Applying Systems of Inequalities to Real-World Scenarios

A farmer has a total of 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 3 hours of labor. If he has a maximum of 240 hours of labor available, write a system of inequalities to represent the situation and determine the feasible region for planting corn and wheat.

• x + y <= 80, 2x + 3y <= 300, x >= 0, y >= 0

• x + y <= 100, 2x + 3y <= 240, x >= 0, y >= 0

• x + y <= 120, 2x + 3y <= 200, x >= 0, y >= 0

• x + y >= 100, 2x + 3y >= 240, 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 while staying within budget. How many students can they take?

• 15

• 25

• 10

• 20

A gym offers two types of memberships: a basic membership for $30 per month and a premium membership for $50 per month. The gym wants to make at least $1,200 in membership fees each month. Write a system of inequalities to represent the number of basic and premium memberships they need to sell to meet their goal.

• 30x + 50y ≤ 1200, x ≥ 0, y ≥ 0

• 30x + 50y ≥ 1200, x ≥ 0, y ≥ 0

• 20x + 40y ≥ 1200, x ≥ 0, y ≥ 0

• 30x + 50y = 1200, x ≥ 0, y ≥ 0

A bakery sells two types of cakes: chocolate and vanilla. Each chocolate cake requires 2 hours to bake, and each vanilla cake requires 1 hour. If the bakery has 10 hours available for baking, write a system of inequalities to represent the number of each type of cake they can bake in a day.

• x + 2y ≤ 10, x ≥ 0, y ≤ 0

• 2x + y ≥ 10, x ≥ 0, y ≥ 0

• 3x + y ≤ 10, x ≥ 0, y ≥ 5

• 2x + y ≤ 10, x ≥ 0, y ≥ 0

A concert venue can hold a maximum of 1,000 people. Tickets for adults cost $15, and tickets for children cost $10. If the venue wants to make at least $12,000 from ticket sales, write a system of inequalities to represent the number of adult and child tickets that can be sold.

• x + y ≤ 1200, 15x + 10y ≥ 10000

• x + y ≥ 1000, 15x + 10y ≤ 12000

• x + y ≤ 1000, 15x + 10y ≥ 12000

• x + y ≤ 500, 15x + 10y ≥ 15000

A company produces two products, A and B. Each product A requires 3 hours of labor and each product B requires 2 hours of labor. The company has a maximum of 60 hours of labor available. Write a system of inequalities to represent the production limits for products A and B.

• 3x + 2y ≤ 50, x ≥ 0, y ≥ 0

• 2x + 3y ≤ 60, x ≥ 0, y ≥ 0

• 3x + 2y ≤ 60, x ≥ 0, y ≥ 0

• 3x + 2y ≤ 60, x < 0, y ≥ 0

A local charity is organizing a food drive. They want to collect at least 500 cans of food. Each can of soup counts as 2 cans, and each can of vegetables counts as 1 can. Write a system of inequalities to represent the number of soup and vegetable cans they need to collect to meet their goal.

• 2x + y >= 500, x >= 0, y >= 0

• 2x + y <= 500, x >= 0, y >= 0

• x + 2y >= 500

• x + y >= 500

A clothing store sells shirts for $20 and pants for $30. They want to make at least $1,000 in sales. Write a system of inequalities to represent the number of shirts and pants they need to sell to reach their sales goal, considering they have a maximum of 50 items in stock.

• 10x + 20y >= 1000, x + y <= 30

• 25x + 35y >= 1000, x + y <= 60

• 15x + 25y >= 1000, x + y <= 40

• 20x + 30y >= 1000, x + y <= 50

A pet shelter has a limit of 40 animals they can take care of. They have dogs and cats, with each dog requiring 3 units of space and each cat requiring 2 units of space. Write a system of inequalities to represent the maximum number of dogs and cats they can house.

• 3x + 2y >= 40, x >= 0, y >= 0

• 3x + 2y <= 40, x >= 0, y >= 0

• 4x + y <= 40, x >= 0, y >= 0

• 2x + 3y <= 40, x >= 0, y >= 0

A restaurant offers two types of meals: vegetarian and non-vegetarian. The vegetarian meal costs $12 and the non-vegetarian meal costs $15. If the restaurant wants to earn at least $1,500 in a day, write a system of inequalities to represent the number of each type of meal they need to sell to meet their earnings goal.

• 10x + 15y >= 1500, x >= 0, y >= 0

• 12x + 10y >= 1500, x >= 0, y >= 0

• 12x + 15y >= 1500, x >= 0, y >= 0

• 12x + 15y <= 1500, x >= 0, y >= 0