Interpreting Systems of Inequalities in Real-World Scenarios

A farmer has a total of 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 maximum of 240 hours of labor available, write a system of inequalities to represent the situation and determine the feasible planting options.

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

• x + y <= 100, 2x + 3y <= 200, x >= 0, y >= 0

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

• x + y <= 80, 2x + 3y >= 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 $200 per month on memberships, create a system of inequalities to represent the situation and determine how many of each type of membership can be purchased.

• x >= 0, y >= 0, 30x + 50y = 250; Possible combinations: (5, 0), (0, 5)

• x < 0, y < 0, 30x + 50y <= 150; Possible combinations: (0, 3), (1, 2)

• x >= 0, y >= 0, 30x + 50y >= 200; Possible combinations: (0, 0), (1, 1), (2, 3)

• x >= 0, y >= 0, 30x + 50y <= 200; Possible combinations: (0, 4), (1, 3), (2, 2), (3, 1), (6, 0)

A local bakery sells two types of pastries: muffins and croissants. Each muffin costs $2 and each croissant costs $3. The bakery wants to make at least 50 pastries but can spend no more than $120 on ingredients. Write a system of inequalities to model this scenario and find the possible combinations of muffins and croissants that can be made.

• The system of inequalities is: x + y >= 50 and 2x + 3y <= 120.

• x + y <= 50 and 2x + 3y <= 100

• x + y >= 50 and 2x + 3y = 120

• x + y <= 50 and 2x + 3y >= 120

A concert venue has a seating capacity of 500. Tickets for the front row cost $100 each, while tickets for the back row cost $50 each. If the venue wants to make at least $30,000 from ticket sales, create a system of inequalities to represent the ticket sales and analyze the possible combinations of front and back row tickets that can be sold.

• x + y <= 600, 100x + 50y >= 35000

• x + y <= 500, 100x + 50y >= 30000

• x + y <= 400, 100x + 50y >= 25000

• x + y >= 500, 100x + 50y <= 30000

A charity event is selling two types of tickets: VIP tickets for $75 and regular tickets for $30. They want to raise at least $5,000 and can sell no more than 200 tickets in total. Formulate a system of inequalities to represent the ticket sales and determine the feasible combinations of VIP and regular tickets that can be sold.

• x + y >= 200

• 75x + 30y <= 5000

• The feasible combinations of VIP and regular tickets can be represented by the inequalities: 75x + 30y >= 5000, x + y <= 200, x >= 0, y >= 0.

• x <= 0, y <= 0

A clothing store sells shirts for $25 and pants for $40. They want to sell at least 30 items but can spend no more than $1,200 on inventory. Write a system of inequalities to model this situation and analyze the possible combinations of shirts and pants that can be purchased.

• x + y <= 25, 25x + 40y <= 1500

• The system of inequalities is: { x + y >= 30, 25x + 40y <= 1200 }

• x + y <= 30, 25x + 40y >= 1200

• x + y >= 40, 25x + 40y <= 1000

A pet shelter has a limit of 50 animals they can care for. They have dogs and cats, with each dog requiring 3 units of space and each cat requiring 2 units of space. If they have a total of 120 units of space available, create a system of inequalities to represent the situation and determine the possible combinations of dogs and cats they can take in.

• x + y = 50, 3x + 2y = 100

• x + y <= 50, 3x + 2y <= 120

• x + y <= 30, 3x + 2y <= 80

• x + y >= 50, 3x + 2y >= 120

A restaurant offers two types of meal plans: a vegetarian plan for $15 and a non-vegetarian plan for $20. They want to serve at least 100 meals but can spend no more than $1,800 on meal plans. Formulate a system of inequalities to represent the meal plan options and determine the possible combinations of vegetarian and non-vegetarian meals that can be served.

• x + y <= 80, 15x + 20y <= 2000

• x + y >= 100, 15x + 20y <= 1800

• x + y >= 150, 15x + 20y <= 1500

• x + y <= 100, 15x + 20y >= 1800