Analyzing Systems of Inequalities and 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 feasible region for planting corn and wheat. Identify the boundary lines.

• x + y >= 120

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

• 2x + 3y <= 120

• x + 2y <= 100

A school is planning a field trip and has a budget of $500. The cost per student for transportation is $10, and for admission, it is $15. Write a system of inequalities to represent the number of students that can attend the trip. What are the boundary lines for this situation?

• 0 ≤ x ≤ 15; Boundary lines: x = 0 and x = 15.

• 0 ≤ x ≤ 25; Boundary lines: x = 0 and x = 25.

• 0 ≤ x ≤ 10; Boundary lines: x = 0 and x = 10.

• 0 ≤ x ≤ 20; Boundary lines: x = 0 and x = 20.

A company produces two types of gadgets: Type A and Type B. Each Type A gadget requires 3 hours of assembly, and each Type B gadget requires 2 hours. The company has 30 hours of assembly time available. Write a system of inequalities to represent the production limits for Type A and Type B gadgets. Identify the feasible region.

• 4x + y ≤ 30, x ≥ 0, y ≥ 0

• 3x + 2y ≤ 30, x ≥ 0, y ≥ 0

• 3x + 3y ≤ 30, x ≥ 0, y ≥ 0

• 2x + 3y ≤ 30, 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. If the bakery has 20 pounds of flour, write a system of inequalities to represent the maximum number of cakes that can be made. What are the boundary lines?

• 2x + 3y ≤ 20, x ≥ 0, y ≥ 0; Boundary lines: 2x + 3y = 20, x = 0, y = 0.

• 3x + y ≤ 20, x ≥ 0, y ≥ 0; Boundary lines: 3x + y = 20, x = 0, y = 0.

• 2x + y ≤ 20, x ≥ 0, y ≥ 0; Boundary lines: 2x + y = 20, x = 0, y = 0.

• x + 2y ≤ 20, x ≥ 0, y ≥ 0; Boundary lines: x + 2y = 20, x = 0, y = 0.

A gym offers two types of memberships: basic and premium. The basic membership costs $20 per month, and the premium membership costs $30 per month. If the gym wants to earn at least $600 in a month, write a system of inequalities to represent the number of basic and premium memberships sold. Identify the feasible region.

• 20x + 30y ≤ 600, x ≥ 0, y ≥ 0

• The system of inequalities is: 20x + 30y ≥ 600, x ≥ 0, y ≥ 0.

• 20x + 30y = 600, x ≥ 0, y ≥ 0

• 10x + 15y ≥ 600, x ≥ 0, y ≥ 0

A concert venue has a seating capacity of 500. If tickets for the front row cost $50 and tickets for the back row cost $30, and the venue wants to earn at least $15,000 from ticket sales, write a system of inequalities to represent the ticket sales. What are the boundary lines?

• x + y ≥ 500, 50x + 30y ≤ 15000

• x + y ≤ 500, 50x + 30y ≥ 15000

• x + y ≤ 400, 50x + 30y ≥ 20000

• x + y = 500, 50x + 30y = 15000

A clothing store sells shirts and pants. Each shirt costs $15, and each pair of pants costs $25. If the store wants to make at least $1,000 in sales, write a system of inequalities to represent the sales of shirts and pants. Identify the feasible region.

• 10x + 20y >= 1000, x >= 0, y >= 0

• 15x + 25y <= 1000, x >= 0, y >= 0

• 15x + 30y >= 1000, x >= 0, y >= 0

• 15x + 25y >= 1000, x >= 0, y >= 0

A charity organization is organizing a fundraiser. They plan to sell two types of items: cookies and brownies. Each cookie costs $2, and each brownie costs $3. If they want to raise at least $200, write a system of inequalities to represent the sales of cookies and brownies. What are the boundary lines?

• x + 2y >= 200; Boundary line: x + 2y = 200.

• 2x + y >= 200; Boundary line: 2x + y = 200.

• 3x + 2y >= 200; Boundary line: 3x + 2y = 200.

• 2x + 3y >= 200; Boundary line: 2x + 3y = 200.

A tech company is developing two products: a smartphone and a tablet. Each smartphone requires 4 hours of development, and each tablet requires 3 hours. If the company has 60 hours available for development, write a system of inequalities to represent the production limits for smartphones and tablets. Identify the feasible region.

• 2x + 5y ≤ 60, x ≥ 0, y ≥ 0

• 4x + 3y ≤ 60, x ≥ 0, y ≥ 0

• 3x + 4y ≤ 60, x ≥ 0, y ≥ 0

• 4x + 2y ≤ 60, x ≥ 0, y ≥ 0

A restaurant offers two types of meals: vegetarian and non-vegetarian. Each vegetarian meal costs $10, and each non-vegetarian meal costs $15. If the restaurant wants to earn at least $800 in a week, write a system of inequalities to represent the number of vegetarian and non-vegetarian meals sold. What are the boundary lines?

• 10x + 20y >= 800; Boundary line: 10x + 20y = 800.

• 15x + 10y >= 800; Boundary line: 15x + 10y = 800.

• 10x + 15y >= 800; Boundary line: 10x + 15y = 800.

• 5x + 10y >= 800; Boundary line: 5x + 10y = 800.