Solving Real-World Problems with Graphing Inequalities

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 inequalities to represent the situation and graph the feasible region.

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

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

• x + y ≤ 80, 2x + y ≤ 100, x ≥ 0, y ≥ 0

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

A school is planning 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 that can attend the trip and graph the solution set.

• x >= 0 and x <= 15

• x >= 0 and x <= 10

• x >= 0 and x <= 20

• x >= 5 and x <= 25

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 available for assembly. Write a system of inequalities to represent the production limits and graph the feasible region.

• 2x + 3y ≤ 30, x ≥ 0, y ≥ 0

• 3x + 2y ≤ 30, x ≥ 0, y ≥ 0

• 3x + 3y ≤ 30, x ≥ 0, y ≥ 0

• 4x + y ≤ 30, x ≥ 0, y ≥ 0

A local 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 $1,200 in a month, write a system of inequalities to represent the number of each type of membership sold and graph the solution set.

• 30x + 50y >= 1200, x >= 0, y >= 0

• 30x + 50y >= 1000, x >= 0, y >= 0

• 30x + 50y <= 1200, x >= 0, y >= 0

• 20x + 40y >= 1200, x >= 0, y >= 0

A 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 has 10 hours available for baking, write a system of inequalities to represent the maximum number of cakes that can be baked and graph the feasible region.

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

• 3x + y ≤ 10, x ≥ 0, y ≥ 0

• 2x + y ≥ 10, x ≤ 0, y ≤ 0

• x + 2y ≤ 10, x ≥ 0, y ≤ 0

A charity organization is collecting donations. They have a goal of raising at least $1,000. Each small donation is $10, and each large donation is $50. Write a system of inequalities to represent the number of small and large donations needed to meet the goal and graph the solution set.

• 10x + 50y >= 1000, x >= 0, y >= 0

• 10x + 50y = 1000, x >= 0, y >= 0

• 5x + 25y >= 1000, x >= 0, y >= 0

• 10x + 50y <= 1000, x >= 0, y >= 0

A concert venue has a seating capacity of 500. Tickets for the front row cost $100 each, and tickets for the back row cost $50 each. If the venue wants to make at least $30,000 from ticket sales, write a system of inequalities to represent the ticket sales and graph the feasible region.

• x + y ≤ 500, 100x + 50y ≥ 30000

• x + y = 500, 100x + 50y = 30000

• x + y ≤ 400, 100x + 50y ≥ 25000

• x + y ≥ 500, 100x + 50y ≤ 30000

A clothing store sells shirts and pants. Each shirt costs $20, and each pair of pants costs $30. The store wants to make at least $1,500 in sales. Write a system of inequalities to represent the number of shirts and pants that need to be sold and graph the solution set.

• 20x + 30y = 1500, x >= 0, y >= 0

• 20x + 30y <= 1500, x >= 0, y >= 0

• The system of inequalities is: { 20x + 30y >= 1500, x >= 0, y >= 0 }

• 10x + 15y >= 1500, x >= 0, y >= 0

A pet shelter has a limit of 50 animals. They can take in cats and dogs, but each cat requires 1 unit of space and each dog requires 2 units of space. If the shelter has 80 units of space available, write a system of inequalities to represent the number of cats and dogs they can take in and graph the feasible region.

• x + 2y <= 50, 2x + y <= 80

• x + y <= 40, x + 2y <= 60

• x + y <= 50, x + 2y <= 80

• 2x + y <= 50, x + y <= 80

A tech company is developing two products, X and Y. Each product X requires 4 hours of development, and each product Y requires 2 hours. The company has 40 hours available for development. Write a system of inequalities to represent the production limits and graph the feasible region.

• 4x + 3y ≤ 40, x ≥ 0, y ≥ 0

• 2x + 4y ≤ 40, x ≥ 0, y ≥ 0

• 3x + 2y ≤ 40, x ≥ 0, y ≥ 0

• 4x + 2y ≤ 40, x ≥ 0, y ≥ 0