Mastering Systems of Inequalities: Graphing & Solving

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

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

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

A school is planning a field trip. The cost per student for the trip is $30 for a bus and $20 for food. If the school has a budget of $600, write a system of inequalities to represent the maximum number of students that can attend the trip and graph the inequalities.

• x ≤ 20, x ≥ 0

• x ≤ 10, x ≥ 5

• x ≤ 15, x ≥ 0

• x ≤ 12, x ≥ 0

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 24 hours of assembly time available. Write a system of inequalities to represent the production limits and graph the feasible region.

• 2x + 3y ≤ 24, x ≥ 0, y ≥ 0

• 3x + 2y ≥ 24, x ≥ 0, y ≥ 0

• The system of inequalities is: {3x + 2y ≤ 24, x ≥ 0, y ≥ 0}.

• x + y ≤ 12, x ≥ 0, y ≥ 0

A gym offers two types of memberships: basic and premium. The basic membership costs $25 per month, and the premium membership costs $40 per month. If the gym wants to earn at least $500 in membership fees, write a system of inequalities to represent the situation and graph it.

• 25x + 40y >= 500, x >= 0, y >= 0

• 25x + 40y <= 500, x >= 0, y >= 0

• 30x + 35y >= 500, x >= 0, y >= 0

• 25x + 40y = 500, x >= 0, y >= 0

A bakery sells two types of cakes: chocolate and vanilla. Each chocolate cake requires 2 cups of flour and each vanilla cake requires 3 cups. If the bakery has 12 cups of flour, write a system of inequalities to represent the maximum number of cakes that can be made and graph the inequalities.

• 2x + 3y ≤ 15, x ≥ 0, y ≥ 0

• x + y ≤ 12, x ≥ 0, y ≥ 0

• The system of inequalities is: 2x + 3y ≤ 12, x ≥ 0, y ≥ 0.

• 2x + 3y ≤ 10, x ≥ 0, y ≥ 0

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

• x + y ≤ 500, 50x + 30y ≥ 15000

• x + y = 500, 50x + 30y = 15000

• x + y ≥ 500, 50x + 30y ≤ 15000

• x + y ≤ 400, 50x + 30y ≥ 20000

A pet store has a limit of 20 animals it can house. They have dogs and cats. Each dog takes up 2 spaces and each cat takes up 1 space. If the store has 30 spaces available, write a system of inequalities to represent the situation and graph the feasible region.

• The system of inequalities is: 2x + y <= 20 and x + y <= 30, with x >= 0 and y >= 0.

• x + y >= 20 and x >= 0, y >= 0

• 2x + y >= 20 and x + y <= 30

• 2x + y <= 30 and x + y <= 20

A clothing store sells shirts and pants. Each shirt costs $15 and each pair of pants costs $25. If the store wants to make at least $1,000 in sales, write a system of inequalities to represent the sales goals and graph the inequalities.

• 15x + 25y ≥ 1000, x ≥ 0, y ≥ 0

• 15x + 25y ≤ 1000, x ≥ 0, y ≥ 0

• 15x + 30y ≥ 1000, x ≥ 0, y ≥ 0

• 10x + 20y ≥ 1000, x ≥ 0, y ≥ 0

A local charity is organizing a fundraiser. They plan to sell cookies and brownies. Each cookie costs $2 and each brownie costs $3. If they want to raise at least $200, write a system of inequalities to represent the fundraising goals and graph the feasible region.

• 2x + 3y = 200, x ≥ 0, y ≥ 0

• The system of inequalities is: 2x + 3y ≥ 200, x ≥ 0, y ≥ 0.

• 2x + 3y ≤ 200, x ≥ 0, y ≥ 0

• x + y ≥ 100, x ≥ 0, y ≥ 0

A tech company is developing two products: software and hardware. Each software product requires 4 hours of development and each hardware product requires 3 hours. If the company has 36 hours available for development, write a system of inequalities to represent the production limits and graph the inequalities.

• 4x + 3y ≥ 36, x ≥ 0, y ≥ 0

• 4x + 3y ≤ 36, x ≥ 0, y ≥ 0

• 2x + 5y ≤ 36, x ≥ 0, y ≥ 0

• x + y ≤ 12, x ≥ 0, y ≥ 0