Graphing and Analyzing Systems of Inequalities

A farmer has 100 meters of fencing to create a rectangular pen for animals. If the length of the pen is represented by x and the width by y, write a system of inequalities to represent the constraints on the dimensions of the pen. What is the feasible region for the dimensions?

• The system of inequalities is: {x + y ≤ 50, x ≥ 0, y ≥ 0}. The feasible region is the area in the first quadrant bounded by the line x + y = 50 and the axes.

• x + y = 50

• x + y ≥ 100

• x ≤ 0, y ≤ 0

A school is planning a field trip and has a budget of $500. Each student ticket costs $15 and each adult ticket costs $20. Write a system of inequalities to represent the number of student (x) and adult (y) tickets that can be purchased. What is the feasible region for ticket purchases?

• 10x + 15y ≤ 500, x ≥ 0, y ≥ 0

• 20x + 15y ≤ 500, x ≥ 0, y ≥ 0

• 15x + 25y ≤ 500, x ≥ 0, y ≥ 0

• 15x + 20y ≤ 500, x ≥ 0, y ≥ 0

A bakery sells two types of cookies: chocolate chip and oatmeal raisin. The bakery can make a maximum of 200 cookies in a day. If the number of chocolate chip cookies is represented by x and oatmeal raisin cookies by y, write a system of inequalities to represent the production limits. What does the feasible region look like?

• x + y = 200, x ≥ 0, y ≤ 0

• x + y < 200, x > 0, y > 0

• x + y ≥ 200, x ≤ 0, y ≤ 0

• The system of inequalities is: x + y ≤ 200, x ≥ 0, y ≥ 0.

A gym has a maximum capacity of 150 members. If the number of adult members is represented by x and the number of child members by y, write a system of inequalities to represent the membership limits. How can you graphically analyze the solutions?

• x + y < 150, x > 0, y > 0

• x + y = 150, x < 0, y < 0

• x + y ≥ 150, x ≤ 0, y ≤ 0

• x + y ≤ 150, x ≥ 0, y ≥ 0

A concert venue can accommodate a maximum of 300 people. If tickets for adults cost $25 and tickets for children cost $10, and the total revenue must be at least $5000, write a system of inequalities to represent the situation. What is the feasible region for ticket sales?

• x + y >= 300, 25x + 10y <= 5000

• x + y <= 300, 25x + 10y = 5000

• x + y <= 300, 25x + 10y >= 5000

• x + y <= 250, 25x + 10y >= 6000

A local charity is organizing a food drive and has a goal to collect at least 300 cans of food. If they collect 5 cans for every adult volunteer (x) and 3 cans for every child volunteer (y), write a system of inequalities to represent the collection goal. How can you analyze the solutions graphically?

• 10x + 2y ≥ 300

• 2x + 4y ≥ 300

• 5x + 3y ≥ 300

• 5x + 3y < 300

A company produces two products, A and B. Each product A requires 2 hours of labor and each product B requires 3 hours of labor. If the company has 60 hours of labor available, write a system of inequalities to represent the production limits. What does the feasible region indicate?

• 2x + 3y ≥ 60, x ≤ 0, y ≤ 0

• 2x + 3y ≤ 60, x ≥ 0, y ≥ 0

• x + y ≤ 20, x ≥ 0, y ≥ 0

• 3x + 2y ≤ 60, x ≥ 0, y ≤ 0

A pet store has a limit of 50 animals it can house. If the number of dogs is represented by x and the number of cats by y, write a system of inequalities to represent the housing limit. How can you graphically analyze the solutions?

• x + y < 50, x > 0, y > 0

• x + y = 50, x < 0, y < 0

• x + y ≤ 50, x ≥ 0, y ≥ 0

• x + y ≥ 50, x ≤ 0, y ≤ 0

A clothing store has a sale on shirts and pants. Each shirt costs $20 and each pair of pants costs $30. If the store wants to sell at least $600 worth of clothing, write a system of inequalities to represent the sales goal. What is the feasible region for the sales?

• System of inequalities: 20x + 30y >= 600, x >= 0, y >= 0.

• 10x + 15y >= 600, x >= 0, y >= 0.

• 20x + 30y <= 600, x >= 0, y >= 0.

• 20x + 30y = 600, x >= 0, y >= 0.

A community garden has space for a maximum of 40 plants. If the number of tomato plants is represented by x and the number of pepper plants by y, write a system of inequalities to represent the planting limits. How can you analyze the solutions graphically?

• x + y ≥ 40, x ≤ 0, y ≤ 0

• x + y < 40, x > 0, y > 0

• x + y = 40, x < 0, y < 0

• x + y ≤ 40, x ≥ 0, y ≥ 0