Graphing and Solving Real-World Linear Inequalities

A local 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 to represent the number of classes they can attend. Graph the inequality.

• x ≤ 10

• x ≤ 20

• x ≤ 12

• x ≤ 14

A school is planning a field trip and has a budget of $500. The cost per student is $20, and there is a fixed cost of $100 for transportation. Write a linear inequality to represent the maximum number of students that can attend. Graph the inequality and explain what it means.

• x ≤ 20

• x ≥ 15

• x ≤ 25

• x < 20

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 a maximum of 30 hours of labor available, write a linear inequality to represent the production limits. Graph the inequality and interpret the feasible region.

• 2x + 3y ≤ 30

• x + 2y ≤ 30

• 2x + 3y ≥ 30

• 4x + y ≤ 30

A concert venue can hold a maximum of 500 people. If tickets for adults cost $15 and tickets for children cost $10, write a linear inequality to represent the total revenue generated if the venue is at full capacity. Graph the inequality and discuss the implications.

• 15x + 10y, where x + y = 500

• 10x + 15y, where x + y < 500

• 15x + 10y, where x + y ≤ 500

• 20x + 5y, where x + y ≤ 500

A restaurant offers two types of meal plans: Plan X costs $25 per meal and Plan Y costs $15 per meal. If a customer wants to spend no more than $200 on meals, write a linear inequality to represent the number of meals they can purchase. Graph the inequality and analyze the results.

• 20x + 10y ≤ 200

• 30x + 5y ≤ 200

• 25x + 20y ≤ 200

• 25x + 15y ≤ 200

A charity event aims to raise at least $1,000. If each ticket sold is $50 and there is a fixed cost of $200 for the venue, write a linear inequality to represent the number of tickets that need to be sold. Graph the inequality and interpret the solution.

• x > 30

• x <= 20

• x >= 24

• x < 24

A factory produces two products, P1 and P2. Each product P1 requires 4 hours of machine time and each product P2 requires 2 hours. If the factory has 40 hours of machine time available, write a linear inequality to represent the production constraints. Graph the inequality and explain the feasible solutions.

• 4x + 3y ≤ 40

• 2x + 4y ≤ 40

• 4x + 2y ≤ 40

• 3x + 2y ≤ 40

A student has a budget of $150 for school supplies. If notebooks cost $5 each and pens cost $2 each, write a linear inequality to represent the maximum number of notebooks and pens the student can buy. Graph the inequality and interpret the results.

• 5x + 2y ≤ 150

• 5x + 3y ≤ 150

• 3x + 4y ≤ 150

• 2x + 5y ≤ 150