Mastering Linear Inequalities: Graphing & Solving Systems

A local gym charges a monthly fee of $30 plus $5 for each class attended. Write a linear inequality to represent the total cost for attending classes, where x is the number of classes attended. Graph the inequality for a budget of $100.

• C = 30 + 5x ≤ 100; x ≤ 14

• C = 30 + 5x = 100; x = 14

• C = 30 + 5x ≥ 100; x ≥ 14

• C = 30 + 5x < 100; x < 14

A farmer has 200 feet of fencing to create a rectangular pen for sheep. 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. Solve the system to find possible dimensions.

• Possible dimensions include (x, y) such that x > 100 and y > 100.

• Possible dimensions include (x, y) such that x = 100 and y = 100.

• Possible dimensions include any pairs (x, y) such that 0 < x < 100 and 0 < y < 100 - x.

• Possible dimensions include (x, y) such that x + y = 200 and x > 0.

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 the bus. Write a linear inequality to represent the maximum number of students that can attend the trip. Graph the inequality.

• x ≥ 25

• x < 20

• x = 15

• 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 30 hours of labor available, write a system of inequalities to represent the production limits. Solve the system to find the maximum number of each gadget that can be produced.

• Maximum of 15 gadgets A or 10 gadgets B.

• Maximum of 20 gadgets A or 5 gadgets B.

• Maximum of 10 gadgets A or 15 gadgets B.

• Maximum of 12 gadgets A or 8 gadgets B.

A concert venue can hold a maximum of 500 people. If tickets for adults cost $15 and tickets for children cost $10, write a system of inequalities to represent the ticket sales if the total revenue must be at least $6000. Solve the system to find possible combinations of adult and child tickets sold.

• (600, 0)

• (100, 400)

• (250, 300)

• Possible combinations of adult and child tickets sold are: (200, 300), (250, 250), (300, 200), (350, 150), (400, 100), (450, 50), (500, 0).

A bakery sells two types of cakes: chocolate and vanilla. Each chocolate cake requires 3 eggs and each vanilla cake requires 2 eggs. If the bakery has 60 eggs, write a system of inequalities to represent the number of cakes that can be made. Solve the system to find the maximum number of cakes.

• 25

• 35

• 40

• 30

A student has a budget of $50 to spend on books. If each book costs $12 and there is a $5 shipping fee, write a linear inequality to represent the maximum number of books the student can buy. Graph the inequality and determine the number of books.

• 2

• 3

• 4

• 5

A clothing store has a sale where jeans are $40 each and shirts are $25 each. If a customer wants to spend no more than $200, write a linear inequality to represent the number of jeans (x) and shirts (y) that can be purchased. Graph the inequality and find possible combinations.

• 50x + 15y ≤ 200

• 30x + 20y ≤ 200

• 40x + 25y ≤ 200

• 40x + 30y ≤ 200

A charity event aims to raise at least $1000. If each ticket sold is $25 and there is a fixed cost of $200 for the venue, write a linear inequality to represent the number of tickets (x) that need to be sold. Graph the inequality and determine the minimum number of tickets that must be sold.

• 60

• 40

• 75

• 48

A delivery service charges a base fee of $10 plus $2 per mile driven. Write a linear inequality to represent the total cost (C) for a delivery that must not exceed $50. Graph the inequality and find the maximum number of miles that can be driven.

• 15 miles

• 30 miles

• 25 miles

• 20 miles