Graphing Systems of Inequalities in Real-World Scenarios

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 feasible region for the number of acres of corn (x) and wheat (y) he can plant.

• x + 2y ≤ 100

• x + y ≤ 80

• x + y ≤ 100, 2x + y ≤ 120, x ≥ 0, y ≥ 0

• 2x + 3y ≤ 120

A school is organizing a field trip and has a budget of $500. The cost per student for the trip is $20, and the bus rental costs $200. Write a system of inequalities to represent the number of students (x) that can attend the trip while staying within budget.

• 15x + 200 <= 500, x >= 0

• 20x + 200 <= 500, x >= 0

• 20x + 150 <= 500, x >= 0

• 20x + 200 >= 500, x <= 0

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 feasible production of Type A (x) and Type B (y) gadgets.

• 3x + 2y ≥ 30, x ≥ 0, y ≥ 0

• 2x + 3y ≤ 30, x ≥ 0, y ≥ 0

• 3x + 2y ≤ 30, x ≥ 0, y ≥ 0

• x + y ≤ 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 10 pounds of flour, write a system of inequalities to represent the maximum number of chocolate (x) and vanilla (y) cakes that can be made.

• x + 2y <= 10, x >= 0, y >= 0

• 2x + y <= 10, x >= 0, y >= 0

• 2x + y >= 10, x >= 0, y >= 0

• 3x + y <= 10, x >= 0, y >= 0

A gym offers two types of memberships: basic and premium. The basic membership costs $30 per month, and the premium membership costs $50 per month. If the gym wants to earn at least $1,000 in a month, write a system of inequalities to represent the number of basic (x) and premium (y) memberships sold.

• 20x + 40y >= 1000, x >= 0, y >= 0

• 30x + 50y <= 1000, x >= 0, y >= 0

• 30x + 50y >= 500, x >= 0, y >= 0

• 30x + 50y >= 1000, x >= 0, y >= 0

A local theater has 200 seats. Tickets for adults cost $15, and tickets for children cost $10. If the theater wants to make at least $1,500 from ticket sales, write a system of inequalities to represent the number of adult (x) and child (y) tickets sold.

• x + y <= 150, 15x + 10y >= 2000

• x + y >= 200, 15x + 10y <= 1500

• x + y = 200, 15x + 10y = 1500

• x + y <= 200, 15x + 10y >= 1500

A charity event is selling two types of tickets: VIP and regular. The VIP ticket costs $100, and the regular ticket costs $50. If the charity wants to raise at least $2,000, write a system of inequalities to represent the number of VIP (x) and regular (y) tickets sold.

• 100x + 50y = 2000, x >= 0, y >= 0

• 100x + 50y >= 2000, x >= 0, y >= 0

• 50x + 100y >= 2000, x >= 0, y >= 0

• 100x + 50y <= 2000, 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 $1,000 in sales, write a system of inequalities to represent the number of shirts (x) and pants (y) sold.

• 20x + 30y >= 1000, x >= 0, y >= 0

• 10x + 15y >= 1000, x >= 0, y >= 0

• 25x + 35y >= 1000, x >= 0, y >= 0

• 20x + 30y <= 1000, x >= 0, y >= 0

A pet store has a limit of 50 animals it can house. They have dogs and cats. Each dog takes up 2 spaces, and each cat takes up 1 space. Write a system of inequalities to represent the maximum number of dogs (x) and cats (y) the store can have.

• 2x + 3y ≤ 50

• x + y ≤ 50

• x + 2y ≤ 50

• 2x + y ≤ 50, x ≥ 0, y ≥ 0