Mastering Linear Inequalities: Feasible Regions & Solutions

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

• x + y < 50, x > 0, y > 0

• x + y = 100, x ≥ 0, y ≤ 0

• x + y ≤ 100, x ≥ 0, y ≥ 0

• x + y ≤ 50, x ≥ 0, y ≥ 0

A school is planning 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 number of students that can attend the trip. What is the feasible region for the number of students?

• x >= 0 and x <= 20

• x >= 0 and x <= 5

• x >= 0 and x <= 10

• x >= 0 and x <= 14, where x is the number of students.

A factory produces two types of toys, A and B. Each toy A requires 2 hours of labor and each toy B requires 3 hours. If the factory has 60 hours of labor available, write the inequality that represents the production limits. What is the feasible region for the number of toys produced?

• 2x + 3y ≥ 60

• 2x + 3y ≤ 60

• 4x + y ≤ 60

• x + 2y ≤ 60

A restaurant sells two types of sandwiches: chicken and veggie. The profit from each chicken sandwich is $3 and from each veggie sandwich is $2. If the restaurant wants to make at least $60 in profit, write the inequality that represents this situation. What are the possible combinations of sandwiches they can sell?

• 5x + 2y ≥ 60, where x, y ≥ 0.

• 3x + 4y ≥ 60, where x, y ≥ 0.

• 3x + 2y ≥ 60, where x, y ≥ 0.

• 3x + 2y ≤ 60, where x, y ≥ 0.

A gym has a maximum capacity of 200 members. If the number of adult members is represented by x and the number of youth members by y, write the inequality that represents the membership limit. What are the feasible values for x and y?

• x + y ≥ 200, x < 0, y < 0

• x + y = 200, x ≤ 0, y ≤ 0

• x + y < 200, x > 0, y > 0

• x + y ≤ 200, x ≥ 0, y ≥ 0

A company produces two products, P1 and P2. Each product P1 requires 4 hours of machine time and each product P2 requires 2 hours. If the machine is available for 40 hours, write the system of inequalities that represents the production limits. What is the feasible region for the number of products produced?

• 4x + 3y ≤ 40, x ≥ 0, y ≥ 0

• 3x + 2y ≤ 40, x ≥ 0, y ≥ 0

• The system of inequalities is: 4x + 2y ≤ 40, x ≥ 0, y ≥ 0.

• 4x + 2y ≤ 30, x ≥ 0, y ≥ 0

A local bakery sells cakes and cookies. Each cake requires 3 cups of flour and each cookie requires 1 cup. If the bakery has 15 cups of flour, write the inequality that represents the flour constraint. What are the feasible values for the number of cakes and cookies?

• 2x + y ≤ 15, where x ≥ 0 and y ≥ 0.

• x + y ≤ 15, where x ≥ 0 and y ≥ 0.

• 3x + 2y ≤ 15, where x ≥ 0 and y ≥ 0.

• 3x + y ≤ 15, where x ≥ 0 and y ≥ 0.

A concert hall has a seating capacity of 500. If the number of adult tickets sold is represented by x and student tickets by y, write the inequality that represents the seating limit. What are the possible combinations of adult and student tickets that can be sold?

• x + y ≤ 500, where x ≥ 0 and y ≥ 0.

• x + y ≥ 500, where x ≤ 500 and y ≤ 500.

• x + y < 500, where x ≥ 0 and y ≥ 0.

• x + y = 500, where x > 0 and 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 keep the total cost under $600, write the inequality that represents this situation. What are the feasible combinations of shirts and pants they can sell?

• 15x + 25y < 500

• 25x + 20y < 700

• 10x + 40y < 600

• 20x + 30y < 600; feasible combinations are (x, y) such that y < (600 - 20x) / 30 for non-negative integers x and y.

A charity organization is collecting donations. Each donation from an individual is $10 and from a corporation is $50. If they want to raise at least $1000, write the inequality that represents the donation limits. What are the possible combinations of individual and corporate donations?

• 15x + 30y ≥ 1000

• 5x + 25y ≥ 1000

• 10x + 50y ≤ 1000

• 10x + 50y ≥ 1000; Possible combinations include (x, y) such that x ≥ 0 and y ≥ 0.