Exploring Feasible Regions with Linear Inequalities

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 3 hours of labor. If he has a total of 240 hours of labor available, write a system of linear inequalities to represent the feasible region for the number of acres of corn (x) and wheat (y) he can plant.

• x + y ≤ 100, 2x + 3y ≤ 240, x ≥ 0, y ≥ 0

• x + y ≥ 120

• 2x + 3y ≤ 300

• x + y ≤ 80

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 linear inequalities to represent the number of students (x) that can attend the trip while staying within budget.

• x >= 0 and x <= 15

• x >= 0 and x <= 20

• x >= 10 and x <= 30

• x >= 0 and x <= 25

A company produces two types of gadgets, A and B. Each gadget A requires 3 hours of assembly and each gadget B requires 2 hours. The company has 60 hours of assembly time available. Write the inequalities that represent the feasible region for the number of gadgets A (x) and B (y) that can be produced.

• 4x + 2y ≤ 60, x ≥ 0, y ≥ 0

• 2x + 3y ≤ 60, x ≥ 0, y ≥ 0

• 3x + y ≤ 60, x ≥ 0, y ≥ 0

• 3x + 2y ≤ 60, x ≥ 0, y ≥ 0

A bakery sells two types of cakes: chocolate and vanilla. Each chocolate cake requires 2 pounds of flour and each vanilla cake requires 1 pound. If the bakery has 20 pounds of flour, write a system of inequalities to represent the number of chocolate (x) and vanilla (y) cakes that can be made.

• 2x + y >= 20, x >= 0, y >= 0

• x + 2y <= 20, x >= 0, y >= 0

• 3x + y <= 20, x >= 0, y >= 0

• 2x + y <= 20, x >= 0, y >= 0

A gym has a maximum capacity of 150 members. If each membership costs $30 and the gym wants to earn at least $3000 in membership fees, write a system of inequalities to represent the number of members (x) and the total fees (y) they can collect.

• x <= 100, 30x >= 4000

• x <= 150, 30x >= 3000

• x < 150, 30x <= 3000

• x >= 150, 30x < 3000

A local theater has 200 seats. Tickets for adults cost $15 and tickets for children cost $10. If the theater wants to earn at least $2500 from ticket sales, write a system of inequalities to represent the number of adult (x) and child (y) tickets sold.

• x + y ≥ 200, 15x + 10y = 2500

• x + y ≤ 200, 15x + 10y ≥ 2500

• x + y = 200, 15x + 10y ≤ 2500

• x + y ≤ 150, 15x + 10y ≥ 3000

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 make at least $1000 in sales, write a system of inequalities to represent the number of vegetarian (x) and non-vegetarian (y) meals sold.

• 5x + 10y >= 1000, x >= 0, y >= 0

• 10x + 15y = 1000, x >= 0, y >= 0

• 10x + 15y <= 1000, x >= 0, y >= 0

• 10x + 15y >= 1000, x >= 0, y >= 0

A clothing store sells shirts and pants. Each shirt costs $20 and each pair of pants costs $30. If the store wants to earn at least $2000 in sales, write a system of inequalities to represent the number of shirts (x) and pants (y) sold.

• 25x + 35y >= 2000, x >= 0, y >= 0

• 10x + 15y >= 2000, x >= 0, y >= 0

• 20x + 30y <= 2000, x >= 0, y >= 0

• 20x + 30y >= 2000, x >= 0, y >= 0

A charity event is selling tickets for adults at $25 and for children at $15. If they want to raise at least $3000, write a system of inequalities to represent the number of adult (x) and child (y) tickets sold, considering they can sell a maximum of 200 tickets in total.

• 20x + 10y >= 3000, x + y <= 150

• 30x + 20y >= 3000, x + y <= 250

• 25x + 15y <= 3000, x + y >= 200

• 25x + 15y >= 3000, x + y <= 200

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. If the company has 120 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 + 2y ≤ 120, x ≥ 0, y ≥ 0

• 3x + 4y ≤ 120, x ≥ 0, y ≥ 0

• 4x + 3y ≤ 120, x ≥ 0, y ≥ 0

• 5x + 3y ≤ 120, x ≥ 0, y ≥ 0