5-4 Solving Compound Inequalities. Solve each compound inequality. Then graph the solution set. 6. f 6 < 5 and f 4 2 SOLUTION: and

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1 Solve each compound inequality. Then graph the solution set. 6. f 6 < 5 f y 1 7 y + 3 < 1 The solution set is {f 6 f < 11}. To graph the solution set, graph 6 f graph f < 11. Then find the intersection. The solution set is {y y 8 y < 4}. Notice that the graphs do not intersect. To graph the solution set, graph y 8 graph y < 4. Then find the union. {f 6 f < 11} {y y 8 y < 4} 7. n n t t 9 < 10 The solution set is {n 12 n 7}. To graph the solution set, graph 12 n graph n 7. Then find the intersection. The solution set is {t t 1 t < 1}. Notice that the graphs do not intersect. To graph the solution set, graph t 1 t < 1. Then find the union. {n 12 n 7} {t t 1 t < 1} esolutions Manual - Powered by Cognero Page 1

2 10. 5 < 3p h 4 6 7h + 11 < 32 The solution set is {p 4 < p 5}. To graph the solution set, graph 4 < p graph p 5. Then find the intersection. The solution set is {h 2 h < 3}. To graph the solution set, graph 2 h graph h < 3. Then find the intersection. {p 4 < p 5} {h 2 h < 3} c + 4 < m 2 5 3m 13 The solution set is {c 1 c < 2}. To graph the solution set, graph 1 c graph c < 2. Then find the intersection. {c 1 c < 2} Notice that the two inequalities overlap all real numbers are solutions. The solution set is {m m is a real number}. To graph the solution set, graph all points. {m m is a real number.} esolutions Manual - Powered by Cognero Page 2

3 14. 4a < 6a SPEED The posted speed limit on an interstate highway is shown. Write an inequality that represents the sign. Graph the inequality. Notice that the two inequalities do not overlap. So, the solution set is empty, ø. The graph is also empty. ø 15. y y + 4 < 5 Sample answer: Let r = rate of speed. The lowest speed you can go is 40 mph, while the highest speed is 70 mph. Therefe, the inequality is 40 r 70. Sample answer: Let r = rate of speed, then 40 r NUMBER THEORY Find all sets of two consecutive positive odd integers with a sum that is at least 8 less than 24. Sample answer: Let x = the smaller of two consecutive odd numbers, then 8 2x Notice that the two inequalities overlap at y < 3, so the solution set is {y y < 3}. To graph the solution set, graph y < 3. {y y < 3} List every combination of numbers in which the smaller number is 3 x 11. 3, 5; 5, 7; 7, 9; 9, 11; 11, 13 Sample answer: Let x = the smaller of two consecutive odd numbers, then 8 2x ; 3 x 11; 3, 5; 5, 7; 7, 9; 9, 11; 11, 13 esolutions Manual - Powered by Cognero Page 3

4 18. Write a compound inequality f each graph. This graph represents an intersection. Both endpoints are closed circles which include the endpoints. The compound inequality is 1 x 4. 1 x The endpoint on the left is a closed endpoint, which represents less than equal to. The endpoint on the right is only a point. The inequalities are x 3 x 6. x 3 x This graph represents an intersection. The left endpoint is an open circle which represents greater than. The right endpoint is a closed circle which represents less than equal to. The compound inequality is 3 < x 2. 3 < x The endpoint on the right is an open endpoint, which represents greater than. The endpoint on the left is only a point. The inequalities are x -3 x > 0. x -3 x > 0 Solve each compound inequality. Then graph the solution set b + 2 < 5b 6 2b + 9 The endpoint on the left is an open endpoint, which represents less than. The endpoint on the right is a closed endpoint, which represents greater than equal to. The inequalities are x < 0 x 3. x < 0 x Both endpoints are open, so the endpoints are not included. The inequalities are x < 4 x 3. x < 4 x > 3 The solution set is {b 4 < b 5}. To graph the solution set, graph 4 < b graph b 5. Then find the intersection. {b 4 < b 5} esolutions Manual - Powered by Cognero Page 4

5 25. 2a + 3 6a 1 > 3a n 1 < 16 3n 1 < 8 The solution set is. To graph the solution set, graph. Then find the intersection. 4 < b grap Notice that the two inequalities overlap, but the point 3 is not included. The solution set is {n n < 3 n > 3}. To graph the solution set, graph all points except f 3. {n n < 3 n > 3} m 7 < 17m 6m > 36 The solution set is {m m < 6 m > 1}. Notice that the graphs do not intersect. To graph the solution set, graph m < 6 graph m > 1. Then find the union. {m m < 6 m > 1} esolutions Manual - Powered by Cognero Page 5

6 28. COUPON Juanita has a coupon f 10% off any digital camera at a local electronics ste. She is looking at digital cameras that range in price from $100 to $250. a. How much are the cameras after the coupon is used? b. If the tax amount is 6.5%, how much should Juanita expect to spend? a. The least expensive camera is $100. The 10% coupon is wth $10. After the coupon is used, the least expensive camera is $100 $10 $90. The most expensive camera is $250. The 10% coupon is wth $25. After the coupon is used, the most expensive camera is $250 $25 $225. Define a variable, write an inequality, solve each problem. Then check your solution. 29. Eight less than a number is no me than 14 no less than 5. Let n = the number. The solution set is {n 13 n 22}. To check this answer, substitute a number greater than equal to 13 less than equal to 22 into the iginal inequality. Let n = 15. The cameras are between $90 $225 inclusive. b. The most expensive camera is $90 after the coupon, so add 6.5% tax. So, the solution checks. Sample answers given. Let n = the number. 5 n 8 14; {n 13 n 22} So, the least expensive camera will cost $ The most expensive camera is $225 after the coupon, so add 6.5% tax. So the most expensive camera will cost $ Juanita should expect to spend between $95.85 $ inclusive. a. from $90 to $225 inclusive b. from $95.85 to $ inclusive esolutions Manual - Powered by Cognero Page 6

7 30. The sum of 3 times a number 4 is between Let n = the number. The solution set is {n 4 < n < 2}. To check this answer, substitute a number greater than 4 less than 2 into the iginal inequality. Let n = 0. So, the solution checks. Sample answers given. Let n = the number. 8 < 3n + 4 < 10; {n 4 < n < 2} esolutions Manual - Powered by Cognero Page 7