Solving Systems of Inequalities: Real-World Applications

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 the inequality that represents the maximum area of the pen. What is the feasible region for the dimensions of the pen?

• x + y ≤ 100, x ≥ 0, y ≥ 0

• x + y = 100, x ≥ 0, y ≥ 0

• x + y ≤ 50, 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 number of students that can attend the trip. Identify the feasible region for the number of students.

• x ≥ 0 and x ≤ 14

• x ≥ 0 and x ≤ 10

• x ≥ 5 and x ≤ 15

• x ≥ 0 and x ≤ 20

A factory produces two types of gadgets, A and B. Each gadget A requires 2 hours of labor and gadget 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 gadgets produced?

• x + y ≤ 20; x ≥ 0; y ≥ 0

• 2x + 3y ≥ 60; x ≥ 0; y ≥ 0

• 2x + 3y ≤ 60; x ≥ 0; y ≥ 0

• 4x + 2y ≤ 60; 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 per month. If a customer wants to spend no more than $200 on memberships, write an inequality to represent this situation. What is the feasible region for the number of each type of membership?

• 30x + 50y ≤ 200, x ≥ 0, y ≥ 0

• 30x + 50y ≤ 250, x ≥ 0, y ≥ 0

• 30x + 50y < 200, x ≥ 0, y ≥ 0

• 30x + 50y = 200, 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 24 cups of flour, write the inequality that represents the maximum number of cakes and cookies that can be made. Identify the feasible region for the production of cakes and cookies.

• 3x + 2y ≤ 24

• 2x + y ≤ 24

• x + 4y ≤ 24

• 3x + y ≤ 24

A company produces two products, X and Y. Each product X requires 4 hours of labor and each product Y requires 2 hours. If the company has a maximum of 40 hours of labor available, write a system of inequalities to represent the production limits. Identify the feasible region for the production of products X and Y.

• 3x + 4y ≤ 40

• 2x + 3y ≤ 40

• 4x + 2y ≤ 40, x ≥ 0, y ≥ 0

• x + y ≤ 20

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

• w > 12

• w < 12

• w <= 12

• w >= 12

A restaurant has a special offer where a meal costs $15 and a drink costs $5. If a customer wants to spend no more than $60, write an inequality to represent the number of meals and drinks they can buy. Identify the feasible region for their choices.

• 10x + 5y ≤ 50

• 15x + 5y ≤ 60

• 20x + 5y ≤ 80

• 15x + 10y ≤ 60

A concert venue has a seating capacity of 500. If tickets are sold for $25 each and VIP tickets for $50 each, write a system of inequalities to represent the maximum number of each type of ticket that can be sold without exceeding capacity. Identify the feasible region for ticket sales.

• x + 2y ≤ 500, x ≥ 0, y ≥ 0

• 2x + y ≤ 500, x ≥ 0, y ≥ 0

• x + y < 500, x > 0, y > 0

• x + y ≤ 500, x ≥ 0, y ≥ 0