Solving Real-World Linear Inequalities and Feasible Regions

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 the inequality that represents the maximum area of the pen. What are the feasible dimensions?

• The feasible dimensions for maximum area are 25 meters by 25 meters.

• 15 meters by 35 meters

• 20 meters by 30 meters

• 10 meters by 40 meters

A school is planning a field trip and has a budget of $500. The cost per student is $20, and the cost for the bus is a flat fee of $200. Write an inequality to represent the number of students that can attend the trip. What is the feasible region for the number of students?

• x ≤ 10

• x ≤ 15

• x ≤ 5

• x ≤ 20

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. The factory has a maximum of 60 hours of labor available per week. Write a system of inequalities to represent the production limits. What is the feasible region for the number of toys produced?

• 2x + y ≤ 60, x ≥ 0, y ≥ 0

• x + y ≤ 60, x ≥ 0, y ≥ 0

• 2x + 3y ≥ 60, x ≥ 0, y ≥ 0

• 2x + 3y ≤ 60, x ≥ 0, y ≥ 0

A restaurant sells two types of sandwiches: vegetarian and meat. The profit from each vegetarian sandwich is $3 and from each meat sandwich is $5. If the restaurant wants to make at least $100 in profit, write an inequality to represent this situation. What are the possible combinations of sandwiches they can sell?

• 3x + 2y >= 100

• 3x + 5y >= 100, where x >= 0 and y >= 0.

• 5x + 3y >= 100

• 2x + 4y >= 100

A gym has a maximum capacity of 150 members. If they currently have x members enrolled and plan to add y new members, write an inequality to represent the situation. What is the feasible region for the number of new members they can enroll?

• x + y < 150; y ≤ 0

• x + y ≥ 150; y < 0

• x + y = 150; y > 0

• x + y ≤ 150; y ≥ 0

A local bakery sells cakes and cookies. Each cake requires 3 cups of flour and each cookie requires 1 cup of flour. If the bakery has 30 cups of flour available, write a system of inequalities to represent the maximum number of cakes and cookies they can make. What is the feasible region for their production?

• 2x + y <= 30, x >= 0, y >= 0

• 3x + y <= 30, x >= 0, y >= 0

• x + y <= 30, x >= 0, y >= 0

• 3x + 2y <= 30, x >= 0, y >= 0

A concert hall has 200 seats. If tickets for adults cost $15 and tickets for children cost $10, and the total revenue must be at least $1500, write an inequality to represent this situation. What are the feasible combinations of adult and child tickets sold?

• x + y ≥ 200 and 15x + 10y ≤ 1500

• x + y ≤ 150 and 15x + 10y ≥ 2000

• x + y = 200 and 15x + 10y = 1500

• x + y ≤ 200 and 15x + 10y ≥ 1500

A charity organization is collecting donations. They want to collect at least $1000. If they receive $50 from each individual donor and $100 from each corporate donor, write an inequality to represent the total donations. What are the feasible combinations of individual and corporate donors?

• 50x + 100y = 1000, where x > 0 and y > 0.

• 50x + 100y ≥ 1000, where x ≥ 0 and y ≥ 0.

• 50x + 100y ≤ 1000, where x ≥ 0 and y ≥ 0.

• 25x + 75y ≥ 1000, where x ≥ 0 and y ≥ 0.

A student is saving money for a new laptop that costs $800. If they save $50 per week from their allowance and $20 from doing chores, write an inequality to represent the number of weeks needed to save enough money. What is the feasible region for their savings?

• w < 12

• w <= 12

• w > 12

• w >= 12