Activities

1. Determine if the following coordinate points are solutions to the system of linear inequalities graphed here.

(0, 0), (5, 0), (5, 9), (-4, 4), (-100, -100), (-8, 0), (5, 5), and (2, 6)

2. Johnny and Jimmy are both thinking of a number between one and ten inclusive. They win the game if the sum of their numbers is less than 12 and the difference is greater than five. What are five solutions to their system of inequalities that would win the game?

3. Sam is buying more than 8 apples and bananas from the grocery store, and wants to spend less than $15. Apples are $2 each and bananas are $1 each. What is the maximum number of bananas Sam can buy?

a + b > 8   and    2a + b < 15

4. Fill in the blanks on the system of inequalities so that the solutions are in area 4.

y ____ 5   and   y ____ x + 2

5. Marsha is buying plants and soil for her garden. The soil cost $4 per bag, and the plants cost $10 each. She wants to buy at least 5 plants and can spend no more than $100. What are three solutions to this system of inequalities?

6. Find three coordinate points that are solutions and three coordinate points that are not solutions to the following system of inequalities.

y ≤ 5    and      y < -x – 3

7. A mother told her son that he must eat at least three carrots before he could have desert. She also said that he could eat one less than twice as many cookies as carrots. What are 5 values in the solution set of this system of inequalities?

8. A homeowner is deciding on the species of tree to plant in her front yard. She wants two trees so that the sum of the mature heights for both trees is greater than 100 feet. She also wants the difference between the mature heights of the trees to be less than 20 feet. Write a system of inequalities that represents this situation. Then determine three different solutions that would work for the homeowner and one possibility that would not work.

9. Explain in words when to use a dotted line and when to use a solid line when graphing the solutions to linear inequalities. Then explain what each type of line means.

10. Filipe exercises several days each week. He either rides his bike for 20 miles or runs for 5 miles. Filipe wants to cover at least 70 miles per week, running or riding. The following system of inequalities represents this situation:

R + B ≤ 7 and 5R + 20B ≥ 70

If Filipe runs four days of the week, how many days must he exercise that week to meet his goal?

11. At a local donut shop, donuts cost d dollars each and muffins cost m dollars each. The total cost of 2 donuts and 5 muffins is more than $8. The total cost of 9 donuts and 12 muffins is less than $32. What could be the total cost of one donut and one muffin?

12. A planning committee is trying to determine how much water they need to provide for a 5K race they are in charge of. They know that each runner needs at least 8 ounces of water. They also know that their containers can only hold 10,000 ounces of water. What is the range of runners that can participate in the race?

13. An entrepreneur is starting an ice cream business. He knows that he needs to make at least $400 in profits each week. He uses the equations p > 0.25s – 100 and p ≥ 400 to evaluate his business. What is the smallest amount of sales that would guarantee the entrepreneur could continue with his business?

14. Micah wants to buy some T-shirts and hoodies for his friends for their birthdays. On a certain website he can get a discount if he buys at least 5 T-shirts and hoodies in all. The discount price is $8 for a T-shirt and $20 for a hoodie. If Micah has a total of $150 to spend, what is the least and greatest amount of money he can spend ordering from the website?

15. The solution to the system of inequalities graphed below represents the profit based on income versus expenditures of a certain small business (in thousands of dollars) since incorporating. By which month did the business start making a profit? 

16. Create a system of equations that has (3, 3), (0, 5) and (2, 0) as solutions, but (0, 0), (2, -2) and (-4, 5) are not solutions.

17. A 20 by 20 unit game board looks exactly like a coordinate plane with the origin at the center, but it does not continue past 10 or -10 in the x or y direction. Jethro creates the following system of equations to plot on the game board:  y ≥ 0 and y ≤ x. What is the probability that a random coordinate point will land in the solution area of Jethro’s system of inequalities?

18. The system of linear inequalities below represents the amount of quarters and dimes that Ella has in her purse. What is the largest difference between the number of quarters and dimes in her purse?

0.10d + 0.25q ≥ 3.50 and 0.25q – 0.10d > 0.75

19. A system of inequalities represents the possible ages of a mother and her child. The daughter is between what two ages?

M + D ≤ 40     M – D ≥ 25

20. A bank is working with an investor regarding the percent interest the investor will receive on his investment over a period of time. The bank suggests that they give 8.5% or less plus a $400 bonus fee. The investor, on the other hand, asked for at least 8.75% return and the same $400 bonus. Give an example solution that the bank and investor agree upon.

21. You have been given a $50 gift certificate to a local fast food restaurant. You only have one day left to use the certificate and it only covers the cost of $5 burgers and $3 fries. There is also a stipulation that you cannot buy only burgers, there must be some fries in your order. Write down three solutions that would allow you to use all $50 on your gift certificate.

22. The solutions to a system of linear inequalities all lie along the line y = 2x – 5. What is the system of linear inequalities?

23. What is the only integer solution to the following system of three inequalities?

y ≥ x + 2  and   y < -x + 5    and    y < 10x – 4

24. The system of equations below represents the ages of two children in a day care. What is the largest possible sum of their ages?

25. The sum of two numbers is greater than a positive number. The difference of the same two numbers is less than a negative number. Sketch the solution set to this system of inequalities on the coordinate plane below. 

Answer Key/Rubric

1. (0, 0) NO, (5, 0) YES, (5, 9) NO, (-4, 4) NO, (-100, -100) NO, (-8, 0) NO, (5, 5) YES, (2, 6) NO

2. Various solutions including: (1, 6), (2, 7), (3, 8), (7, 2), (6, 1)

3. The max number of bananas Sam can buy is 14

4. y ≤ 5   and   y > x + 2

5. (plants, soil): (5, 4), (6, 4), (7, 4)

6. Various solutions including: Solutions: (0, -4), (0, -5), (0, -6) Not Solutions: (2, 5), (3, 6), (4, 7)

7. Various answers including (carrots, cookies): (3, 5), (4, 7), (4, 6), (5, 9), (5, 8)

8. T₁ + T₂ > 100, T₁ – T₂ < 20, (T₁, T₂): (80, 70), (90, 75), (100, 100).  Not solution (40, 15)

9. A dotted line is used in a linear inequality to represent that the line is not included. It is used with a greater than or less than sign. The solid line is used to represent that the line is part of the solution and is used with a greater than or equal to or less than or equal to sign.

10. 7 days of that week (he must bike for 2.5 days, therefore exercising 7 days that week)

11. Various answers including: $2.25

12. 0 to 1250 runners can participate

13. $2000.00

14. Least amount of money is $40, Greatest amount of money is $148.00

15. 6 months after incorporating

16. y > -x + 1 and y > -2x + 2

17. 12.5%

18. 14 quarters and 0 dimes

19. 0 to 7.5 years old

20. There is no solution that works for both

21. 5 fries and 7 burgers, 10 fries and 4 burgers, 15 fries and 1 burger

22. y ≥ 2x – 5 and y ≤ 2x – 5

23. (1, 3)

24. 7 years, either 4 + 3 or 5 + 2
    
    
25.