Graphing and Analyzing Constraints in Linear 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, formulate a system of linear inequalities to represent the situation. What are the constraints on the number of acres of corn and wheat that can be planted?

• x + y <= 100, 2x + y <= 150, x >= 0, y >= 0

• x + y >= 100, 2x + y >= 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

A school is organizing 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 linear inequalities to represent the maximum number of students that can attend the trip. How would you graph the solution?

• x ≤ 20, x ≥ 0

• x ≤ 14, x ≥ 0

• x ≤ 10, x ≥ 1

• x ≤ 25, x ≥ 5

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 of assembly time available. Additionally, they want to produce at least 5 gadgets A. Create a system of linear inequalities to represent this situation and analyze the constraints.

• 3x + 2y ≤ 25, x ≥ 5, y ≥ 0

• 3x + 2y ≤ 30, x ≥ 10, y ≥ 0

• 3x + 2y ≤ 30, x ≥ 5, x ≥ 0, y ≥ 0

• 3x + 2y ≤ 30, x ≥ 5, y ≤ 2

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 20 pounds of flour, write a system of linear inequalities to represent the maximum number of cakes that can be made. How can you interpret the solution graphically?

• 2x + y ≤ 20, x ≥ 0, y ≥ 0

• x + 2y ≤ 20, x ≥ 0, y ≥ 0

• 3x + y ≤ 20, x ≥ 0, y ≥ 0

• 2x + y < 20, x ≥ 0, y ≥ 0

A gym has a maximum capacity of 200 members. Each membership costs $30 per month, and the gym wants to ensure that they earn at least $3000 per month. Write a system of linear inequalities to represent the situation. What are the constraints on the number of members?

• m >= 100 and m <= 250

• m >= 200 and m <= 300

• m >= 50 and m <= 150

• m >= 100 and m <= 200

A concert hall has 300 seats. Tickets for the concert are sold at $25 for adults and $15 for students. The hall wants to earn at least $5000 from ticket sales. Formulate a system of linear inequalities to represent the constraints on ticket sales. How would you graph the solution?

• x + y >= 300, 25x + 15y <= 5000

• x + y <= 250, 25x + 15y >= 6000

• x + y <= 300, 25x + 15y >= 5000

• x + y = 300, 25x + 15y = 5000

A local restaurant offers two types of meals: vegetarian and non-vegetarian. Each vegetarian meal costs $10 and each non-vegetarian meal costs $15. The restaurant has a budget of $300 for ingredients. Write a system of linear inequalities to represent the maximum number of meals that can be prepared. What are the constraints?

• 10x + 15y ≤ 300, x ≥ 0, y ≥ 0

• 10x + 15y ≤ 200, x ≥ 0, y ≥ 0

• 10x + 20y ≤ 300, x ≥ 0, y ≥ 0

• 5x + 10y ≤ 300, x ≥ 0, y ≥ 0

A charity organization is planning a fundraising event. They need to sell at least 100 tickets, and each ticket costs $50. However, they also have a maximum capacity of 150 attendees. Create a system of linear inequalities to represent this situation. How can you analyze the constraints?

• 100 <= x <= 150

• 50 <= x <= 100

• x > 150

• x < 100

A clothing store sells shirts and pants. Each shirt costs $20 and each pair of pants costs $30. The store has a budget of $600 for inventory. Write a system of linear inequalities to represent the maximum number of shirts and pants that can be purchased. How would you interpret the solution graphically?

• 15x + 25y ≤ 600, x ≥ 0, y ≥ 0

• 10x + 20y ≤ 600, x ≥ 0, y ≥ 0

• 25x + 35y ≤ 600, x ≥ 0, y ≥ 0

• 20x + 30y ≤ 600, x ≥ 0, y ≥ 0

A student is saving money for a new laptop that costs $800. They can save $50 per week from their allowance and $20 per week from doing chores. Write a system of linear inequalities to represent the number of weeks needed to save enough money. What are the constraints on their savings?

• w < 12, w >= 0

• w <= 16, w >= 0

• w >= 10, w <= 20

• w >= 12, w >= 0