Finding Feasible Regions in Linear Inequalities

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 are the feasible dimensions?

• x = 40 meters and y = 10 meters

• The feasible dimensions are x = 25 meters and y = 25 meters.

• x = 30 meters and y = 20 meters

• x = 50 meters and y = 5 meters

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. What is the feasible region for the number of students?

• x ≥ 0 and x ≤ 10

• x ≥ 5 and x ≤ 25

• x ≥ 0 and x ≤ 20

• x ≥ 0 and x ≤ 14

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

• 2x + 3y ≥ 60

• x + y ≤ 20

• 2x + 3y = 60

• 2x + 3y ≤ 60

A restaurant sells two types of sandwiches: turkey and veggie. The profit from each turkey sandwich is $3 and from each veggie sandwich is $2. If the restaurant wants to make at least $60 in profit, write the inequality that represents this situation. What combinations of sandwiches can they sell?

• 3x + 2y = 60

• 4x + 2y ≥ 60

• 3x + 2y ≥ 60

• 3x + 2y ≤ 60

A charity event aims to raise at least $1000. Each ticket sold for the event costs $25, and each donation made is at least $50. Write a system of inequalities to represent the number of tickets sold and donations received. What is the feasible region for their fundraising goal?

• x + y >= 40, x >= 0, y >= 0

• The system of inequalities is: 25x + 50y >= 1000, x >= 0, y >= 0.

• 25x + 50y = 1000, x >= 0, y >= 0

• 25x + 50y <= 1000, x >= 0, y >= 0

A gym has a maximum capacity of 200 members. If they currently have x members enrolled in yoga classes and y members in pilates classes, write the inequality that represents the maximum number of members. What combinations of yoga and pilates members are feasible?

• x + y ≤ 200, where x ≥ 0 and y ≥ 0.

• x + y < 200, where x ≥ 0 and y ≥ 0.

• x + y = 200, where x > 0 and y > 0.

• x + y ≥ 200, where x ≤ 0 and y ≤ 0.

A local bakery sells muffins and cookies. Each muffin costs $2 and each cookie costs $1. If the bakery wants to earn at least $100 in one day, write the inequality that represents their sales goal. What combinations of muffins and cookies can they sell?

• x + 2y >= 100

• 2x + y >= 100, where x >= 0 and y >= 0.

• 3x + y >= 100

• x + y >= 50

A concert venue has a seating capacity of 300. If x represents the number of VIP tickets sold and y represents the number of general admission tickets sold, write the inequality that represents the seating limit. What are the feasible combinations of ticket sales?

• x + y = 300, where x, y > 0

• x + y < 300, where x, y ≥ 0

• x + y ≤ 300, where x, y ≥ 0

• x + y ≥ 300, where x, y ≤ 0

A clothing store has a sale on shirts and pants. Each shirt costs $15 and each pair of pants costs $25. If a customer wants to spend no more than $200, write the inequality that represents their spending limit. What combinations of shirts and pants can they buy?

• 15x + 25y ≤ 200, where x ≥ 0 and y ≥ 0.

• 15x + 30y ≤ 200

• 10x + 20y ≤ 200

• 20x + 25y ≤ 200

A tech company produces laptops and tablets. Each laptop requires 4 hours of assembly and each tablet requires 2 hours. If the company has 40 hours available for assembly, write the inequality that represents the production limits. What is the feasible region for the number of laptops and tablets produced?

• 5x + y ≤ 40; feasible region is bounded by x ≥ 0, y ≥ 0, and the line y ≤ 40 - 5x.

• 4x + 2y ≤ 40; feasible region is bounded by x ≥ 0, y ≥ 0, and the line y ≤ 20 - 2x.

• 3x + 4y ≤ 40; feasible region is bounded by x ≥ 0, y ≥ 0, and the line y ≤ 10 - 3x.

• 2x + 3y ≤ 40; feasible region is bounded by x ≥ 0, y ≥ 0, and the line y ≤ 15 - 2x.