Linear Inequalities: Feasible Regions & Boundary Lines

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 linear inequalities to represent the situation and find the feasible region for planting corn and wheat.

• x + y <= 100, 2x + y <= 150, x >= 0, y >= 0

• x + y <= 80, 2x + y <= 100, x >= 0, y >= 0

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

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

A school is organizing a field trip and has a budget of $500. Each student ticket costs $10, and each adult ticket costs $15. If the school can take a maximum of 40 people, create a system of linear inequalities to represent the situation and identify the boundary lines for the number of student and adult tickets that can be purchased.

• 5x + 10y ≤ 500, x + y ≤ 40

• 10x + 15y ≤ 500, x + y ≤ 40

• 10x + 20y ≤ 500, x + y ≤ 30

• 15x + 10y ≤ 500, x + y ≤ 50

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 30 hours of assembly time available. Write a system of linear inequalities to represent the production limits and find the feasible region for the number of gadgets A and B that can be produced.

• x + y ≤ 30, x ≥ 0, y ≥ 0

• 2x + 3y ≤ 30, x ≥ 0, y ≥ 0

• The system of linear inequalities is: 3x + 2y ≤ 30, x ≥ 0, y ≥ 0.

• 3x + 2y ≤ 20, 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. Write a system of linear inequalities to represent the cake production and identify the boundary lines for the number of chocolate and vanilla cakes that can be made.

• 2x + 3y <= 20, x >= 0, y >= 0

• 2x + y >= 20, x <= 0, y <= 0

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

• x + 2y <= 20, x >= 0, y <= 0

A gym offers two types of fitness classes: yoga and spinning. Each yoga class can accommodate 15 people, and each spinning class can accommodate 10 people. If the gym can host a maximum of 100 people in total, create a system of linear inequalities to represent the class sizes and find the feasible region for the number of yoga and spinning classes.

• 15x + 5y <= 100, x >= 0, y >= 0

• 10x + 15y <= 100, x >= 0, y >= 0

• 20x + 10y <= 100, x >= 0, y >= 0

• The system of inequalities is: 15x + 10y <= 100, 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. If the venue wants to make at least $15,000 from ticket sales, write a system of linear inequalities to represent the ticket sales and identify the boundary lines for the number of front and back row tickets sold.

• x + y >= 500, 50x + 30y <= 15000

• x + y <= 400, 50x + 30y >= 20000

• x + y <= 500, 50x + 30y <= 10000

• x + y <= 500, 50x + 30y >= 15000

A local restaurant offers two types of meals: vegetarian and non-vegetarian. Each vegetarian meal costs $12, and each non-vegetarian meal costs $15. The restaurant has a budget of $600 for ingredients. Write a system of linear inequalities to represent the meal preparation and find the feasible region for the number of vegetarian and non-vegetarian meals that can be prepared.

• 12x + 10y ≤ 600, x ≥ 0, y ≥ 0

• 12x + 15y ≤ 600, x ≥ 0, y ≥ 0

• 12x + 15y ≤ 500, x ≥ 0, y ≥ 0

• 10x + 15y ≤ 600, x ≥ 0, y ≥ 0

A clothing store sells shirts and pants. Each shirt costs $20, and each pair of pants costs $30. The store has a budget of $1,200 for inventory. If the store wants to have at least 20 items in stock, create a system of linear inequalities to represent the inventory and identify the boundary lines for the number of shirts and pants that can be purchased.

• The system of inequalities is: 20x + 30y <= 1200 and x + y >= 20.

• 25x + 35y <= 1200 and x + y <= 15

• 10x + 15y <= 1200 and x + y >= 30

• 20x + 30y >= 1200 and x + y <= 20

A charity event is selling tickets for a dinner. Each ticket costs $25, and the event has a goal of selling at least 200 tickets. If the venue can accommodate a maximum of 300 people, write a system of linear inequalities to represent the ticket sales and find the feasible region for the number of tickets sold.

• 200 <= x <= 300

• 150 <= x <= 250

• x > 300

• x < 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. The company has a total of 36 hours available for development. Write a system of linear inequalities to represent the development time and identify the boundary lines for the number of smartphones and tablets that can be developed.

• 2x + 5y ≤ 36, x ≥ 0, y ≥ 0

• 4x + 2y ≤ 36, x ≥ 0, y ≥ 0

• 3x + 4y ≤ 36, x ≥ 0, y ≥ 0

• 4x + 3y ≤ 36, x ≥ 0, y ≥ 0