Real-World Systems of Inequalities: Finding Feasible Regions

A farmer has 100 meters of fencing to create a rectangular pen for sheep. If the length of the pen is represented by x and the width by y, write the inequalities that represent the constraints on the dimensions of the pen. Identify the feasible region for the dimensions.

• x + y < 50

• x ≤ 0, y ≤ 0

• The inequalities are: x + y ≤ 50, x ≥ 0, y ≥ 0.

• x + y ≥ 100

A school is planning a field trip and has a budget of $500. Each student ticket costs $15 and each adult ticket costs $20. Write a system of inequalities to represent the number of student (x) and adult (y) tickets that can be purchased. What is the feasible region for ticket purchases?

• 15x + 20y ≤ 500, x ≥ 0, y ≥ 0

• 10x + 15y ≤ 500

• 20x + 15y ≤ 500

• 15x + 25y ≤ 500

A bakery sells two types of cookies: chocolate chip and oatmeal. The bakery can make a maximum of 200 cookies in a day. If the number of chocolate chip cookies is represented by x and oatmeal cookies by y, write the inequality for the total number of cookies. Identify the feasible region for cookie production.

• x + y ≤ 200; feasible region: x ≥ 0, y ≥ 0, x + y ≤ 200

• x + y ≥ 200; feasible region: x ≥ 0, y ≥ 0, x + y ≥ 200

• x + y = 200; feasible region: x ≤ 0, y ≤ 0, x + y = 200

• x + y < 200; feasible region: x < 0, y < 0, x + y < 200

A gym has a maximum capacity of 150 members. If the number of adult members is represented by x and the number of youth members by y, write the inequality that represents the gym's capacity. What is the feasible region for the number of members?

• x + y = 150; feasible region is x ≤ 0, y ≤ 0, and x + y = 150.

• x + y ≥ 150; feasible region is x ≥ 0, y ≥ 0, and x + y ≥ 150.

• x + y ≤ 150; feasible region is x ≥ 0, y ≥ 0, and x + y ≤ 150.

• x + y < 150; feasible region is x < 0, y < 0, and x + y < 150.

A concert venue can hold a maximum of 300 people. If tickets for adults cost $25 and tickets for children cost $15, and the total revenue must be at least $5000, write a system of inequalities to represent this situation. Identify the feasible region for ticket sales.

• x + y ≤ 300, 25x + 15y ≥ 5000

• x + y ≥ 300, 25x + 15y = 5000

• x + y ≤ 250, 25x + 15y ≥ 6000

• x + y < 300, 25x + 15y < 5000

A company produces two products, A and B. Each product A requires 2 hours of labor and each product B requires 3 hours. The company has a maximum of 60 hours of labor available. Write the inequality for the labor constraint and identify the feasible region for production.

• 2x + 3y ≤ 60

• 4x + 2y ≤ 60

• x + y ≤ 20

• 2x + 3y ≥ 60

A local charity is organizing a food drive. They want to collect at least 200 cans of food. If they collect 3 cans for every adult volunteer (x) and 2 cans for every youth volunteer (y), write the inequality representing this situation. Identify the feasible region for volunteers.

• x + y ≥ 100

• 4x + y ≥ 200

• 3x + 2y ≥ 200

• 3x + 2y ≤ 200

A pet store has a limit of 50 animals it can house. If the number of dogs is represented by x and the number of cats by y, write the inequality for the total number of animals. What is the feasible region for the number of pets?

• x + y ≥ 50; feasible region is x ≥ 0, y ≥ 0, and x + y ≥ 50.

• x + y = 50; feasible region is x ≤ 0, y ≤ 0, and x + y = 50.

• x + y < 50; feasible region is x < 0, y < 0, and x + y < 50.

• x + y ≤ 50; feasible region is x ≥ 0, y ≥ 0, and x + y ≤ 50.

A school is selling tickets for a play. They have a goal to sell at least 200 tickets. If student tickets are $5 (x) and adult tickets are $10 (y), write the inequality for the total ticket sales. Identify the feasible region for ticket sales.

• x + y ≤ 200

• x + y ≥ 200

• x + 2y ≥ 200

• 5x + 10y ≥ 200

A factory produces two types of toys: dolls and cars. Each doll requires 4 hours to produce and each car requires 2 hours. The factory has a maximum of 80 hours available for production. Write the inequality for the production time and identify the feasible region for toy production.

• 4x + 2y ≤ 80

• x + 2y ≤ 80

• 2x + y ≤ 40, x ≥ 0, y ≥ 0

• 2x + y ≥ 40