Real-World Linear Inequalities: Graphing Solutions

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 3 hours of labor. If the farmer has a total of 240 hours of labor available, write a system of linear inequalities to represent the situation. Graph the inequalities and interpret the feasible region.

• x + y <= 100, 2x + 3y <= 300, x >= 0, y >= 0

• The system of linear inequalities is: x + y <= 100, 2x + 3y <= 240, x >= 0, y >= 0.

• x + y >= 100, 2x + 3y >= 240, x <= 0, y <= 0

• x + y <= 80, 2x + 3y <= 200, x >= 0, y >= 0

A company produces two types of gadgets: Type A and Type B. Each Type A gadget requires 4 hours of assembly, and each Type B gadget requires 2 hours. The company has 40 hours of assembly time available. Write a system of linear inequalities to represent the production limits. Graph the inequalities and discuss the possible production combinations.

• 4x + 2y ≥ 40, x ≤ 0, y ≤ 0

• The system of linear inequalities is: 4x + 2y ≤ 40, x ≥ 0, y ≥ 0.

• 2x + 4y ≤ 40, x ≥ 0, y ≤ 0

• 3x + 3y ≤ 40, x ≥ 0, y ≥ 0

A restaurant offers two types of meals: vegetarian and non-vegetarian. The vegetarian meal costs $8, and the non-vegetarian meal costs $12. The restaurant wants to make at least $300 in a day. Write a system of linear inequalities to represent the meal sales. Graph the inequalities and analyze the feasible solutions.

• 8x + 12y = 300, x >= 0, y >= 0

• The system of inequalities is: 8x + 12y >= 300, x >= 0, y >= 0.

• 8x + 12y <= 300, x >= 0, y >= 0

• 8x + 12y >= 200, x >= 0, y >= 0

A local gym has a maximum capacity of 150 members. Each membership costs $30 per month. The gym wants to earn at least $3000 in monthly revenue. Write a system of linear inequalities to represent the membership situation. Graph the inequalities and interpret the results.

• x <= 150 and x >= 120

• The system of inequalities is: x <= 150 and x >= 100.

• x <= 200 and x >= 50

• x <= 100 and x >= 150

A clothing store sells shirts and pants. Each shirt costs $20, and each pair of pants costs $30. The store wants to make at least $600 in sales. Write a system of linear inequalities to represent the sales goal. Graph the inequalities and explain the meaning of the solution set.

• 20x + 30y >= 600, x >= 0, y >= 0

• 20x + 30y <= 600, x >= 0, y >= 0

• 10x + 15y >= 600, x >= 0, y >= 0

• 20x + 30y = 600, x >= 0, y >= 0

A tech company is developing two products: a smartphone and a tablet. Each smartphone requires 5 hours of development, and each tablet requires 3 hours. The team has 60 hours available for development. Write a system of linear inequalities to represent the development time. Graph the inequalities and interpret the feasible solutions.

• 4x + 3y <= 60, x >= 0, y >= 0

• 5x + 2y <= 60, x >= 0, y >= 0

• 5x + 4y <= 60, x >= 0, y >= 0

• The system of linear inequalities is: 5x + 3y <= 60, x >= 0, y >= 0.

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 linear inequalities to represent the baking time. Graph the inequalities and analyze the feasible region.

• 2x + y ≥ 10, x ≤ 0, y ≤ 0

• The system of linear inequalities is: 2x + y ≤ 10, x ≥ 0, y ≥ 0.

• x + 2y ≤ 10, x ≥ 0, y ≤ 0

• 3x + y ≤ 10, x ≥ 0, y ≥ 0

A concert venue has a seating capacity of 500. Tickets for the front row cost $50, and tickets for the back row cost $30. The venue wants to earn at least $15,000 from ticket sales. Write a system of linear inequalities to represent the ticket sales. Graph the inequalities and interpret the feasible solutions.

• x + y <= 500, 50x + 30y >= 15000

• x + y = 500, 50x + 30y = 15000

• x + y <= 400, 50x + 30y >= 20000

• x + y >= 500, 50x + 30y <= 15000