Graphing Systems of Inequalities: Finding Feasible Regions

A farmer has 100 acres of land. He wants 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 he has a total of 120 hours of labor available, write a system of inequalities to represent the feasible region for the number of acres of corn (x) and wheat (y) he can plant. Graph the inequalities and identify the feasible region.

• The system of inequalities is: { x + y <= 100, 2x + y <= 120, x >= 0, y >= 0 }.

• x + 2y <= 100

• x + y >= 100

• 2x + y >= 120

A school is organizing 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 (x) that can attend the trip. Graph the inequalities and determine the feasible region for the number of students.

• x >= 0 and x <= 15

• x >= 5 and x <= 20

• x >= 0 and x <= 10

• x >= 0 and x <= 20

A company produces two types of gadgets: A and B. Each gadget A requires 3 hours of assembly and each gadget B requires 2 hours. The company has a maximum of 30 hours available for assembly. Additionally, they want to produce at least 5 gadgets A. Write a system of inequalities to represent the situation and graph the feasible region.

• 3x + 2y ≤ 25, x ≥ 5, x ≥ 0, y ≥ 0

• 2x + 3y ≤ 30, x ≥ 5, x ≥ 0, y ≥ 0

• 3x + 2y ≤ 30, x ≥ 5, x ≥ 0, y ≥ 0

• 3x + 2y ≤ 30, x ≥ 3, x ≥ 0, y ≥ 0

A bakery sells two types of cakes: chocolate and vanilla. Each chocolate cake requires 2 pounds of flour and each vanilla cake requires 1 pound. The bakery has 20 pounds of flour available. If they want to make at least 5 chocolate cakes, write a system of inequalities and graph the feasible region.

• 2x + y >= 20, x <= 5, x >= 0, y >= 0

• x + y <= 20, x >= 5, y >= 0

• The system of inequalities is: 2x + y <= 20, x >= 5, x >= 0, y >= 0.

• 2x + y <= 15, x >= 5, y >= 0

A gym offers two types of memberships: basic and premium. The basic membership costs $30 per month, and the premium membership costs $50 per month. The gym wants to earn at least $1,000 in membership fees each month. Write a system of inequalities to represent the number of basic (x) and premium (y) memberships sold. Graph the inequalities and identify the feasible region.

• 30x + 50y = 1000, x >= 0, y >= 0

• The system of inequalities is: 30x + 50y >= 1000, x >= 0, y >= 0.

• 30x + 50y <= 1000, x >= 0, y >= 0

• 20x + 40y >= 1000, x >= 0, y >= 0

A local theater has 200 seats. They sell tickets for $10 for adults and $5 for children. They want to make at least $1,500 from ticket sales. Write a system of inequalities to represent the number of adult (x) and child (y) tickets sold. Graph the inequalities and determine the feasible region.

• x + y = 200, 10x + 5y = 1500

• x + y ≤ 150, 10x + 5y ≥ 2000

• x + y ≤ 200, 10x + 5y ≥ 1500

• x + y ≥ 200, 10x + 5y ≤ 1500

A charity event is planning to sell two types of items: T-shirts and mugs. Each T-shirt costs $15 and each mug costs $10. They want to raise at least $1,200. Additionally, they can only produce a maximum of 100 items. Write a system of inequalities and graph the feasible region for the number of T-shirts (x) and mugs (y) they can sell.

• x + y >= 100

• x <= 0, y <= 0

• The system of inequalities is: 15x + 10y >= 1200, x + y <= 100, x >= 0, y >= 0.

• 15x + 10y <= 1200

A restaurant offers two types of meals: vegetarian and non-vegetarian. Each vegetarian meal costs $12 and each non-vegetarian meal costs $15. The restaurant wants to earn at least $800 in one day. Write a system of inequalities to represent the number of vegetarian (x) and non-vegetarian (y) meals sold. Graph the inequalities and identify the feasible region.

• 10x + 15y ≥ 800, x ≥ 0, y ≥ 0

• 12x + 10y ≥ 800, x ≥ 0, y ≥ 0

• 12x + 15y ≤ 800, x ≥ 0, y ≥ 0

• 12x + 15y ≥ 800, x ≥ 0, y ≥ 0

A clothing store sells two types of shirts: casual and formal. Each casual shirt costs $20 and each formal shirt costs $30. The store wants to make at least $1,000 in sales. Additionally, they can only sell a maximum of 50 shirts in total. Write a system of inequalities and graph the feasible region for the number of casual (x) and formal (y) shirts sold.

• 10x + 20y ≥ 1000

• 20x + 30y ≥ 1000, x + y ≤ 50, x ≥ 0, y ≥ 0

• x + 2y ≤ 50

• x ≤ 0, y ≤ 0

A tech company produces two products: laptops and tablets. Each laptop requires 4 hours of production time and each tablet requires 2 hours. The company has a total of 40 hours available for production. Write a system of inequalities to represent the number of laptops (x) and tablets (y) they can produce. Graph the inequalities and identify the feasible region.

• 4x + 2y ≤ 40, x ≥ 0, y ≥ 0

• 2x + 3y ≤ 40

• 3x + 4y ≤ 40

• 5x + y ≤ 40