Linear Inequalities: Identifying Feasible Regions in Real Life

A farmer has 100 meters of fencing to create a rectangular pen for his animals. If the length of the pen is represented by x and the width by y, write a system of inequalities to represent the constraints on the dimensions of the pen. What is the feasible region for the dimensions?

• x + y ≤ 100, x ≥ 0, y ≥ 0

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

• x + y ≤ 75, x ≥ 0, y ≤ 0

• x + y ≥ 50, x ≤ 0, y ≤ 0

A school is organizing a field trip and has a budget of $500. The cost per student is $20 for transportation and $15 for admission. Write a system of inequalities to represent the maximum number of students that can attend. Identify the feasible region for the number of students.

• 0 ≤ x ≤ 5

• 0 ≤ x ≤ 20

• 0 ≤ x ≤ 10

• 0 ≤ x ≤ 14

A factory produces two types of toys: type A and type B. Each type A toy requires 2 hours of labor and each type B toy requires 3 hours. If the factory has 60 hours of labor available, write a system of inequalities to represent the production limits. What is the feasible region for the number of toys produced?

• 2x + 3y ≤ 70, x ≥ 0, y ≥ 0

• 2x + 3y ≤ 50, x ≥ 0, y ≥ 0

• x + y ≤ 60, x ≥ 0, y ≥ 0

• 2x + 3y ≤ 60, x ≥ 0, y ≥ 0

A bakery sells two types of cakes: chocolate and vanilla. Each chocolate cake requires 3 cups of flour and each vanilla cake requires 2 cups. If the bakery has 30 cups of flour, write a system of inequalities to represent the maximum number of cakes that can be made. What is the feasible region for the cakes?

• 4x + y ≤ 30, x ≥ 0, y ≥ 0

• 3x + 2y ≤ 30, x ≥ 0, y ≥ 0

• 3x + 3y ≤ 30, x ≥ 0, y ≥ 0

• 2x + 3y ≤ 30, x ≥ 0, y ≥ 0

A gym offers two types of memberships: basic and premium. The basic membership costs $30 per month and the premium costs $50. If a customer wants to spend no more than $200 a month, write a system of inequalities to represent the number of each type of membership they can purchase. Identify the feasible region.

• 30x + 50y = 200, x ≥ 0, y ≥ 0

• The system of inequalities is: 30x + 50y ≤ 200, x ≥ 0, y ≥ 0.

• 30x + 50y ≤ 150, x ≥ 5, y ≥ 0

• 30x + 50y ≥ 200, x ≤ 0, y ≤ 0

A concert hall has a seating capacity of 500. If tickets for the front row cost $50 and tickets for the back row cost $30, and the total revenue must be at least $15,000, write a system of inequalities to represent the ticket sales. What is the feasible region for ticket sales?

• x + y <= 500, 50x + 30y = 10000

• x + y <= 400, 50x + 30y >= 20000

• x + y <= 500, 50x + 30y >= 15000

• x + y >= 500, 50x + 30y <= 15000

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 a system of inequalities to represent the production limits. What is the feasible region for the products?

• 5x + y ≤ 40

• 4x + 2y ≤ 40, x ≥ 0, y ≥ 0

• 3x + 4y ≤ 40

• 2x + 3y ≤ 40

A charity event is selling two types of tickets: VIP tickets for $100 and regular tickets for $50. If the total revenue from ticket sales must be at least $2,000, write a system of inequalities to represent the ticket sales. Identify the feasible region for ticket sales.

• 50x + 100y ≥ 2000, x ≥ 0, y < 0

• 100x + 50y ≤ 2000, x ≥ 0, y ≥ 0

• 100x + 50y ≥ 2000, x ≥ 0, y ≥ 0

• 100x + 50y = 2000, 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 make at least $1,000 in sales, write a system of inequalities to represent the meal sales. What is the feasible region for the meals?

• 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 two types of shirts: casual and formal. Each casual shirt costs $20 and each formal shirt costs $30. If the store wants to earn at least $1,500 in sales, write a system of inequalities to represent the shirt sales. Identify the feasible region for the shirts.

• 20x + 30y <= 1500, x >= 0, y >= 0

• 20x + 30y = 1500, x >= 0, y >= 0

• 10x + 15y >= 1500, x >= 0, y >= 0

• System of inequalities: 20x + 30y >= 1500, x >= 0, y >= 0.