Word Problems with Inequalities
- 1. Word Problems with Inequalities Don’t panic! Remember to: • Read the questions carefully. • Define the variable. • Write an inequality. • Solve the inequality. • Check that your answer is reasonable. • Answer in a complete sentence.
- 3. Example 1 Eight less than the product of -3 and a number is greater than -26. Write and solve an inequality to represent this relationship. Graph the solution set, and check your answer. Let x = the number -3x – 8 > -26 +8 +8 -3x > -18 -3 -3 x < 6 4 5 6 7 8 CHECK @ x = 5 -3x – 8 > -26 -3(5) – 8 > -26 -15 – 8 > -26 Don’t forget to reverse the inequality sign -23 > -26 when dividing by a negative!
- 4. Example 2 The sum of two consecutive integers is at least negative fifteen. What are the smallest values of consecutive integers that will make this true? Let x = the first integer x + 1 = the next consecutive integer x + (x + 1) ≥ -15 2x + 1 ≥ -15 - 1 - 1 2x ≥ -16 2 2 x ≥ -8 x + 1 ≥ -7 The smallest values of consecutive integers that will add to at least -15 are -8 and -7.
- 5. Example 3 Connor went to the carnival with $22.50. He bought a hot dog and a drink for $3.75, and he wanted to spend the rest of his money on ride tickets which cost $1.25 each. What is the maximum number of ride tickets that he can buy? Let r = the number of ride tickets he can buy cost of food + cost of rides ≤ $22.50 3.75 + 1.25r ≤ 22.50 -3.75 -3.75 1.25r ≤ 18.75 1.25 1.25 r ≤ 15 tickets Connor can buy a maximum of 15 ride tickets.
- 6. Example 4 Stan earned $7.55 per hour plus an additional $100 in tips waiting tables on Saturday evening. He earned $160 in all. To the nearest hour, what is the least number of hours Stan would have to work to earn this much money? Let h = the number of hours Stan will have to work tips + hourly wages ≥ $160 100 + 7.55h ≥ 160 -100 -100 7.55h ≥ 60 7.55 7.55 x ≥ 7.9 hours To the nearest hour, Stan would have to work at least 8 hours to earn $160.
- 7. Example 5 Brenda has $500 in her bank account. Every week, she withdraws $40 for expenses. Without making any deposits, how many weeks can she withdraw this money if she wants to maintain a balance of at least $200? Let w = the number of weeks Brenda withdraws money starting account balance – money withdrawn ≥ $200 500 – 40w ≥ 200 -500 -500 – 40w ≥ -300 – 40 – 40 w ≤ 7.5 weeks Don’t forget to reverse the inequality sign when dividing by a negative! Brenda can withdraw $40 from the account for 7 full weeks and still have at least $200 in the account.
- 8. Sean earns $6.70 per hour working after school. He needs at least $155 (≥ 155) for a stereo system. Write an inequality that describes how many hours he must work to reach his goal. $6.70 h ≥ 155 Per means to multiply To figure your pay check: You multiply rate times hours worked $6.70 $6.70 h 23.1343 So 23 hours is not enough So Sean must Work 24 hours To earn enough Money. h 24
- 9. Sean earns $6.85 per hour working after school. He needs at least $310 (≥ 310) for a stereo system. Write an inequality that describes how many hours he must work to reach his goal. $6.85 h ≥ 310 Per means to multiply To figure your pay check: You multiply rate times hours worked $6.85 $6.85 h 45.25547 So 45 hours is not enough So Sean must Work 46 hours To earn enough Money. h 46
- 10. The width of a rectangle is 11 cm. Find all possible values for the length of the rectangle if the perimeter is at least 300 cm. ( ≥ 300 cm) • 2L + 2W = p 2L 2(11) 300 2L 22 300 22 22 2L 278 2 2 L 139
- 11. The width of a rectangle is 31 cm. Find all possible values for the length of the rectangle if the perimeter is at least 696 cm. ( ≥ 696 cm) • 2L + 2W = p 2 2(31) L 696 2L 62 696 62 62 2L 634 2 2 L 317
- 12. The perimeter of a square is to be between 11 and 76 feet, inclusively. Find all possible values for the length of its sides. • 4 equal sides • Perimeter – add up sides s s s s P = s +s+s+s=4s 11 p 76 11 4s 76 4 4 4 2.75 s 19
- 13. The perimeter of a square is to be between 14 and 72 feet, inclusively. Find all possible values for the length of its sides. • 4 equal sides • Perimeter – add up sides s s s s P = s +s+s+s=4s 14 p 72 14 4s 72 4 4 4 3.5 s 18
- 14. Three times the difference of a number and 12 is at most 87. Let x represent the number and find all possible values for the number. 3( 12) x 87 3x 36 87 36 36 3x 123 3 3 x 41
- 15. Marlee rented a paddleboat at the Park for a fixed charge of $2.50 plus $1.50 per hour. She wants to stay out on the water as long as possible. How many hours can she use the boat without spending more than $7.00? $2.50 $1.50h $7.00 $2.50 $2.50 $1.50h $4.50 $1.50 $1.50 h 3