Analyzing Systems of Inequalities in Real-World Scenarios

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 linear inequality to represent the maximum area of the pen. Graph the feasible region.

• x + y ≤ 50, 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 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. Identify the feasible region on a graph.

• x ≥ 0 and x ≤ 14

• x ≥ 0 and x ≤ 10

• x ≥ 0 and x ≤ 20

• x ≥ 5 and x ≤ 15

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. If the factory has 60 hours of labor available, write a linear inequality to represent the production limits. Graph the feasible region.

• 2x + 3y ≥ 60

• 2x + 3y ≤ 60

• 2x + 3y = 60

• x + y ≤ 20

A local gym offers two types of memberships: basic and premium. The basic membership costs $30 per month, while the premium membership costs $50 per month. If a customer can spend no more than $200 per month on memberships, write a linear inequality and graph the feasible region.

• 30x + 50y = 200

• 30x + 50y ≥ 200

• 30x + 50y ≤ 200

• 20x + 40y ≤ 200

A bakery sells two types of cakes: chocolate and vanilla. Each chocolate cake requires 3 cups of flour and each vanilla cake requires 2 cups. If the bakery has 24 cups of flour, write a system of inequalities to represent the maximum number of cakes that can be made. Graph the feasible region.

• 4x + y ≤ 24, x ≥ 0, y ≥ 0

• 3x + 2y ≤ 24, x ≥ 0, y ≥ 0

• 2x + 3y ≤ 24, x ≥ 0, y ≥ 0

• 3x + 3y ≤ 24, x ≥ 0, y ≥ 0

A company produces two products, P1 and P2. Each product requires different amounts of resources: P1 requires 4 units of resource A and 2 units of resource B, while P2 requires 3 units of resource A and 5 units of resource B. If the company has 60 units of resource A and 40 units of resource B, write a system of inequalities and graph the feasible region.

• 4x + 2y <= 60, 3x + 5y <= 40, x >= 0, y >= 0

• 4x + 3y <= 60, 2x + 5y <= 40, x >= 0, y >= 0

• 4x + 3y <= 40, 2x + 5y <= 60, x >= 0, y >= 0

• 4x + 3y <= 60, 2x + 5y <= 30, x >= 0, y >= 0

A charity event is selling tickets for adults and children. Adult tickets cost $15 and children tickets cost $10. If the total revenue from ticket sales cannot exceed $600, write a linear inequality to represent this situation and graph the feasible region.

• 5x + 3y ≤ 200

• 4x + y ≤ 150

• 3x + 2y ≤ 120

• 2x + 3y ≤ 100

A restaurant has a limited number of tables. Each table can seat up to 4 people. If the restaurant has 10 tables, write a linear inequality to represent the maximum number of customers that can be seated. Graph the feasible region.

• x <= 40

• x < 40

• x >= 40

• x = 30

A student is saving money for a new laptop that costs $800. If they save $50 per week from their allowance, write a linear inequality to represent the number of weeks needed to save enough money. Graph the feasible region.

• w > 16

• w <= 16

• w >= 16

• w < 16

A clothing store has a sale on shirts and pants. Shirts are $20 each and pants are $30 each. If a customer wants to spend no more than $150, write a linear inequality to represent the spending limit and graph the feasible region.

• 20x + 30y ≥ 150

• 10x + 15y ≤ 150

• 20x + 30y = 150

• 20x + 30y ≤ 150