Graphing and Interpreting Systems of Inequalities in Context

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 >= 100, 2x + y >= 120, x <= 0, y <= 0

• x + y <= 80, 2x + y <= 100, x >= 0, y >= 0

• x + y <= 120, 2x + y <= 150, x >= 10, y >= 10

• x + y <= 100, 2x + y <= 120, x >= 0, y >= 0

A school is planning 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.

• x ≥ 0 and x ≤ 10

• x ≥ 0 and x ≤ 15

• x ≥ 5 and x ≤ 25

• x ≥ 0 and x ≤ 20

A company produces two types of gadgets: Type A and Type B. Each Type A gadget requires 3 hours of labor and each Type B gadget requires 2 hours. The company has 60 hours of labor available. Write a system of inequalities to represent the production limits and graph the feasible region.

• 4x + 2y ≤ 60, x ≥ 0, y ≥ 0

• 3x + y ≤ 60, x ≥ 0, y ≥ 0

• 2x + 3y ≤ 60, x ≥ 0, y ≥ 0

• 3x + 2y ≤ 60, x ≥ 0, y ≥ 0

A restaurant offers two types of meal packages: Package X and Package Y. Package X costs $15 and Package Y costs $10. The restaurant has a budget of $300 for meals and can serve a maximum of 30 packages. Write a system of inequalities and graph the solution.

• 15x + 10y ≤ 250

• x + y ≤ 25

• 15x + 10y ≤ 300, x + y ≤ 30, x ≥ 0, y ≥ 0

• x ≤ 0, y ≤ 0

A factory produces two products: Product A and Product B. Each Product A requires 4 hours of machine time and each Product B requires 2 hours. The factory has 80 hours of machine time available. Write a system of inequalities to represent the production limits and graph the feasible region.

• 2x + 4y ≤ 80

• 3x + 4y ≤ 80

• 4x + 2y ≤ 80, x ≥ 0, y ≥ 0

• x + y ≤ 40

A clothing store sells shirts for $25 and pants for $40. The store has a budget of $1000 for inventory and wants to stock at least 20 items. Write a system of inequalities to represent the inventory situation and graph the solution.

• 25x + 40y ≥ 1000, x + y ≤ 20, x ≤ 0, y ≤ 0

• 25x + 40y ≤ 500, x + y ≥ 30, x ≥ 0, y ≥ 0

• The system of inequalities is: 25x + 40y ≤ 1000, x + y ≥ 20, x ≥ 0, y ≥ 0.

• 25x + 40y ≤ 1000, x + y = 15, x ≥ 0, y ≥ 0

A concert venue has a seating capacity of 500. Tickets for the front row cost $50 and for the back row cost $30. The venue wants to earn at least $15,000 from ticket sales. Write a system of inequalities to represent the ticket sales and graph the feasible region.

• x + y = 500, 50x + 30y = 15000

• x + y ≤ 400, 50x + 30y ≥ 20000

• x + y ≤ 500, 50x + 30y ≥ 15000

• x + y ≥ 500, 50x + 30y ≤ 15000

A bakery makes two types of cakes: chocolate and vanilla. Each chocolate cake requires 3 eggs and each vanilla cake requires 2 eggs. The bakery has 30 eggs available. Write a system of inequalities to represent the cake production limits and graph the solution.

• 4x + y ≤ 30, x ≥ 0, y ≥ 0

• 3x + 3y ≤ 30, x ≥ 0, y ≥ 0

• 2x + 3y ≤ 30, x ≥ 0, y ≥ 0

• 3x + 2y ≤ 30, x ≥ 0, y ≥ 0