Mastering Inequalities: Graphing & Writing Scenarios

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 <= 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

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 ≤ 25 and x ≥ 0

• x ≤ 15 and x ≥ 0

• x ≤ 20 and x ≥ 5

• x ≤ 10 and x ≥ 0

A bakery sells two types of cakes: chocolate and vanilla. Each chocolate cake requires 3 eggs and each vanilla cake requires 2 eggs. If the bakery has 30 eggs available, write a system of inequalities to represent the number of cakes that can be made and graph the feasible region.

• 3x + 2y ≤ 30, x ≥ 0, y ≥ 0

• 2x + 3y ≤ 30

• 4x + y ≤ 30

• x + y ≤ 30

A gym offers two types of memberships: basic and premium. The basic membership costs $30 per month, while the premium membership costs $50 per month. If the gym wants to earn at least $1,000 in membership fees, write a system of inequalities to represent the situation and graph the feasible region.

• 30x + 50y <= 1000, x >= 0, y >= 0

• 30x + 50y = 1000, x >= 0, y >= 0

• 30x + 50y >= 500, x >= 0, y >= 0

• 30x + 50y >= 1000, x >= 0, y >= 0

A company produces two products, A and B. Each product A requires 4 hours of labor and each product B requires 2 hours of labor. If the company has a maximum of 40 hours of labor available, write a system of inequalities to represent the production limits and graph the feasible region.

• 4x + 2y ≤ 40, x ≥ 0, y ≥ 0

• 5x + y ≤ 40, x ≥ 0, y ≥ 0

• 2x + 3y ≤ 40, x ≥ 0, y ≥ 0

• 3x + 4y ≤ 40, 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 to represent the ticket sales and graph the feasible region.

• x + y ≤ 400, 50x + 30y ≥ 20000

• x + y ≤ 500, 50x + 30y ≥ 15000

• x + y = 500, 50x + 30y = 15000

• x + y ≥ 500, 50x + 30y ≤ 15000

A clothing store sells shirts and pants. Each shirt costs $20 and each pair of pants costs $30. If the store wants to make at least $1,200 in sales and has a maximum of 50 items to sell, write a system of inequalities to represent the sales and graph the feasible region.

• 20x + 30y >= 1200, x + y <= 40, x >= 0, y >= 0

• The system of inequalities is: 20x + 30y >= 1200, x + y <= 50, x >= 0, y >= 0.

• 20x + 30y <= 1200, x + y >= 50, x >= 0, y >= 0

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

A charity organization is collecting donations. They want to collect at least $2,000 and have a maximum of 100 donors. If each donor contributes at least $20, write a system of inequalities to represent the donations and graph the feasible region.

• y <= 20x

• y >= 2000, x <= 100, y >= 20x

• y <= 2000

• x >= 100

A restaurant offers two types of meals: vegetarian and non-vegetarian. Each vegetarian meal costs $15 and each non-vegetarian meal costs $25. If the restaurant wants to earn at least $1,500 and can serve a maximum of 80 meals, write a system of inequalities to represent the meal sales and graph the feasible region.

• 15x + 30y ≥ 1500, x + y ≤ 60, x ≥ 0, y ≥ 0

• 15x + 25y ≥ 1500, x + y ≤ 80, x ≥ 0, y ≥ 0

• 20x + 25y ≥ 1500, x + y ≤ 80, x ≥ 0, y ≥ 10

• 10x + 20y ≥ 1500, x + y ≤ 100, x ≥ 0, y ≥ 0

A tech company is developing two products: a smartphone and a tablet. Each smartphone requires 5 hours of development and each tablet requires 3 hours. If the company has a maximum of 60 hours available for development, write a system of inequalities to represent the development limits and graph the feasible region.

• 6x + 3y ≤ 60, x ≥ 0, y ≥ 0

• 4x + 2y ≤ 60, x ≥ 0, y ≥ 0

• 5x + 4y ≤ 60, x ≥ 0, y ≥ 0

• 5x + 3y ≤ 60, x ≥ 0, y ≥ 0