Graphing Inequalities: Finding Feasible Regions in Context

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. Graph the inequalities and identify the feasible region.

• x + y ≤ 75, x ≥ 10, y ≥ 10

• x + y ≥ 50, x ≤ 0, y ≤ 0

• x + y ≤ 100, x ≥ 0, y ≥ 0

• The system of inequalities is: { 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. Graph the inequalities and identify the feasible region.

• 0 ≤ x ≤ 20

• 0 ≤ x ≤ 10

• 0 ≤ x ≤ 14

• 0 ≤ x ≤ 5

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. The factory has a maximum of 60 hours of labor available. Write a system of inequalities to represent the production limits. Graph the inequalities and identify the feasible region.

• x + y ≤ 20, x ≥ 0, y ≥ 0

• 2x + 3y ≥ 60, x ≤ 0, y ≤ 0

• The system of inequalities is: 2x + 3y ≤ 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 3 eggs and each vanilla cake requires 2 eggs. The bakery has a total of 30 eggs available. Write a system of inequalities to represent the number of cakes that can be made. Graph the inequalities and identify the feasible region.

• 3x + 2y ≤ 30, x ≥ 0, y ≥ 0

• x + y ≤ 15

• 2x + 3y ≤ 30

• 4x + y ≤ 30

A gym offers two types of memberships: basic and premium. The basic membership costs $30 per month and the premium membership costs $50 per month. If the gym wants to earn at least $1,200 in a month, write a system of inequalities to represent the number of each type of membership sold. Graph the inequalities and identify the feasible region.

• 30x + 50y <= 1200, x >= 0, y >= 0

• 30x + 50y >= 1000, x >= 0, y >= 0

• 30x + 50y >= 1200, x >= 0, y >= 0

• 20x + 40y >= 1200, 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. Graph the inequalities and identify the feasible region.

• x + y <= 500, 50x + 30y >= 15000

• x + y >= 500, 50x + 30y <= 15000

• x + y = 500, 50x + 30y = 15000

• x + y <= 400, 50x + 30y >= 20000

A clothing store sells shirts and pants. Each shirt costs $20 and each pair of pants costs $30. If the store wants to make at least $1,000 in sales, write a system of inequalities to represent the number of shirts and pants sold. Graph the inequalities and identify the feasible region.

• 20x + 30y <= 1000, x >= 0, y >= 0

• 10x + 15y >= 1000, x >= 0, y >= 0

• 20x + 30y = 1000, x >= 0, y >= 0

• The system of inequalities is: 20x + 30y >= 1000, x >= 0, y >= 0.

A charity organization is organizing a fundraiser. They plan to sell tickets for a dinner event at $25 each and for a raffle at $10 each. If they want to raise at least $2,000, write a system of inequalities to represent the number of tickets sold. Graph the inequalities and identify the feasible region.

• 25x + 10y ≤ 2000, x ≥ 0, y ≥ 0

• 25x + 10y ≥ 1000, x ≥ 0, y ≥ 0

• 25x + 10y = 2000, x < 0, y < 0

• The system of inequalities is: 25x + 10y ≥ 2000, x ≥ 0, y ≥ 0.

A pet store has a limit of 50 animals that can be housed. If they have dogs and cats, and each dog takes up 2 spaces while each cat takes up 1 space, write a system of inequalities to represent the maximum number of dogs and cats. Graph the inequalities and identify the feasible region.

• x + 2y <= 50, x >= 0, y <= 0

• 2x + y <= 30, x >= 0, y >= 0

• 2x + y >= 50, x <= 0, y <= 0

• The system of inequalities is: 2x + y <= 50, x >= 0, y >= 0.

A local gym has a maximum capacity of 200 members. If they have a mix of basic and premium memberships, where basic memberships are limited to 150 and premium memberships to 100, write a system of inequalities to represent the membership limits. Graph the inequalities and identify the feasible region.

• x + y >= 200, x >= 150, y >= 100

• x + y <= 300, x <= 200, y <= 150

• x + y <= 150, x <= 100, y <= 50

• The system of inequalities is: x + y <= 200, x <= 150, y <= 100.