Real-Life Applications of Linear Inequalities and Graphing

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 3 hours of labor. If he has a total of 240 hours of labor available, write a system of inequalities to represent the situation and graph the solution.

• x + y ≤ 100, 2x + 4y ≤ 240, x ≥ 0, y ≥ 0

• x + y ≤ 120, 2x + 3y ≤ 300, x ≥ 0, y ≥ 0

• x + y ≤ 100, 2x + 3y ≤ 240, x ≥ 0, y ≥ 0

• x + y ≤ 80, 2x + 2y ≤ 240, x ≥ 0, y ≥ 0

A gym offers two types of memberships: a basic membership for $30 per month and a premium membership for $50 per month. If a customer can spend no more than $300 on memberships, write a system of inequalities to represent the situation and find the possible combinations of memberships they can purchase.

• (0, 5)

• (2, 3)

• The possible combinations of memberships are: (0, 6), (1, 5), (2, 4), (3, 3), (4, 2), (5, 1), (6, 0) where (x, y) represents (basic memberships, premium memberships).

• (4, 4)

A local bakery sells two types of cakes: chocolate and vanilla. Each chocolate cake requires 2 hours to bake, and each vanilla cake requires 1 hour. If the bakery can only bake for 10 hours a day, write a system of inequalities to represent the baking time and graph the solution.

• 2x + y ≥ 10, x ≥ 0, y ≥ 0

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

• x + 2y ≤ 10, x ≥ 0, y ≥ 0

• 2x + 3y ≤ 10, x ≥ 0, y ≥ 0

A concert hall has a seating capacity of 500. Tickets for the front row cost $50, and tickets for the back row cost $30. If the total revenue from ticket sales must be at least $15,000, write a system of inequalities to represent the ticket sales and analyze the solution graphically.

• x + y <= 500, 50x + 30y >= 15000

• x + y <= 400, 50x + 30y >= 20000

• x + y >= 500, 50x + 30y <= 15000

• x + y <= 500, 50x + 30y = 10000

A company produces two products, A and B. Each product A requires 3 hours of labor and each product B requires 2 hours. The company has a maximum of 60 hours of labor available. Write a system of inequalities to represent the production limits and graph the solution.

• 3x + 2y ≤ 60, x ≥ 0, y ≥ 0

• 3x + 3y ≤ 60, x ≥ 0, y ≥ 0

• 4x + y ≤ 60, x ≥ 0, y ≥ 0

• 2x + 3y ≤ 60, x ≥ 0, y ≥ 0

A charity event is selling two types of tickets: VIP tickets for $100 and regular tickets for $50. If they want to raise at least $5,000 and can sell no more than 100 tickets, write a system of inequalities to represent the ticket sales and analyze the solution graphically.

• 100x + 50y <= 5000

• x <= 0, y <= 0

• The system of inequalities is: 100x + 50y >= 5000, x + y <= 100, x >= 0, y >= 0.

• x + y >= 100

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 make at least $1,000 in a week and can serve no more than 150 meals, write a system of inequalities to represent the meal sales and graph the solution.

• 10x + 15y <= 1000

• x + y >= 150

• x <= 0, y <= 0

• 10x + 15y >= 1000, x + y <= 150, x >= 0, y >= 0

A clothing store sells shirts and pants. Each shirt costs $20 and each pair of pants costs $30. If the store wants to make at least $1,200 in sales and can sell no more than 60 items, write a system of inequalities to represent the sales and analyze the solution graphically.

• The system of inequalities is: 20x + 30y >= 1200, x + y <= 60, x >= 0, y >= 0.

• 20x + 30y <= 1200

• x <= 0, y <= 0

• x + y >= 60

A student is saving money for a new laptop that costs $800. He saves $50 a week from his allowance and $20 a week from his part-time job. Write a system of inequalities to represent how many weeks it will take him to save enough money and graph the solution.

• w <= 10

• w >= 12

• w = 15

• w < 12