Exploring Real-Life Systems of Inequalities Graphically

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 1 hour of labor. If he has a total of 120 hours of labor available, write a system of inequalities to represent the situation and graph the solution.

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

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

• x + y <= 50, 2x + y <= 60, x >= 0, y >= 0

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

A school is planning a field trip and has a budget of $500. The cost per student is $20 for transportation and $15 for admission. Write a system of inequalities to determine the maximum number of students that can attend the trip, and analyze the solution graphically.

• The maximum number of students that can attend the trip is 5.

• The maximum number of students that can attend the trip is 14.

• The maximum number of students that can attend the trip is 10.

• The maximum number of students that can attend the trip is 20.

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, write a system of inequalities to represent the situation and find the feasible region graphically.

• 30x + 50y = 200, x >= 0, y <= 0

• The system of inequalities is: 30x + 50y <= 200, x >= 0, y >= 0.

• 30x + 50y <= 150, x >= 0, y >= 0

• 30x + 50y >= 200, x <= 0, y <= 0

A company produces two types of gadgets: Type A and Type B. Each Type A gadget requires 3 hours of labor and each Type B gadget 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 analyze the solution graphically.

• 2x + 3y ≤ 60, x ≥ 0, y ≤ 0

• x + y ≤ 30, x ≥ 0, y ≥ 0

• 3x + 2y ≥ 60, x ≥ 0, y ≥ 0

• 3x + 2y ≤ 60, x ≥ 0, y ≥ 0

A restaurant has a seating capacity of 150. They want to serve at least 30 tables of 4 and at least 20 tables of 6. Write a system of inequalities to represent the seating arrangement and graph the solution to find the possible combinations of tables.

• 4x + 6y <= 100, x >= 40, y >= 10

• The system of inequalities is: 4x + 6y <= 150, x >= 30, y >= 20.

• 4x + 6y = 150, x >= 20, y >= 30

• 4x + 6y >= 150, x <= 30, y <= 20

A charity event aims to raise at least $1,000. They sell tickets for $15 and donations are accepted. If they expect to sell at least 50 tickets, write a system of inequalities to represent the fundraising goal and analyze the solution graphically.

• 15x + y <= 1000, x <= 50

• 15x + y >= 500, x >= 30

• 15x + y >= 1000, x >= 50

• 10x + y >= 1000, x >= 50

A clothing store sells shirts for $25 and pants for $40. They want to make at least $1,200 in sales. If they can sell no more than 50 items in total, write a system of inequalities to represent the sales goal and graph the solution to find the possible combinations of shirts and pants sold.

• 20x + 30y ≥ 1000, x + y ≤ 60, x ≥ 0, y ≥ 0

• 30x + 50y ≥ 1200, x + y ≤ 40, x ≥ 0, y ≥ 0

• 25x + 40y ≤ 1200, x + y ≥ 50, x ≥ 0, y ≥ 0

• 25x + 40y ≥ 1200, x + y ≤ 50, x ≥ 0, y ≥ 0

A local bakery makes 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 has 10 hours available for baking, write a system of inequalities to represent the baking limits and analyze the solution graphically.

• 2x + y ≥ 10, x ≥ 0, y ≥ 0

• x + 2y ≤ 10, x ≥ 0, y ≥ 0

• x + y ≤ 10, x ≥ 0, y ≥ 0

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

A concert venue has a maximum capacity of 500 people. They are selling tickets for $20 each and VIP tickets for $50 each. If they want to make at least $10,000 in ticket sales, write a system of inequalities to represent the ticket sales and graph the solution to find the possible combinations of regular and VIP tickets sold.

• x + y = 500, 20x + 50y = 10000

• x + y ≤ 500, 20x + 50y ≥ 10000

• x + y ≤ 600, 20x + 50y ≥ 8000

• x + y < 500, 20x + 50y < 10000

A pet shelter has a limit of 40 animals they can care for. They have dogs and cats, with each dog taking up 2 spaces and each cat taking up 1 space. If they want to have at least 10 dogs, write a system of inequalities to represent the shelter's capacity and analyze the solution graphically.

• 2x + y ≤ 30 and x ≥ 15

• 2x + y ≥ 40 and x ≤ 10

• x + y ≤ 40 and x ≥ 5

• The system of inequalities is: 2x + y ≤ 40 and x ≥ 10.