actual water depths if the depth finder s reading is 50 feet? MIXED REVIEW PREVIEW Prepare for Lesson.1 in Exs. 83 94. Plot the points in the same coordinate plane. (p. 987) 83. (4, 4) 84. (7, 8) 85. (3, 0) 86. (0, 6) 87. (, 3) 88. (5, ) Evaluate the expression for the given value of the variable. (p. 10) 89. 6m 10; m 5 4 90. 4n 1 18; n 5 3 91. 5p 1 17; p 5 0 9. 7q 1 3; q 5 4 93. r 3; r 5 7 94. 10t 5; t 5 3 Solve the equation for y. Then find the value of y for the given value of x. (p. 6) 95. 5x 1 y 5 14; x 5 8 96. 3x 1 y 5 1; x 5 9 97. 8x 4y 5 3; x 5 3 98. 6x 1 15y 5 33; x 5 10 QUIZ for Lessons 1.6 1.7 Solve the inequality. Then graph the solution. (p. 41) 1. 4k 17 < 7. 14n 8 90 3. 9p 1 15 96 4. 8r 11 > 45 5. 3(x 7) < 6(10 x) 6. 5 4z > 66 17z Solve the equation or inequality. (p. 51) 7. x 6 5 9 8. 3y 1 3 5 1 9. z 1 5 5 9z 10. p 1 7 > 11. q 3 3 1. 5 r 4 13. TEST SCORES Your final grade in a course is 80% of your current grade, plus 0% of your final exam score. Your current grade is 83 and your goal is to get a final grade of 85 or better. Write and solve an inequality to find the final exam scores that will meet your goal. (p. 41) 14. GROCERY WEIGHTS A container of potato salad from your grocer s deli is supposed to weigh 1.5 pounds, with a tolerance of 0.05 pound. Write and solve an absolute value inequality that describes the acceptable weights for the container of potato salad. (p. 51) 58 Equations EXTRA PRACTICE and Inequalities for Lesson 1.7, p. 1010 ONLINE QUIZ at classzone.com

MIXED REVIEW of Problem Solving STATE TEST PRACTICE classzone.com Lessons 1.5 1.7 1. MULTI-STEP PROBLEM A hybrid car gets about 60 miles per gallon of gas in the city and about 51 miles per gallon on the highway. During one week, the hybrid uses 1 gallons of gas and travels 675 miles. a. Write a verbal model for the total distance driven. Then write an equation based on the verbal model. b. Solve the equation to find the amounts of gas used in the city and on the highway. c. Tell how many miles were driven in the city and on the highway.. MULTI-STEP PROBLEM A popcorn manufacturer s ideal weight for a bag of microwave popcorn is 3.5 ounces, with a tolerance of 0.5 ounce. a. Write an absolute value inequality for the acceptable weights of a bag of popcorn. b. Solve the inequality. What is the range of acceptable weights? 3. EXTENDED RESPONSE You are draining a swimming pool. The table shows the depth of the water at different times. Time (h), t 0 1 3 Depth (ft), d 1 10.5 9 7.5 a. Write an equation for the depth. b. Use your equation to find how long it will take for the pool to be empty. c. Does your equation make sense for times greater than the value you found in part (b)? Explain. 4. OPEN-ENDED For a rope trick, a magician cuts a 7 inch piece of rope into three pieces of different lengths. The length of one piece must be the mean of the lengths of the other two pieces. short long medium 1 a b (a 1 b) 7 in. 5. SHORT RESPONSE A video store rents movies for $.95 each. Recently, the store has added a special deal that allows you to rent an unlimited number of movies for $15.95 per month. Explain when the special deal is less expensive than renting movies at the usual price. Write and solve an inequality to justify your answer. 6. EXTENDED RESPONSE The triangle inequality relationship from geometry states that the sum of the lengths of any two sides of a triangle is greater than the length of the third side. x 9 a. Write three different inequalities for the triangle. b. Solve the three inequalities for x. c. Based on your results from part (b), what is the range of possible values for x? d. Use your results to draw three different triangles that meet the conditions shown in the diagram. 7. MULTI-STEP PROBLEM Oxygen exists as a liquid between 3698F and 978F, inclusive. Write the temperature range for liquid oxygen as a compound inequality in degrees Fahrenheit. Then rewrite the temperature range in degrees Celsius. x 8. GRIDDED ANSWER A football kicker scores 1 point for each extra point and 3 points for each field goal. One season, a kicker made 34 extra points and scored a total of 11 points. How many field goals did the kicker make? Find the length of the medium piece. Then give possible lengths for the short and long pieces. Mixed Review of Problem Solving 59