Graphing Linear Inequalities: Finding Feasible Regions

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

• x + y ≤ 100, x ≥ 0, y ≥ 0

• x + y ≤ 75, x ≥ 0, y ≤ 0

• The system of linear inequalities is: { x + y ≤ 50, 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 linear inequalities to represent the number of students that can attend the trip. Graph the inequalities and identify the feasible region.

• 0 ≤ x ≤ 10

• 0 ≤ x ≤ 14

• 0 ≤ x ≤ 20

• 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 per week. Write a system of linear inequalities to represent the production limits. Graph the inequalities and identify the feasible region.

• 2x + 3y ≥ 60, x ≤ 0, y ≤ 0

• 3x + 2y ≤ 60, x ≥ 0, y ≤ 0

• The system of inequalities is: 2x + 3y ≤ 60, x ≥ 0, y ≥ 0.

• x + y ≤ 20, 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 linear inequalities to represent the number of cakes that can be made. Graph the inequalities and identify the feasible region.

• x + y <= 30, x >= 0, y <= 0

• The system of linear inequalities is: 3x + 2y <= 30, x >= 0, y >= 0.

• 3x + 2y >= 30, x <= 0, y <= 0

• 2x + 3y <= 30, x >= 0, y >= 0

A gym offers two types of classes: yoga and spinning. Each yoga class can accommodate 15 people, and each spinning class can accommodate 10 people. If the gym can host a maximum of 100 people in total, write a system of linear inequalities to represent the class sizes. Graph the inequalities and identify the feasible region.

• 15x + 10y >= 100, x <= 0, y <= 0

• 10x + 15y <= 100, x >= 0, y <= 0

• 15x + 10y <= 50, x >= 0, y >= 0

• The system of linear inequalities is: 15x + 10y <= 100, x >= 0, y >= 0.

A concert venue has a seating capacity of 300. Tickets for the front row cost $50 each, and tickets for the back row cost $30 each. If the venue wants to make at least $10,000 from ticket sales, write a system of linear inequalities to represent the ticket sales. Graph the inequalities and identify the feasible region.

• x + y >= 300, 50x + 30y <= 10000, x <= 0, y <= 0

• x + y <= 300, 50x + 30y >= 5000, x >= 0, y <= 0

• x + y <= 300, 50x + 30y >= 10000, x >= 0, y >= 0

• x + y <= 250, 50x + 30y >= 15000, 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 earn at least $300 in a day, write a system of linear inequalities to represent the meal sales. Graph the inequalities and identify the feasible region.

• 10x + 15y <= 300, x >= 0, y >= 0

• 5x + 10y >= 300, x >= 0, y >= 0

• 10x + 15y = 300, x >= 0, y >= 0

• 10x + 15y >= 300, 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 make at least $600 in sales, write a system of linear inequalities to represent the sales. Graph the inequalities and identify the feasible region.

• The system of inequalities is: 20x + 30y >= 600, x >= 0, y >= 0.

• 20x + 30y = 600, x >= 0, y >= 0

• 20x + 30y <= 600, x >= 0, y >= 0

• 10x + 15y >= 600, x >= 0, y >= 0

A charity organization is organizing a fundraiser. They plan to sell two types of items: cookies and brownies. Each cookie costs $2 and each brownie costs $3. If they want to raise at least $200, write a system of linear inequalities to represent the sales. Graph the inequalities and identify the feasible region.

• 2x + 3y >= 100, x >= 0, y >= 0

• x + y >= 200, x >= 0, y >= 0

• 2x + 3y >= 200, x >= 0, y >= 0

• 2x + 3y <= 200, x >= 0, y >= 0