Real-Life Systems of Inequalities: Finding Feasible 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 feasible region for the number of acres of corn (x) and wheat (y) he can plant.

• 2x + 3y ≤ 120

• x + y ≤ 80

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

• x + 2y ≤ 100

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 maximum number of students (x) that can attend the trip while staying within budget.

• x <= 25, x >= 0

• x <= 20, x >= 5

• x <= 10, x >= 0

• x <= 15, x >= 0

A bakery sells two types of cookies: chocolate chip and oatmeal. Each chocolate chip cookie requires 3 ounces of flour, and each oatmeal cookie requires 2 ounces. If the bakery has 30 ounces of flour, write a system of inequalities to represent the number of chocolate chip (x) and oatmeal (y) cookies that can be made.

• 3x + 3y ≤ 30, x ≥ 0, y ≥ 0

• 2x + 3y ≤ 30, x ≥ 0, y ≥ 0

• 4x + y ≤ 30, x ≥ 0, y ≥ 0

• 3x + 2y ≤ 30, x ≥ 0, y ≥ 0

A gym offers two types of memberships: basic and premium. The basic membership costs $30 per month, and the premium membership costs $50 per month. If the gym wants to make at least $1,000 in membership fees, write a system of inequalities to represent the number of basic (x) and premium (y) memberships sold.

• 30x + 50y >= 1000, x >= 0, y >= 0

• 20x + 40y >= 1000, x >= 0, y >= 0

• 30x + 50y = 1000, x >= 0, y >= 0

• 30x + 50y <= 1000, 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 make at least $15,000 from ticket sales, write a system of inequalities to represent the number of front row (x) and back row (y) tickets sold.

• x + y ≥ 500, 50x + 30y ≤ 15000

• x + y = 500, 50x + 30y = 15000

• x + y ≤ 400, 50x + 30y ≥ 20000

• x + y ≤ 500, 50x + 30y ≥ 15000

A company produces two products, A and B. Each product A requires 4 hours of labor and each product B requires 2 hours. If the company has 40 hours of labor available, write a system of inequalities to represent the number of products A (x) and B (y) that can be produced.

• 2x + 3y ≤ 40, x ≥ 0, y ≥ 0

• 3x + 4y ≤ 40, x ≥ 0, y ≥ 0

• 5x + y ≤ 40, x ≥ 0, y ≥ 0

• 4x + 2y ≤ 40, x ≥ 0, y ≥ 0

A local charity is organizing a food drive. They can collect a maximum of 200 cans of food. Each can of soup counts as 1 can, and each can of vegetables counts as 2 cans. Write a system of inequalities to represent the number of cans of soup (x) and vegetables (y) they can collect.

• x + 2y ≤ 200, x ≥ 0, y ≥ 0

• 2x + y ≤ 200, x ≥ 0, y ≥ 0

• x + y ≤ 200, x ≥ 0, y ≥ 0

• x + 3y ≤ 200, x ≥ 0, y ≥ 0

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 wants to maximize the number of animals, write a system of inequalities to represent the number of dogs (x) and cats (y) they can have.

• 4x + y ≤ 50, x ≥ 0, y ≥ 0

• 3x + 4y ≤ 50, x ≥ 0, y ≥ 0

• 3x + 2y ≤ 50, x ≥ 0, y ≥ 0

• 2x + 3y ≤ 50, x ≥ 0, y ≥ 0

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 $1,200 in sales, write a system of inequalities to represent the number of shirts (x) and pants (y) sold.

• 15x + 25y >= 1200, x >= 0, y >= 0

• 15x + 30y >= 1500, x >= 0, y >= 0

• 10x + 20y >= 1000, x >= 0, y >= 0

• 20x + 25y >= 1300, x >= 0, y >= 0

A community garden has 60 plots available. Each vegetable plot requires 2 plots of land, and each flower plot requires 1 plot. If the garden wants to plant at least 30 plots of vegetables, write a system of inequalities to represent the number of vegetable (x) and flower (y) plots they can plant.

• 2x + y ≥ 60

• x + y ≤ 60

• x ≤ 30

• { "system_of_inequalities": ["2x + y ≤ 60", "x ≥ 30"] }