Exploring Inequalities: Word Problems & Feasible Regions

A farmer has 100 meters of fencing to create a rectangular pen for his sheep. If the length of the pen is represented by x meters, write an inequality to represent the maximum area of the pen. Graph the inequality and identify the feasible region.

• x + y ≤ 100, x ≥ 0, y ≥ 0

• x + y ≤ 75, x ≥ 0, y ≥ 0

• x + y ≤ 50, x ≥ 0, y ≥ 0

• x + y ≤ 25, x ≥ 0, y ≥ 0

A school is planning a field trip and has a budget of $500. The cost per student is $20. Write an inequality to represent the maximum number of students that can attend the trip. Graph the inequality and identify the feasible region.

• x ≤ 25

• x ≤ 20

• x < 25

• x ≥ 30

A company produces two types of gadgets, A and B. Each gadget A requires 2 hours of labor and gadget B requires 3 hours. If the company has 30 hours of labor available, write an inequality to represent the production limits. Graph the inequality and identify the feasible region.

• 4x + y ≤ 30

• 2x + 3y ≤ 30

• 2x + 3y ≥ 30

• x + 2y ≤ 30

A concert venue can hold a maximum of 200 people. If tickets are sold for $15 each, write an inequality to represent the total revenue generated if x is the number of tickets sold. Graph the inequality and identify the feasible region.

• 0 <= x <= 150

• 15x >= 3000

• 15x <= 3000, 0 <= x <= 200

• 10x <= 2000

A bakery sells cookies for $2 each and muffins for $3 each. If the bakery wants to make at least $60 in sales, write an inequality to represent the relationship between the number of cookies (c) and muffins (m) sold. Graph the inequality and identify the feasible region.

• 4c + m ≥ 60

• c + m ≥ 20

• 2c + 3m ≥ 60

• 2c + 3m ≤ 60

A gym has a maximum capacity of 50 members. If x represents the number of adult members and y represents the number of child members, write an inequality to represent the gym's capacity. Graph the inequality and identify the feasible region.

• x + y < 50

• x + y ≥ 50

• x + y = 50

• x + y ≤ 50

A local charity is organizing a fundraiser and needs to raise at least $1,000. If they earn $50 for each ticket sold, write an inequality to represent the number of tickets (t) they need to sell. Graph the inequality and identify the feasible region.

• t <= 25

• t < 20

• t >= 20

• t > 15

A factory produces two products, X and Y. Each product X requires 4 hours of machine time and each product Y requires 2 hours. If the factory has 40 hours of machine time available, write an inequality to represent the production limits. Graph the inequality and identify the feasible region.

• 2x + 4y ≤ 40

• 4x + 3y ≤ 40

• 4x + 2y ≤ 40

• 3x + 2y ≤ 40

A restaurant has a maximum seating capacity of 80 people. If x represents the number of tables for two and y represents the number of tables for four, write an inequality to represent the seating capacity. Graph the inequality and identify the feasible region.

• 2x + 4y ≤ 80

• 2x + 4y < 80

• x + 2y ≤ 80

• 3x + 2y ≤ 80

A student is saving money for a new laptop that costs $800. If they save $50 each week, write an inequality to represent the number of weeks (w) they need to save to afford the laptop. Graph the inequality and identify the feasible region.

• w < 16

• w > 16

• w >= 16

• w <= 16