Mastering Systems of Inequalities: Graphing & Solutions

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 >= 0, y >= 0

• x + y <= 100, 2x + y <= 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 feasible region.

• x >= 0 and x <= 10

• x >= 0 and x <= 20

• x >= 5 and x <= 20

• x >= 0 and x <= 15

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.

• 2x + 3y ≤ 60, x ≥ 0, y ≥ 0

• x + 2y ≤ 60, x ≥ 0, y ≥ 0

• 3x + 2y ≤ 60, x ≥ 0, y ≥ 0

• 4x + y ≤ 60, x ≥ 0, y ≥ 0

A restaurant offers two types of meals: vegetarian and non-vegetarian. The vegetarian meal costs $10 and the non-vegetarian meal costs $15. If the restaurant wants to make at least $300 in a day and can serve a maximum of 40 meals, write a system of inequalities and graph the feasible region.

• 10x + 15y ≤ 300, x + y ≥ 40, x ≥ 0, y ≥ 0

• 10x + 15y ≥ 300, x + y = 40, x ≥ 0, y ≥ 0

• 10x + 15y = 300, x + y ≤ 40, x ≥ 0, y ≥ 0

• The system of inequalities is: 10x + 15y ≥ 300, x + y ≤ 40, x ≥ 0, y ≥ 0.

A local gym has a maximum capacity of 50 members. Each membership costs $30 per month. The gym wants to earn at least $1200 in membership fees. Write a system of inequalities to represent the situation and graph the feasible region.

• x >= 40 and x <= 50

• x >= 20 and x <= 30

• x >= 50 and x <= 60

• x >= 30 and x <= 40

A charity event is selling tickets for $25 each and has a goal to raise at least $2000. If they can sell a maximum of 100 tickets, write a system of inequalities to represent the ticket sales and graph the feasible region.

• x >= 50 and x <= 80

• x >= 0 and x <= 100

• x >= 80 and x <= 100

• x >= 80 and x <= 120

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

• 3x + 4y ≤ 80, x ≥ 0, y ≥ 0

• 2x + 3y ≤ 80, x ≥ 0, y ≥ 0

• 4x + 2y ≤ 80, x ≥ 0, y ≥ 0

• 5x + y ≤ 80, x ≥ 0, y ≥ 0

A bookstore sells fiction and non-fiction books. The fiction books are priced at $12 and the non-fiction books at $15. If the bookstore wants to earn at least $600 and can sell a maximum of 50 books, write a system of inequalities and graph the feasible region.

• 12x + 15y <= 600, x + y <= 50, x <= 0, y >= 0

• 12x + 15y <= 600, x + y >= 50, x >= 0, y >= 0

• The system of inequalities is: 12x + 15y >= 600, x + y <= 50, x >= 0, y >= 0.

• 12x + 15y >= 600, x + y >= 50, x >= 0, y <= 0

A concert venue has a seating capacity of 200. Tickets for the front row cost $50 and tickets for the back row cost $30. If the venue wants to earn at least $8000, write a system of inequalities to represent the ticket sales and graph the feasible region.

• x + y = 200, 50x + 30y = 8000

• x + y ≤ 150, 50x + 30y ≥ 10000

• x + y ≤ 200, 50x + 30y ≥ 8000

• x + y ≥ 200, 50x + 30y ≤ 8000

A clothing store sells shirts and pants. Each shirt costs $20 and each pair of pants costs $30. The store wants to make at least $1000 in sales and can sell a maximum of 60 items. Write a system of inequalities to represent the sales and graph the feasible region.

• The system of inequalities is: 20x + 30y >= 1000, x + y <= 60, x >= 0, y >= 0.

• 20x + 30y >= 500, x + y <= 40, x >= 0, y >= 0

• 20x + 30y >= 1000, x + y <= 80, x >= 0, y >= 0

• 20x + 30y <= 1000, x + y >= 60, x >= 0, y >= 0