Real-World Systems of Inequalities: Finding Feasible Regions

A farmer has 100 acres of land 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 the farmer has a maximum of 240 hours of labor available, write a system of inequalities to represent the situation and identify the feasible region for planting corn and wheat.

• The system of inequalities is: x + y ≤ 100, 2x + 3y ≤ 240, x ≥ 0, y ≥ 0.

• x + y ≤ 50, 2x + 3y ≤ 180, x ≥ 0, y ≥ 0

• x + y ≤ 100, 2x + 2y ≤ 240, x ≥ 0, y ≥ 0

• x + y ≤ 200, 2x + 3y ≤ 300, 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 identify the feasible region for the number of students.

• x >= 0 and x <= 20

• x >= 5 and x <= 25

• x >= 0 and x <= 15

• x >= 0 and x <= 10

A company produces two types of gadgets: Type A and Type B. Each Type A gadget requires 4 hours of labor and each Type B gadget requires 2 hours of labor. The company has a maximum of 40 hours of labor available. Write a system of inequalities to represent the production limits and identify the feasible region for the number of gadgets produced.

• 3x + 2y ≤ 40, x ≥ 0, y ≥ 0

• 4x + 3y ≤ 40, x ≥ 0, y ≥ 0

• 4x + 2y ≤ 30, x ≥ 0, y ≥ 0

• 4x + 2y ≤ 40, x ≥ 0, y ≥ 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 identify the feasible region for the production of cakes.

• 3x + 3y ≤ 30, x ≥ 0, y ≥ 0

• 4x + y ≤ 30, x ≥ 0, y ≥ 0

• 2x + 3y ≤ 30, x ≥ 0, y ≥ 0

• 3x + 2y ≤ 30, x ≥ 0, y ≥ 0

A gym offers two types of classes: yoga and spinning. Each yoga class can accommodate 15 people, and each spinning class can accommodate 10 people. If the gym has a maximum capacity of 100 people, write a system of inequalities to represent the class sizes and identify the feasible region for the number of classes offered.

• 10x + 15y ≤ 100, x ≥ 0, y ≥ 0

• 20x + 10y ≤ 100, x ≥ 0, y ≥ 0

• 15x + 10y ≤ 100, x ≥ 0, y ≥ 0

• 15x + 5y ≤ 100, x ≥ 0, y ≥ 0

A concert venue 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 venue wants to make at least $15,000 from ticket sales, write a system of inequalities to represent the ticket sales and identify the feasible region for ticket pricing.

• x + y ≤ 500, 50x + 30y ≥ 15000

• x + y ≤ 400, 50x + 30y ≥ 20000

• 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. The store has a budget of $600 for purchasing inventory. Write a system of inequalities to represent the purchasing limits and identify the feasible region for the number of shirts and pants that can be bought.

• 10x + 20y ≤ 600, x ≥ 0, y ≥ 0

• 15x + 25y ≤ 600, x ≥ 0, y ≥ 0

• 20x + 30y ≤ 600, x ≥ 0, y ≥ 0

• 25x + 35y ≤ 600, x ≥ 0, y ≥ 0

A charity event is selling two types of tickets: VIP and regular. VIP tickets cost $100 each and regular tickets cost $50 each. The charity wants to raise at least $5,000. Write a system of inequalities to represent the ticket sales and identify the feasible region for the number of tickets sold.

• 100x + 50y <= 5000, x >= 0, y >= 0

• x + y >= 100, x >= 0, y >= 0

• 100x + 50y >= 5000, x >= 0, y >= 0

• 100x + 50y = 5000, x >= 0, y >= 0

A pet shelter has a limit of 50 animals it can house. Each dog takes up 2 spaces and each cat takes up 1 space. If the shelter has a maximum of 30 dogs, write a system of inequalities to represent the animal capacity and identify the feasible region for the number of dogs and cats.

• 2x + y ≤ 30, x ≤ 50, x ≥ 0, y ≥ 0

• x + 2y ≤ 50, x ≤ 20, x ≥ 0, y ≥ 0

• 2x + y ≤ 60, x ≤ 25, x ≥ 0, y ≥ 0

• The system of inequalities is: 2x + y ≤ 50, x ≤ 30, x ≥ 0, y ≥ 0.

A restaurant offers two types of meals: vegetarian and non-vegetarian. Each vegetarian meal costs $15 and each non-vegetarian meal costs $25. The restaurant wants to earn at least $1,000 in a day. Write a system of inequalities to represent the meal sales and identify the feasible region for the number of meals sold.

• 20x + 25y >= 1000, x >= 0, y >= 0

• 10x + 20y >= 1000, x >= 0, y >= 0

• 15x + 30y >= 1000, x >= 0, y >= 0

• 15x + 25y >= 1000, x >= 0, y >= 0