Feasible Regions and Boundary Lines 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 is the feasible region for the dimensions of the pen?

• x + y ≤ 50, x ≥ 0, y ≥ 0

• x + y = 100, x ≥ 0, y ≥ 0

• x + y < 50, x > 0, y > 0

• x + y ≤ 100, 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 the inequality that represents the number of students (x) and the total cost (y). Identify the boundary line and feasible region for the number of students.

• 35x ≤ 500, boundary line x = 14.29, feasible region x ≤ 14

• 40x ≤ 500, boundary line x = 12.5, feasible region x ≤ 12

• 30x ≤ 500, boundary line x = 16.67, feasible region x ≤ 15

• 25x ≤ 500, boundary line x = 20, feasible region x ≤ 20

A factory produces two types of toys: plush toys and action figures. Each plush toy requires 2 hours of labor and each action figure 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 each type of toy produced?

• x + y ≤ 60, x ≥ 0, y ≥ 0

• 2x + y ≤ 60, x ≥ 0, y ≥ 0

• 2x + 3y ≥ 60, x ≥ 0, y ≥ 0

• 2x + 3y ≤ 60, x ≥ 0, y ≥ 0

A gym has a maximum capacity of 200 members. If the number of adult members is represented by x and the number of youth members by y, write the inequality that represents the membership limit. What is the feasible region for the number of adult and youth members?

• x + y ≥ 200; feasible region: x ≥ 0, y ≥ 0, x + y ≥ 200.

• x + y < 200; feasible region: x < 0, y < 0, x + y < 200.

• x + y ≤ 200; feasible region: x ≥ 0, y ≥ 0, x + y ≤ 200.

• x + y = 200; feasible region: x ≥ 0, y ≥ 0, x + y = 200.

A concert venue has a seating capacity of 300. If tickets for adults cost $50 and tickets for children cost $30, and the total revenue must be at least $10,000, write the inequality for the number of adult (x) and child (y) tickets sold. Identify the boundary line and feasible region.

• x + y ≤ 300 and 50x + 30y ≥ 10000

• x + y ≤ 250 and 50x + 30y ≥ 12000

• x + y ≥ 300 and 50x + 30y ≤ 10000

• x + y = 300 and 50x + 30y = 10000

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 the inequality for the number of shirts (x) and pants (y) sold. What is the feasible region for the sales?

• 25x + 35y ≥ 600

• 10x + 15y ≥ 600

• 20x + 30y ≥ 600

• 20x + 30y ≤ 600

A local community center has a budget of $1,200 for a new program. If each workshop costs $150 and each event costs $200, write the inequality that represents the number of workshops (x) and events (y) that can be held. Identify the boundary line and feasible region.

• 150x + 200y ≥ 1200

• 150x + 200y ≤ 1200

• 100x + 250y ≤ 1200

• 150x + 200y = 1200

A pet shelter can take care of a maximum of 50 animals. If dogs are represented by x and cats by y, write the inequality that represents the maximum number of animals. What is the feasible region for the number of dogs and cats?

• x + y ≥ 50; feasible region: x ≥ 0, y ≥ 0, x + y ≥ 50

• x + y < 50; feasible region: x < 0, y < 0, x + y < 50

• x + y = 50; feasible region: x > 0, y > 0, x + y = 50

• x + y ≤ 50; feasible region: x ≥ 0, y ≥ 0, x + y ≤ 50

A restaurant has a maximum of 80 tables. If the number of indoor tables is represented by x and outdoor tables by y, write the inequality that represents the seating capacity. Identify the boundary line and feasible region for the number of indoor and outdoor tables.

• x + y < 80; Boundary line: x + y = 80; Feasible region: area on or above the line in the first quadrant.

• x + y = 80; Boundary line: x + y = 80; Feasible region: area on the line in the first quadrant.

• x + y ≤ 80; Boundary line: x + y = 80; Feasible region: area on or below the line in the first quadrant.

• x + y ≥ 80; Boundary line: x + y = 80; Feasible region: area above the line in the first quadrant.