Graphing Inequalities: Real-World Applications for 9th Grade

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. What are the possible combinations of corn and wheat that can be planted?

• The farmer can plant 50 acres of corn and 50 acres of wheat.

• The total labor required is 200 hours for 100 acres of corn.

• The inequalities are x + y ≥ 100 and 2x + y ≥ 120.

• The possible combinations of corn and wheat that can be planted are represented by the feasible region defined by the inequalities: x + y ≤ 100 and 2x + y ≤ 120, with x, y ≥ 0.

A school is organizing a field trip and has a budget of $500. Each student ticket costs $15, and each adult ticket costs $25. Write a system of inequalities to represent the budget constraints if at least 10 students must attend. Graph the inequalities and interpret the solution.

• 15x + 25y ≤ 600 and x ≥ 5

• The system of inequalities is: 15x + 25y ≤ 500 and x ≥ 10.

• 20x + 30y ≤ 500 and x ≥ 10

• 10x + 20y ≤ 500 and x ≥ 15

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. Additionally, the company wants to produce at least 5 gadgets A. Write a system of inequalities and graph the solution. What are the possible production combinations?

• The possible production combinations are (x, y) where x ≥ 5 and 3x + 2y ≤ 30.

• x ≥ 5 and 3x + 2y = 30

• x ≤ 5 and 3x + 2y ≥ 30

• x + y ≤ 15 and x ≥ 5

A restaurant offers two types of meals: vegetarian and non-vegetarian. The vegetarian meal costs $10, and the non-vegetarian meal costs $15. The restaurant wants to make at least $200 in sales and can serve a maximum of 30 meals. Write a system of inequalities to represent this situation, graph it, and interpret the feasible solutions.

• The system of inequalities is: 10x + 15y >= 200, x + y <= 30, x >= 0, y >= 0.

• x <= 0, y <= 0

• x + y >= 30

• 10x + 15y <= 200

A charity event is selling tickets for a concert. Each ticket costs $20, and they want to raise at least $1,000. If they can sell a maximum of 80 tickets, write a system of inequalities to represent the situation. Graph the inequalities and determine the possible number of tickets they can sell to meet their goal.

• 40 <= x <= 70

• 0 <= x <= 50

• 50 <= x <= 80

• 30 <= x <= 90

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. The gym wants to earn at least $2,000 in membership fees and can have a maximum of 100 members. Write a system of inequalities, graph it, and interpret the feasible solutions.

• The gym must earn exactly $2,000 in membership fees.

• The feasible solutions are the combinations of x and y that satisfy the inequalities, representing the number of basic and premium memberships that the gym can sell while meeting its financial and membership constraints.

• The gym can only sell 50 memberships total.

• The basic membership costs $50 and the premium costs $30.

A local bakery sells two types of cakes: chocolate and vanilla. Each chocolate cake requires 2 hours to bake, and each vanilla cake requires 3 hours. The bakery has a total of 12 hours available for baking. Additionally, they want to bake at least 2 chocolate cakes. Write a system of inequalities, graph the solution, and interpret the results.

• The system of inequalities is: 2x + 3y ≤ 12, x ≥ 2, x ≥ 0, y ≥ 0.

• 2x + 3y ≤ 10, x ≥ 1, y ≥ 0

• 2x + 3y ≤ 15, x ≥ 3, y ≥ 0

• x + y ≤ 12, x ≥ 2, y ≥ 1

A tech company is developing two products: a smartphone and a tablet. Each smartphone requires $200 in materials, and each tablet requires $150. The company has a budget of $1,500 for materials and wants to produce at least 5 smartphones. Write a system of inequalities, graph it, and interpret the feasible solutions.

• The budget allows for producing 8 smartphones and 2 tablets.

• The feasible solutions are combinations of smartphones (x) and tablets (y) that satisfy the inequalities, specifically where x >= 5 and 200x + 150y <= 1500.

• The company can produce a maximum of 10 smartphones and 5 tablets.

• Each smartphone requires $150 in materials, and each tablet requires $200.

A community center is planning a summer camp and can accommodate a maximum of 50 children. Each child costs $30 to enroll, and the center wants to raise at least $1,200. Write a system of inequalities to represent the situation, graph the inequalities, and interpret the possible enrollment combinations.

• x >= 30 and x <= 40

• x >= 20 and x <= 30

• x >= 40 and x <= 50

• x >= 50 and x <= 60