Linear Inequalities: Graphing and Real-World Applications

A local bakery sells muffins and cookies. Each muffin costs $2 and each cookie costs $1. If the bakery wants to make at least $50 in sales, write a linear inequality to represent the number of muffins (m) and cookies (c) they need to sell. Graph the inequality.

• 3m + c ≥ 50

• m + 2c ≤ 50

• 2m + c ≥ 50

• m + c ≥ 50

A school is planning a field trip and has a budget of $300. The cost per student is $15 for the trip. Write a linear inequality to represent the maximum number of students (s) that can attend. Interpret the graph of this inequality.

• s ≤ 20

• s ≤ 25

• s ≤ 10

• s ≤ 15

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. If the farmer has 120 hours of labor available, write a linear inequality to represent the situation. Graph the inequality and interpret the feasible region.

• 2x + y >= 120 and x + y >= 100

• 2x + y <= 100 and x + y <= 120

• The linear inequalities are: 2x + y <= 120 and x + y <= 100.

• x + 2y <= 120 and x + y <= 100

A gym charges a monthly fee of $30 plus $5 for each class attended. If a member wants to spend no more than $100 in a month, write a linear inequality for the number of classes (c) they can attend. Graph the inequality and explain the results.

• c ≤ 20

• c ≤ 14

• c ≤ 5

• c ≤ 10

A concert hall has a seating capacity of 500. If tickets for the concert are sold at $20 each and VIP tickets at $50 each, write a linear inequality to represent the number of regular tickets (r) and VIP tickets (v) that can be sold without exceeding capacity. Graph the inequality and discuss the implications.

• r + v ≤ 500

• r + v < 500

• r + v ≥ 500

• r + v = 500

A company produces two types of gadgets: Type A and Type B. Each Type A gadget requires 3 hours of labor and each Type B requires 2 hours. If the company has 60 hours of labor available, write a linear inequality to represent the production limits. Graph the inequality and interpret the results.

• 3x + 2y ≥ 60

• 3x + 2y = 60

• 2x + 3y ≤ 60

• 3x + 2y ≤ 60

A charity event aims to raise at least $2000. If each ticket sold is $25 and each donation is $50, write a linear inequality to represent the relationship between tickets (t) and donations (d). Graph the inequality and analyze the feasible solutions.

• 50t + 25d >= 2000

• 25t + 50d >= 2000

• 25t + 50d = 2000

• 25t + 50d < 2000

A restaurant has a maximum seating capacity of 150. If each table seats 5 people, write a linear inequality to represent the number of tables (x) that can be set up. Graph the inequality and interpret the results in terms of seating arrangements.

• x < 30

• x > 30

• 5x >= 150

• 5x <= 150, or x <= 30

A clothing store sells shirts for $15 and pants for $25. If the store wants to make at least $600 in sales, write a linear inequality to represent the number of shirts (s) and pants (p) they need to sell. Graph the inequality and discuss the possible combinations of sales.

• 10s + 20p ≥ 500

• 15s + 25p ≥ 600

• 20s + 30p ≤ 600

• 15s + 25p < 600

A community center has a budget of $400 for a new program. If each workshop costs $50 and each seminar costs $75, write a linear inequality to represent the number of workshops (w) and seminars (s) that can be held. Graph the inequality and interpret the feasible solutions.

• 25w + 50s ≤ 400

• 50w + 75s ≥ 400

• 50w + 75s ≤ 400

• 50w + 75s = 400