Maximizing Resources: Exploring Linear Inequalities & Constraints

A farmer has 100 meters of fencing to create a rectangular pen for sheep. If the length of the pen is represented by x and the width by y, write the inequality that represents the maximum area of the pen. What are the constraints?

• x + y < 100, x > 0, y > 0

• x + y ≤ 50, x ≥ 0, y ≥ 0

• x + y ≤ 75, x ≥ 10, y ≥ 10

• x + y = 50, x ≤ 0, y ≤ 0

A school is planning a field trip and has a budget of $500. Each student ticket costs $15 and each adult ticket costs $20. Write the inequality that represents the number of tickets they can sell. Identify the constraints on the number of tickets sold.

• 15x + 20y ≤ 500; x ≥ 0; y ≥ 0

• 20x + 15y ≤ 500; x ≥ 0; y ≥ 0

• 10x + 15y ≤ 500; x ≥ 0; y ≥ 0

• 15x + 25y ≤ 500; x ≥ 0; y ≥ 0

A factory produces two types of toys: dolls and cars. Each doll requires 2 hours of labor and each car requires 3 hours. If the factory has 60 hours of labor available, write the inequality that represents the maximum number of toys that can be produced. What are the constraints?

• 2x + 3y ≤ 60; x ≥ 0; y ≥ 0

• 2x + 3y ≤ 30; x ≥ 0; y ≥ 0

• 2x + 3y < 60; x ≥ 0; y ≥ 0

• x + y ≤ 20; x ≥ 0; y ≥ 0

A restaurant offers two types of meals: vegetarian and non-vegetarian. The vegetarian meal costs $10 and the non-vegetarian meal costs $15. If the restaurant wants to make at least $300 in a day, write the inequality that represents the number of each type of meal they need to sell. What are the constraints?

• 10x + 15y <= 300; x >= 0; y >= 0

• 10x + 15y = 300; x >= 0; y >= 0

• 5x + 10y >= 300; x >= 0; y >= 0

• 10x + 15y >= 300; x >= 0; y >= 0

A gym has a maximum capacity of 200 members. If x represents the number of adult members and y represents the number of youth members, write the inequality that represents the maximum number of members allowed. Identify the constraints on membership.

• x + y ≤ 200, x ≥ 0, y ≥ 0

• x + y < 200, x ≥ 0, y ≥ 0

• x + y ≤ 250, x ≥ 0, y ≥ 0

• x + y = 200, x < 0, y < 0

A local charity is organizing a fundraiser and needs to sell at least 150 tickets. Each ticket costs $25. Write the inequality that represents the minimum revenue they need to generate. What are the constraints on ticket sales?

• 25x <= 3750, with x <= 150 and x >= 0

• 25x >= 3000, with x >= 100 and x >= 0

• 25x = 3750, with x >= 150 and x >= 0

• 25x >= 3750, with x >= 150 and x >= 0

A company produces two products, A and B. Each product A requires 4 hours of machine time and each product B requires 2 hours. If the company has 40 hours of machine time available, write the inequality that represents the maximum production capacity. Identify the constraints.

• 3x + 2y ≤ 40; x ≥ 0; y ≥ 0

• 4x + 3y ≤ 40; x ≥ 0; y ≥ 0

• 4x + 2y ≥ 40; x ≥ 0; y ≥ 0

• 4x + 2y ≤ 40; x ≥ 0; y ≥ 0

A bookstore sells novels and textbooks. Each novel costs $12 and each textbook costs $20. If the bookstore wants to earn at least $600 in sales, write the inequality that represents the sales goal. What are the constraints on the number of books sold?

• 10x + 25y >= 600; x >= 0, y >= 0

• 12x + 20y >= 600; x >= 0, y >= 0

• 12x + 20y = 600; x >= 1, y >= 1

• 12x + 20y <= 600; x >= 0, y >= 0

A concert hall has a seating capacity of 500. If x represents the number of adult tickets sold and y represents the number of child tickets sold, write the inequality that represents the maximum number of tickets that can be sold. Identify the constraints on ticket sales.

• x + y ≤ 500, x ≥ 0, y ≥ 0

• x + y = 500, x < 0, y ≥ 0

• x + y ≥ 500, x ≥ 0, y < 0

• x + y < 500, x < 0, y < 0

A clothing store has a sale on shirts and pants. Each shirt costs $25 and each pair of pants costs $40. If the store wants to make at least $1000 in sales, write the inequality that represents the sales goal. What are the constraints on the number of items sold?

• 30x + 35y >= 1000; x >= 0, y >= 0

• 25x + 40y >= 1000; x >= 0, y >= 0

• 25x + 40y = 1000; x > 0, y > 0

• 25x + 40y <= 1000; x >= 0, y >= 0