Graphing Feasible Regions: Linear Inequalities Challenge

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.

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

• x + y ≤ 80

• 2x + 3y ≤ 120

• x + 2y ≤ 100

A school is organizing a field trip and has a budget of $500. Each student ticket costs $10, and each adult ticket costs $15. If they want to take at least 20 students and no more than 30 adults, write a system of inequalities to represent the feasible region for the number of student tickets (x) and adult tickets (y) they can purchase.

• 10x + 15y ≥ 500, x ≤ 20, y ≥ 30

• 10x + 15y ≤ 500, x ≥ 20, y ≤ 30

• 5x + 10y ≤ 500, x ≥ 10, y ≤ 25

• 20x + 15y ≤ 500, x ≥ 15, y ≤ 35

A company produces two types of gadgets: Type A and Type B. Each Type A gadget requires 3 hours of assembly, and each Type B gadget requires 2 hours. The company has a maximum of 30 hours available for assembly. Write a system of inequalities to represent the feasible region for the number of Type A gadgets (x) and Type B gadgets (y) they can produce.

• 4x + y ≤ 30, x ≥ 0, y ≥ 0

• 3x + 3y ≤ 30, x ≥ 0, y ≥ 0

• 2x + 3y ≤ 30, x ≥ 0, y ≥ 0

• 3x + 2y ≤ 30, x ≥ 0, y ≥ 0

A gym has a maximum capacity of 200 members. They offer two types of memberships: basic and premium. Each basic membership costs $20, and each premium membership costs $30. If the gym wants to earn at least $4000 in membership fees, write a system of inequalities to represent the feasible region for the number of basic memberships (x) and premium memberships (y) sold.

• x + y ≤ 200, 20x + 30y ≥ 4000

• x + y ≤ 150, 20x + 30y ≥ 3000

• x + y ≤ 250, 20x + 30y ≥ 5000

• x + y ≥ 200, 20x + 30y ≤ 4000

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. The bakery has a total of 10 hours available for baking. Write a system of inequalities to represent the feasible region for the number of chocolate cakes (x) and vanilla cakes (y) they can bake.

• 2x + 3y ≤ 10, x ≥ 0, y ≥ 0

• 3x + y ≤ 10, x ≥ 0, y ≥ 0

• x + 2y ≤ 10, x ≥ 0, y ≥ 0

• 2x + y ≤ 10, x ≥ 0, y ≥ 0

A concert venue has a seating capacity of 500. They sell two types of tickets: general admission and VIP. Each general admission ticket costs $50, and each VIP ticket costs $100. If they want to make at least $30,000 from ticket sales, write a system of inequalities to represent the feasible region for the number of general admission tickets (x) and VIP tickets (y) sold.

• x + y ≤ 500, 50x + 100y ≥ 30000, x ≥ 0, y ≥ 0

• x + 2y ≤ 500

• x + y ≥ 500

• 50x + 100y ≤ 30000

A clothing store sells shirts and pants. Each shirt costs $15, and each pair of pants costs $25. The store has a budget of $600 for inventory. If they want to stock at least 10 shirts, write a system of inequalities to represent the feasible region for the number of shirts (x) and pants (y) they can purchase.

• 20x + 25y <= 600, x >= 0, y >= 0

• 15x + 25y <= 600, x >= 10, x >= 0, y >= 0

• 10x + 20y <= 600, x >= 5, y >= 0

• 15x + 30y <= 600, x >= 10, y >= 5

A charity organization is planning a fundraiser. They want to sell two types of items: cookies and brownies. Each cookie costs $2, and each brownie costs $3. If they want to raise at least $500 and sell at least 100 items, write a system of inequalities to represent the feasible region for the number of cookies (x) and brownies (y) they can sell.

• 2x + 3y <= 500, x + y <= 100, x >= 0, y >= 0

• 2x + 3y >= 500, x + y >= 100, x >= 0, y >= 0

• x + 2y >= 500, x + y >= 150, x >= 0, y >= 0

• 2x + 3y >= 600, x + y >= 80, x >= 0, y >= 0

A tech company is developing two products: a smartphone and a tablet. Each smartphone requires 4 hours of development, and each tablet requires 3 hours. The company has a total of 60 hours available for development. Write a system of inequalities to represent the feasible region for the number of smartphones (x) and tablets (y) they can develop.

• 4x + 3y ≤ 60, x ≥ 0, y ≥ 0

• 4x + 2y ≤ 60, x ≥ 0, y ≥ 0

• 5x + 3y ≤ 60, x ≥ 0, y ≥ 0

• 4x + 4y ≤ 60, x ≥ 0, y ≥ 0

A restaurant offers two types of meals: vegetarian and non-vegetarian. Each vegetarian meal costs $10, and each non-vegetarian meal costs $15. If the restaurant wants to earn at least $2000 and serve at least 100 meals, write a system of inequalities to represent the feasible region for the number of vegetarian meals (x) and non-vegetarian meals (y) they can serve.

• 10x + 15y >= 2000, x + y >= 100, x >= 0, y >= 0

• x >= 50, y >= 50

• x + y <= 100

• 10x + 15y <= 2000