Graphing Systems of Inequalities: Real-World Applications

A farmer has 100 acres of land. He wants to plant corn and wheat. Each acre of corn requires 2 hours of labor, and each acre of wheat requires 1 hour of labor. If he has a total of 120 hours of labor available, write a system of inequalities to represent the situation and graph the feasible region.

• x + y ≤ 80, 2x + y ≤ 100, x ≥ 0, y ≥ 0

• x + y ≤ 100, 2x + y ≤ 120, x ≥ 0, y ≥ 0

• x + y ≤ 120, 2x + y ≤ 100, x ≥ 0, y ≥ 0

• x + 2y ≤ 100, 2x + 2y ≤ 120, x ≥ 0, y ≥ 0

A school is organizing a field trip and has a budget of $500. The cost per student for the trip is $20, and the bus rental costs $200. Write a system of inequalities to represent the number of students that can attend the trip and graph the solution set.

• x ≥ 0, x ≤ 15

• x ≥ 0, x ≤ 12

• x ≥ 0, x ≤ 10

• x ≥ 5, x ≤ 20

A bakery produces cookies and cakes. Each cookie requires 1 hour of baking time, and each cake requires 3 hours. The bakery has a maximum of 12 hours available for baking. Write a system of inequalities to represent the number of cookies (x) and cakes (y) that can be made, and graph the feasible region.

• 2x + y ≤ 12, x ≥ 0, y ≥ 0

• x + 2y ≤ 12, x ≥ 0, y ≥ 0

• x + 4y ≤ 12, x ≥ 0, y ≥ 0

• x + 3y ≤ 12, x ≥ 0, y ≥ 0

A gym offers two types of memberships: a basic membership for $30 per month and a premium membership for $50 per month. If a customer can spend no more than $200 per month on memberships, write a system of inequalities to represent the situation and graph the solution set.

• 30x + 50y = 200, x < 0, y < 0

• 30x + 50y >= 200, x >= 0, y >= 0

• 30x + 50y <= 150, x >= 0, y >= 0

• The system of inequalities is: 30x + 50y <= 200, x >= 0, y >= 0.

A company produces two products, A and B. Each product A requires 4 hours of labor and each product B requires 2 hours. The company has a maximum of 40 hours of labor available. Write a system of inequalities to represent the production limits and graph the feasible region.

• 3x + 4y ≤ 40

• 2x + 3y ≤ 40

• x + y ≤ 20

• 4x + 2y ≤ 40, x ≥ 0, y ≥ 0

A local theater has 300 seats. Tickets for adults cost $15 and tickets for children cost $10. If the theater wants to make at least $2000 from ticket sales, write a system of inequalities to represent the situation and graph the solution set.

• x + y ≤ 300, 15x + 10y ≥ 2000

• x + y = 300, 15x + 10y = 2000

• x + y ≥ 300, 15x + 10y ≤ 2000

• x + y ≤ 250, 15x + 10y ≥ 2500

A pet store has a limit of 50 animals it can house. Each dog takes up 3 spaces and each cat takes up 2 spaces. If the store has a maximum of 120 spaces available, write a system of inequalities to represent the number of dogs (x) and cats (y) that can be housed, and graph the feasible region.

• 3x + 2y <= 100, x + y <= 55, x >= 0, y >= 0

• 2x + 3y <= 120, x + y <= 40, x >= 0, y >= 0

• 4x + y <= 100, x + y <= 60, x >= 0, y >= 0

• 3x + 2y <= 120, x + y <= 50, x >= 0, y >= 0

A clothing store sells shirts for $25 and pants for $40. The store has a budget of $1000 for inventory. Write a system of inequalities to represent the number of shirts (x) and pants (y) that can be purchased, and graph the solution set.

• The system of inequalities is: 25x + 40y <= 1000, x >= 0, y >= 0.

• 25x + 40y >= 1000, x <= 0, y <= 0

• 25x + 40y < 1000, x >= 10, y >= 5

• 25x + 40y = 1000, x >= 0, y >= 0

A restaurant has a seating capacity of 100. Each table for four seats 4 people, and each table for two seats 2 people. If the restaurant wants to maximize the number of tables while not exceeding the seating capacity, write a system of inequalities to represent the situation and graph the solution set.

• 4x + 3y <= 100, x >= 0, y >= 0

• 3x + 2y <= 100, x >= 0, y >= 0

• 2x + 4y <= 100, x >= 0, y >= 0

• 4x + 2y <= 100, x >= 0, y >= 0