120 (2-56) Chapter 2 Linear Equations and Inequalities in One Variable x? FIGURE FOR EXERCISE 8 84. Building a ski ramp. Write an inequality in the variable x for the degree measure of the smallest angle of the triangle shown in the figure, given that the degree measure of the smallest angle is at most 0. 180 x (x 8) 0 x 8 x? FIGURE FOR EXERCISE 84 85. Maximum girth. United Parcel Service defines the girth of a box as the sum of the length, twice the width, and twice the height. The maximum girth that UPS will ship is 10 in. If a box has a length of 45 in. and a width of 0 in., then what inequality must be satisfied by the height? 45 2(0) 2h 10 86. Batting average. At one point during the 1997 season, Jay Lopez of the Atlanta Braves had 9 hits in 17 times at bat for an average of 9 17 or 0.29. If he gets x hits in the next 20 times at bat to get his average over 0.00, then what 9 x inequality must x satisfy? 0.00 17 20 Solve. 87. Bicycle gear ratios. The gear ratio r for a bicycle is defined by the formula Nw r, n FIGURE FOR EXERCISE 86 where N is the number of teeth on the chainring (by the pedal), n is the number of teeth on the cog (by the wheel), and w is the wheel diameter in inches (Cycling, Burkett and Darst). The following chart gives uses for the various gear ratios. Ratio r 90 Use hard pedaling on level ground 70 r 90 moderate effort on level ground 50 r 70 mild hill climbing 5 r 50 long hill climbing with load A bicycle with a 27-inch diameter wheel has 50 teeth on the chainring and 17 teeth on the cog. Find the gear ratio and indicate what this gear ratio is good for. 79, moderate effort on level ground In this section Rules for Inequalities Solving Inequalities Applications of Inequalities 2.9 SOLVING INEQUALITIES AND APPLICATIONS To solve equations, we write a sequence of equivalent equations that ends in a very simple equation whose solution is obvious. In this section you will learn that the procedure for solving inequalities is the same. However, the rules for performing operations on each side of an inequality are slightly different from the rules for equations.
2.9 Solving Inequalities and Applications (2-57) 121 helpful hint You can think of an inequality like a seesaw that is out of balance. 50 > 20 Rules for Inequalities Equivalent inequalities are inequalities that have exactly the same solutions. Inequalities such as x > and x 2 5 are equivalent because any number that is larger than certainly satisfies x 2 5 and any number that satisfies x 2 5 must certainly be larger than. We can get equivalent inequalities by performing operations on each side of an inequality just as we do for solving equations. If we start with the inequality 6 10 and add 2 to each side, we get the true statement 8 12. Examine the results of performing the same operation on each side of 6 10. Perform these operations on each side: If the same weight is added to or subtracted from each side, it will remain in the same state of imbalance. study tip Get in the habit of checking your work and having confidence in your answers. The answers to the odd-numbered exercises are in the back of this book, but you should look in the answer section only after you have checked on your own. You will not always have an answer section available. Add 2 Subtract 2 Multiply by 2 Divide by 2 Start with 6 10 8 12 4 8 12 20 5 All of the resulting inequalities are correct. Now if we repeat these operations using 2, we get the following results. Perform these operations on each side: Add 2 Subtract 2 Multiply by 2 Divide by 2 Start with 6 10 4 8 8 12 12 20 5 Notice that the direction of the inequality symbol is the same for all of the results except the last two. When we multiplied each side by 2 and when we divided each side by 2, we had to reverse the inequality symbol to get a correct result. These tables illustrate the rules for solving inequalities. Addition Property of Inequality If we add the same number to each side of an inequality we get an equivalent inequality. If a b, then a c b c. The addition property of inequality also allows us to subtract the same number from each side of an inequality because subtraction is defined in terms of addition. helpful hint Changing the signs of numbers, changes their relative position on the number line. For example, lies to the left of 5 on the number line,but lies to the right of 5. So 5, but 5. Since multiplying and dividing by a negative cause sign changes, these operations reverse the inequality. Multiplication Property of Inequality If we multiply each side of an inequality by the same positive number, we get an equivalent inequality. If a b and c 0, then ac bc. If we multiply each side of an inequality by the same negative number and reverse the inequality symbol, we get an equivalent inequality. If a b and c 0, then ac bc. The multiplication property of inequality also allows us to divide each side of an inequality by a nonzero number because division is defined in terms of multiplication. So if we multiply or divide each side by a negative number, the inequality symbol is reversed.
122 (2-58) Chapter 2 Linear Equations and Inequalities in One Variable E X A M P L E 1 Writing equivalent inequalities Write the appropriate inequality symbol in the blank so that the two inequalities are equivalent. a) x 9, x 6 b) 2x 6, x a) If we subtract from each side of x 9, we get the equivalent inequality x 6. b) If we divide each side of 2x 6 by 2, we get the equivalent inequality x. CAUTION We use the properties of inequality just as we use the properties of equality. However, when we multiply or divide each side by a negative number, we must reverse the inequality symbol. Solving Inequalities To solve inequalities, we use the properties of inequality to isolate x on one side. E X A M P L E 2 Using the properties of inequality Solve and graph the inequality 4x 5 19. 1 2 4 5 6 7 8 FIGURE 2.1 4x 5 19 Original inequality 4x 5 5 19 5 Add 5 to each side. 4x 24 Simplify. x 6 Divide each side by 4. Since the last inequality is equivalent to the first, they have the same solutions and the same graph, which is shown in Fig. 2.1. In the next example we divide each side of an inequality by a negative number. calculator close-up You can use the TABLE feature of a graphing calculator to numerically support the solution to the inequality 4x 5 19 in Example 2. Use the Y = key to enter the equation y 1 4x 5. Next,use TBLSET to set the table so that the values of x start at 4.5 and the change in x is 0.5. Notice that when x is larger than 6, y 1 (or 4x 5) is larger than 19. Note that this table is not a method for solving an inequality, it is merely a way of verifying or supporting the algebraic solution. Finally, press TABLE to see lists of x-values and the corresponding y-values.
2.9 Solving Inequalities and Applications (2-59) 12 E X A M P L E Reversing the inequality symbol Solve and graph the inequality 5 5x 1 2(5 x ). 6 5 4 2 1 0 1 FIGURE 2.14 5 5x 1 2(5 x) Original inequality 5 5x 11 2x Simplify the right side. 5 x 11 Add 2x to each side. x 6 Subtract 5 from each side. x 2 Divide each side by, and reverse the inequality. The inequalities 5 5x 1 2(5 x) and x 2 have the same graph, which is shown in Fig. 2.14. We can use the rules for solving inequalities on the compound inequalities that we studied in Section 2.8. E X A M P L E 4 Solving a compound inequality Solve and graph the inequality 9 2 x 7 5. 9 2 x 7 5 Original inequality 9 7 2 x 7 7 5 7 2 2 x 12 Add 7 to each part. Simplify. 0 6 9 12 15 18 FIGURE 2.15 2 ( 2) 2 2 x 2 12 Multiply each part by 2. x 18 Simplify. Any number that satisfies x 18 also satisfies the original inequality. Figure 2.15 shows all of the solutions to the original inequality. CAUTION There are many negative numbers in Example 4, but the inequality was not reversed, since we did not multiply or divide by a negative number. An inequality is reversed only if you multiply or divide by a negative number. E X A M P L E 5 Reversing inequality symbols in a compound inequality Solve and graph the inequality 5 x 5. 5 x 5 5 5 x 5 5 5 8 x 0 ( 1)( 8) ( 1)( x ) ( 1)(0) 8 x 0 Original inequality Subtract 5 from each part. Simplify. Multiply each part by 1, reversing the inequality symbols.
124 (2-60) Chapter 2 Linear Equations and Inequalities in One Variable It is customary to write 8 x 0 with the smallest number on the left: 0 x 8 Figure 2.16 shows all numbers that satisfy 5 x 5. 1 0 1 2 4 5 6 7 8 FIGURE 2.16 9 Applications of Inequalities The following example shows how inequalities can be used in applications. E X A M P L E 6 Averaging test scores Mei Lin made a 76 on the midterm exam in history. To get a B, the average of her midterm and her final exam must be between 80 and 90. For what range of scores on the final exam will she get a B? helpful hint Remember that all inequality symbols in a compound inequality must point in the same direction. We usually have them all point to the left so that the numbers are increasing in size as you go from left to right in the inequality. Let x represent the final exam score. Her average is then x 76. The inequality 2 expresses the fact that the average must be between 80 and 90: 80 x 76 90 2 2(80) 2 x 76 2 2(90) Multiply each part by 2. 160 x 76 180 Simplify. 160 76 x 76 76 180 76 Subtract 76 from each part. 84 x 104 Simplify. The last inequality indicates that Mei Lin s final exam score must be between 84 and 104. WARM-UPS True or false? Explain your answer. 1. The inequality 2x 18 is equivalent to x 9. True 2. The inequality x 5 0 is equivalent to x 5. False. We can divide each side of an inequality by any real number. False 4. The inequality 2x 6 is equivalent to x. True 5. The statement x is at most 7 is written as x 7. False 6. The sum of x and 0.05x is at least 76 is written as x 0.05x 76. True 7. The statement x is not more than 85 is written as x 85. False 8. The inequality x 9 is equivalent to 9 x. True 9. If x is the sale price of Glen s truck, the sales tax rate is 8%, and the title fee is $50, then the total that he pays is 1.08x 50 dollars. True 10. If the selling price of the house, x, less the sales commission of 6% must be at least $60,000, then x 0.06x 60,000. False
2.9 Solving Inequalities and Applications (2-61) 125 2. 9 EXERCISES Reading and Writing After reading this section, write out the answers to these questions. Use complete sentences. 1. What are equivalent inequalities? Equivalent inequalities are inequalities that have the same solutions. 2. What is the addition property of inequality? The addition property of inequality says that adding any real number to each side of an inequality produces an equivalent inequality.. What is the multiplication property of inequality? According to the multiplication property of inequality, the inequality symbol is reversed when multiplying (or dividing) by a negative number and not reversed when multiplying (or dividing) by a positive number. 4. What similarities are there between solving equations and solving inequalities? For equations or inequalities we try to isolate the variable. The properties of equality and inequality are similar. 5. How do we solve compound inequalities? We solve compound inequalities using the properties of inequality as we do for simple inequalities. 6. How do you know when to reverse the direction of an inequality symbol? The direction of the inequality symbol is reversed when we multiply or divide by a negative number. Write the appropriate inequality symbol in the blank so that the two inequalities are equivalent. See Example 1. 7. x 7 0 8. x 6 0 x 7 x 6 x 7 x 6 9. 9 w 10. 10 5z w z 2 w z 2 11. 4k 4 12. 9t 27 k 1 t k 1 t 1. 1 2 y 4 14. 1 x 4 y 8 x 12 y 8 x 12 Solve and graph each of the following inequalities. See Examples 2 and. 15. x 0 x 16. x 9 8 x 17 17. w 1 w 2 18. 9 w 12 w 21 19. 8 2b b 4 20. 5 7b b 5 21. 4z 8 z 2 22. 5y 20 y 4 2. y 2 7 y 24. 2y 5 9 y 2 25. 7z 17 z 2 26. 5 z 20 z 5 27. 6 r r 28. 6 12 r r 6 29. 5 4p 8 p p 1 0. 7 9p 11 8p p 4 1. 5 q 20 q 24 6 2. 2 q 4 q 6. 1 1 t 2 t 12 4 4. 2 1 t 0 t 6 5. 2x 5 x 6 x 11 6. x 4 2x 9 x 1 7. x 4 2(x ) x 10 8. 2x (x 5) x 18 9. 0.52x 5 0.45x 8 x 614. 40. 8455(x.4) 420 x.91
126 (2-62) Chapter 2 Linear Equations and Inequalities in One Variable Solve and graph each compound inequality. See Examples 4 and 5. 41. 5 x 7 8 x 10 42. 2 x 5 6 7 x 11 4. 2v 1 10 1 v 9 2 44. v 4 7 7 v 1 45. 4 5 k 7 2 k 9 46. 2 k 8 5 k 1 47. 2 7 y 22 5 y 48. 1 1 2y 1 y 1 49. 5 2 u 17 12 u 0 50. 4 u 1 11 4 4 u 16 51. 7 m 1 8 2 5 m 5 52. 0 2m 9 2 1 5 x 2 2 5. 0.02 0.54 0.0048x 0.05 102.1 x 108. 54. 0.44 4.55 22.x 0.76 124.5 2.69 x 0.91 Solve and graph each inequality. 55. 1 2 x 1 4 1 x x 6 y 5 y 1 56. 1 y 8 4 2 4 57. 1 2 x 1 4 1 4 6x 1 2 x 0 58. 1 2 z 2 5 2 4 z 6 5 z 1 59. 1 1 4 x 1 6 7 12 2 x 60. 5 1 5 2 w 1 15 4 w 6 Solve each of the following problems by using an inequality. See Example 6. 61. Boat storage. The length of a rectangular boat storage shed must be 4 meters more than the width, and the perimeter must be at least 120 meters. What is the range of values for the width? At least 28 meters 62. Fencing a garden. Elka is planning a rectangular garden that is to be twice as long as it is wide. If she can afford to buy at most 180 feet of fencing, then what are the possible values for the width? At most 0 feet FIGURE FOR EXERCISE 62 6. Car shopping. Harold Ivan is shopping for a new car. In addition to the price of the car, there is a 5% sales tax and a $144 title and license fee. If Harold Ivan decides that he will spend less than $9970 total, then what is the price range for the car? Less than $958 64. Car selling. Ronald wants to sell his car through a broker who charges a commission of 10% of the selling price. Ronald still owes $11,025 on the car. Ronald must get enough to at least pay off the loan. What is the range of the selling price? At least $12,250 65. Microwave oven. Sherie is going to buy a microwave in a city with an 8% sales tax. She has at most $594 to spend. In what price range should she look? At most $550 66. Dining out. At Burger Brothers the price of a hamburger is twice the price of an order of French fries, and the price of
2.9 Solving Inequalities and Applications (2-6) 127 a Coke is $0.25 more than the price of the fries. Burger Brothers advertises that you can get a complete meal (burger, fries, and Coke) for under $2.00. What is the price range of an order of fries? Less than 44 cents 67. Averaging test scores. Tilak made 44 and 72 on the first two tests in algebra and has one test remaining. For Tilak to pass the course, the average on the three tests must be at least 60. For what range of scores on his last test will Tilak pass the course? At least 64 68. Averaging income. Helen earned $400 in January, $450 in February, and $80 in March. To pay all of her bills, she must average at least $40 per month. For what income in April would her average for the four months be at least $40? At least $490 69. Going for a C. Professor Williams gives only a midterm exam and a final exam. The semester average is computed by taking 1 of the midterm exam score plus 2 of the final exam score. To get a C, Stacy must have a semester average between 70 and 79 inclusive. If Stacy scored only 48 on the midterm, then for what range of scores on the final exam will Stacy get a C? Between 81 and 94.5 inclusive 70. Different weights. Professor Williamson counts his midterm as 2 of the grade and his final as 1 of the grade. Wendy scored only 48 on the midterm. What range of scores on the final exam would put Wendy s average between 70 and 79 inclusive? Compare to the previous exercise. Between 114 and 141 inclusive 71. Average driving speed. On Halley s recent trip from Bangor to San Diego, she drove for 8 hours each day and traveled between 96 and 45 miles each day. In what range was her average speed for each day of the trip? Between 49.5 and 56.625 miles per hour 72. Driving time. On Halley s trip back to Bangor, she drove at an average speed of 55 mph every day and traveled between 0 and 495 miles per day. In what range was her daily driving time? Between 6 and 9 hours 7. Sailboat navigation. As the sloop sailed north along the coast, the captain sighted the lighthouse at points A and B as shown in the figure. If the degree measure of the angle at the lighthouse is less than 0, then what are the possible values for x? Between 55 and 85 74. Flight plan. A pilot started at point A and flew in the direction shown in the diagram for some time. At point B she made a 110 turn to end up at point C, due east of where she started. If the measure of angle C is less than 85, then what are the possible values for x? Between 0 and 65 North A 85 B? x A x B 110? Lighthouse FIGURE FOR EXERCISE 7 FIGURE FOR EXERCISE 74 75. Bicycle gear ratios. The gear ratio r for a bicycle is defined by the formula Nw r, n where N is the number of teeth on the chainring (by the pedal), n is the number of teeth on the cog (by the wheel), and w is the wheel diameter in inches (Cycling, Burkett and Darst). a) If the wheel has a diameter of 27 in. and there are 12 teeth on the cog, then for what number of teeth on the chainring is the gear ratio between 60 and 80? Between 27 and 5 teeth inclusive b) If a bicycle has 48 teeth on the chainring and 17 teeth on the cog, then for what diameter wheel is the gear ratio between 65 and 70? Between 2.02 in. and 24.79 in. c) If a bicycle has a 26-in. diameter wheel and 40 teeth on the chainring, then for what number of teeth on the cog is the gear ratio less than 75? At least 14 teeth 76. Virtual demand. The weekly demand (the number bought by consumers) for the Acme Virtual Pet is given by the formula d 9000 60p where p is the price each in dollars. a) What is the demand when the price is $0 each? 7200 b) In what price range will the demand be above 6000? Less than $50 C