Graphing Inequalities: Feasible Regions in Real Life

A farmer has 100 acres of land 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 the farmer 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 ≤ 120, 2x + y ≤ 100, x ≥ 0, y ≥ 0

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

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

• 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 feasible region.

• x ≥ 0, x ≤ 12

• x ≥ 5, x ≤ 20

• x ≥ 0, x ≤ 10

• x ≥ 0, x ≤ 15

A company produces two types of gadgets, A and B. Each gadget A requires 3 hours of assembly and each gadget B requires 2 hours. The company has 30 hours available for assembly. Write a system of inequalities to represent the production limits and identify the feasible region.

• 3x + 2y ≥ 30, x ≥ 0, y ≥ 0

• x + y ≤ 30, x ≥ 0, y ≥ 0

• 2x + 3y ≤ 30, x ≥ 0, y ≤ 0

• 3x + 2y ≤ 30, x ≥ 0, y ≥ 0

A bakery sells two types of cakes: chocolate and vanilla. Each chocolate cake requires 2 eggs and each vanilla cake requires 1 egg. If the bakery has 50 eggs, write a system of inequalities to represent the maximum number of cakes that can be made and graph the feasible region.

• x + y ≤ 50

• 2x + y ≤ 50, x ≥ 0, y ≥ 0

• 2x + y ≥ 50

• x ≤ 25, y ≤ 25

A gym offers two types of classes: yoga and spinning. Each yoga class can accommodate 15 people, and each spinning class can accommodate 10 people. If the gym has a maximum capacity of 150 people, write a system of inequalities to represent the class sizes and graph the feasible region.

• 20x + 10y ≤ 150, x ≥ 0, y ≥ 0

• 15x + 5y ≤ 150, x ≥ 0, y ≥ 0

• 10x + 15y ≤ 150, x ≥ 0, y ≥ 0

• The system of inequalities is: 15x + 10y ≤ 150, x ≥ 0, y ≥ 0.

A concert venue has a seating capacity of 300. Tickets for the front row cost $50 each, and tickets for the back row cost $30 each. If the venue wants to make at least $10,000 from ticket sales, write a system of inequalities to represent the ticket sales and graph the feasible region.

• x + y <= 300, 50x + 30y = 9000

• x + y >= 300, 50x + 30y <= 10000

• x + y <= 250, 50x + 30y >= 12000

• x + y <= 300, 50x + 30y >= 10000

A clothing store sells shirts and pants. Each shirt costs $20 and each pair of pants costs $30. The store has a budget of $600 for inventory. Write a system of inequalities to represent the maximum number of shirts and pants that can be purchased and graph the feasible region.

• The system of inequalities is: 20x + 30y ≤ 600, x ≥ 0, y ≥ 0.

• 25x + 35y ≤ 600, x ≥ 0, y ≥ 0

• 10x + 15y ≤ 600, x ≥ 0, y ≤ 0

• 20x + 30y ≥ 600, x ≤ 0, y ≤ 0

A charity organization is collecting donations. Each donation of $10 can provide 2 meals, and each donation of $20 can provide 5 meals. If the organization wants to provide at least 100 meals, write a system of inequalities to represent the donations needed and graph the feasible region.

• 3x + 4y >= 100, x >= 0, y >= 0

• x + y >= 50, x >= 0, y >= 0

• 2x + 5y <= 100, x >= 0, y >= 0

• The system of inequalities is: 2x + 5y >= 100, x >= 0, y >= 0.

A local park has a limit of 200 visitors at any given time. If each adult counts as 1 visitor and each child counts as 0.5 visitors, write a system of inequalities to represent the number of adults and children allowed in the park and graph the feasible region.

• x + 0.5y <= 200, x >= 0, y >= 0

• x + 0.5y <= 150, x >= 10, y >= 5

• x + 0.5y = 200, x < 0, y < 0

• x + 0.5y >= 200, x <= 0, y <= 0

A restaurant offers two types of meal combos: Combo A and Combo B. Combo A costs $15 and Combo B costs $10. The restaurant wants to sell at least $1,000 worth of meals in a day. Write a system of inequalities to represent the sales and graph the feasible region.

• 15x + 10y ≥ 1000, x ≥ 0, y ≥ 0

• 10x + 15y ≥ 1000, x ≥ 0, y < 0

• 15x + 10y = 1000, x < 0, y < 0

• 15x + 10y ≤ 1000, x ≥ 0, y ≥ 0