Inequalities in Action: Writing & Analyzing Feasible Regions

A farmer has 100 acres of land to plant corn and wheat. Each acre of corn requires 2 hours of labor, and each acre of wheat requires 1 hour of labor. If the farmer has a total of 120 hours of labor available, write a system of inequalities to represent the situation and find the feasible region.

• x + y >= 100, 2x + y >= 120, x <= 0, y <= 0

• x + y <= 120, 2x + y <= 100, x >= 0, y <= 0

• x + y <= 80, 2x + y <= 100, x >= 10, y >= 10

• x + y <= 100, 2x + y <= 120, x >= 0, y >= 0

A school is planning a field trip and has a budget of $500. The cost per student for the trip is $20, and the bus rental costs $200. Write a system of inequalities to represent the number of students that can attend the trip and find the feasible region.

• x ≥ 0 and x ≤ 12

• x ≥ 5 and x ≤ 20

• x ≥ 0 and x ≤ 10

• x ≥ 0 and x ≤ 15, where x is the number of students.

A company produces two types of gadgets: Type A and Type B. Each Type A gadget requires 3 hours of labor and each Type B gadget requires 2 hours. The company has 60 hours of labor available. Write the inequalities and find the feasible region for the number of gadgets that can be produced.

• The feasible region is bounded by the lines 3x + 2y = 60, x = 0, and y = 0 in the first quadrant.

• The feasible region is defined by the lines 3x + y = 60, x = 0, and y = 0 in the first quadrant.

• The feasible region is unbounded and includes negative values for x and y.

• The feasible region is bounded by the lines 2x + 3y = 60, x = 0, and y = 0 in the first quadrant.

A restaurant offers two types of meals: vegetarian and non-vegetarian. The vegetarian meal costs $10 and the non-vegetarian meal costs $15. The restaurant wants to make at least $300 in sales. Write a system of inequalities to represent the situation and find the feasible region.

• 10x + 15y = 300, x ≥ 0, y ≥ 0

• 5x + 10y ≥ 300, x ≥ 0, y ≥ 0

• 10x + 15y ≥ 300, x ≥ 0, y ≥ 0

• 10x + 15y ≤ 300, x ≥ 0, y ≥ 0

A clothing store sells shirts and pants. Each shirt costs $25 and each pair of pants costs $40. The store wants to make at least $1000 in sales. Write the inequalities and find the feasible region for the number of shirts and pants sold.

• 30x + 35y >= 1000; x >= 0; y >= 0

• 25x + 40y <= 1000; x >= 0; y >= 0

• 25x + 40y >= 1000; x >= 0; y >= 0

• 25x + 40y = 1000; x >= 0; y >= 0

A charity event is selling tickets for adults and children. Adult tickets cost $15 and children tickets cost $10. The goal is to sell at least $600 worth of tickets. Write a system of inequalities and find the feasible region for the number of adult and child tickets sold.

• 15x + 10y <= 600, x >= 0, y >= 0

• 20x + 5y >= 600, x >= 0, y >= 0

• System of inequalities: 15x + 10y >= 600, x >= 0, y >= 0.

• 10x + 15y >= 600, x >= 0, y >= 0

A gym has a maximum capacity of 200 members. Each membership costs $30 per month. The gym wants to earn at least $5000 in monthly revenue. Write the inequalities and find the feasible region for the number of members.

• 100 <= x <= 200

• 180 <= x <= 220

• 150 <= x <= 180

• 167 <= x <= 200

A bakery makes two types of cakes: chocolate and vanilla. Each chocolate cake requires 2 hours to bake and each vanilla cake requires 1 hour. The bakery has 10 hours available for baking. Write a system of inequalities and find the feasible region for the number of cakes that can be made.

• x + 2y ≤ 10, x ≥ 0, y ≤ 0

• The feasible region is defined by the inequalities: 2x + y ≤ 10, x ≥ 0, y ≥ 0.

• 3x + y ≤ 10, x ≥ 0, y ≥ 0

• 2x + y ≥ 10, x ≤ 0, y ≤ 0

A concert venue has a seating capacity of 500. Tickets for adults are $50 and tickets for children are $30. The venue wants to earn at least $20,000 from ticket sales. Write the inequalities and find the feasible region for the number of adult and child tickets sold.

• x + y = 500, 50x + 30y = 20000

• x + y ≤ 500, 50x + 30y ≥ 20000

• x + y ≤ 400, 50x + 30y ≥ 25000

• x + y ≥ 500, 50x + 30y ≤ 20000

A local gym offers two types of classes: yoga and spin. Each yoga class can accommodate 15 people and each spin class can accommodate 20 people. The gym wants to have at least 100 participants in total. Write a system of inequalities and find the feasible region for the number of yoga and spin classes offered.

• 15x + 25y >= 100, x >= 0, y >= 0

• 20x + 15y >= 100, x >= 0, y >= 0

• 10x + 15y >= 100, x >= 0, y >= 0

• 15x + 20y >= 100, x >= 0, y >= 0