Graphing and Comparing Linear Inequalities in Real Life

1. A local gym charges a monthly fee of $30 plus $5 for each class attended. Write a linear inequality in slope-intercept form to represent the total cost (C) for attending x classes if the total cost must be less than $100. Graph the inequality and interpret the solution.

• C < 5x + 30, x < 14

• C > 5x + 30, x < 20

• C < 5x + 50, x < 12

• C < 10x + 30, x < 8

2. A school is planning a field trip and has a budget of $500. The cost per student is $20. Write a linear inequality to represent the maximum number of students (s) that can attend the trip. Compare this function with a scenario where the cost is $25 per student. How does the graph change?

• s ≤ 10 for $20 per student; s ≤ 5 for $25 per student.

• s ≤ 25 for $20 per student; s ≤ 20 for $25 per student.

• s ≤ 20 for $20 per student; s ≤ 25 for $25 per student.

• s ≤ 30 for $20 per student; s ≤ 15 for $25 per student.

3. A car rental company charges a flat fee of $50 plus $0.20 per mile driven. Write a linear inequality to represent the total cost (C) if the total cost must be less than $100. Graph the inequality and explain what the slope and y-intercept represent in this context.

• C < 50 + 0.50m

• C < 100 + 0.20m

• C < 50 + 0.20m

• C < 50 + 0.10m

4. A farmer wants to plant two types of crops. Crop A requires 2 hours of labor per acre, and Crop B requires 3 hours. If the farmer has a maximum of 30 hours available, write a linear inequality to represent the relationship between acres of Crop A (a) and Crop B (b). How would you graph this inequality?

• 2a + 3b < 30

• 2a + 3b = 30

• 2a + 3b ≤ 30

• 2a + 3b ≥ 30

5. A concert venue has a seating capacity of 200. If tickets are sold for $15 each, write a linear inequality to represent the total revenue (R) if the revenue must be at least $1,500. Compare this with a scenario where tickets are sold for $20 each. How does the graph differ?

• 15x ≥ 2000 (x ≥ 133) for $15 tickets

• 20x ≤ 1500 (x ≤ 75) for $20 tickets

• 15x ≤ 1500 (x ≤ 100) for $15 tickets

• 15x ≥ 1500 (x ≥ 100) for $15 tickets; 20x ≥ 1500 (x ≥ 75) for $20 tickets.

6. A company produces two products, A and B. Each product A requires 4 hours of labor, and each product B requires 2 hours. If the company has 40 hours of labor available, write a linear inequality to represent the production limits. Graph the inequality and discuss the feasible region.

• 5x + y ≤ 40

• 3x + 4y ≤ 40

• 2x + 3y ≤ 40

• 4x + 2y ≤ 40

7. A charity event aims to raise at least $2,000. If each ticket sold is $25, write a linear inequality to represent the number of tickets (t) that need to be sold. Graph the inequality and interpret the solution in terms of ticket sales.

• t < 80

• t <= 80

• t > 80

• t >= 80

9. A student has a budget of $100 to spend on school supplies. If notebooks cost $5 each and pens cost $2 each, write a linear inequality to represent the number of notebooks (n) and pens (p) that can be purchased. How would you graph this inequality?

• 3n + 4p ≤ 100

• n + 5p ≤ 100

• 5n + 2p ≤ 100

• 10n + p ≤ 100

10. A restaurant has a special offer where a meal costs $12 plus $3 for each drink. Write a linear inequality to represent the total cost (C) if the total must be less than $60. Graph the inequality and discuss the implications of the solution in terms of meal and drink combinations.

• C < 70; d < 20

• C < 60; d < 16

• C < 60; d < 12

• C < 50; d < 10