Graphing Linear Inequalities: Real-World Applications

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 and the width by y, write the inequality that represents the maximum area of the pen. Graph the inequality and identify the feasible region.

• x + y < 50, x ≥ 0, y ≥ 0

• x + y ≤ 100, x ≥ 0, y ≥ 0

• x + y = 50, x > 0, y > 0

• x + y ≤ 50, x ≥ 0, y ≥ 0

A school is planning a field trip and has a budget of $500. The cost per student is $20 for transportation and $15 for admission. Write an inequality to represent the maximum number of students that can attend the trip. Graph the inequality and interpret the solution.

• x ≤ 14

• x ≤ 12

• x ≤ 20

• x ≤ 10

A factory produces two types of toys: type A and type B. Each type A toy requires 2 hours of labor and each type B toy requires 3 hours. If the factory has 60 hours of labor available, write an inequality to represent the production constraints. Graph the inequality and describe the feasible production combinations.

• x + y ≤ 20

• 2x + 3y ≥ 60

• 2x + 3y = 60

• 2x + 3y ≤ 60

A concert venue can hold a maximum of 300 people. If tickets for adults cost $15 and tickets for children cost $10, write an inequality to represent the total revenue generated if the venue is at full capacity. Graph the inequality and analyze the revenue potential.

• 15x + 10y ≥ R, with x + y ≤ 300.

• x + y ≤ 250, with 15x + 10y ≤ R.

• 15x + 10y = R, with x + y = 300.

• The inequality representing total revenue is 15x + 10y ≤ R, with x + y ≤ 300.

A bakery sells cakes and cookies. Each cake requires 3 cups of flour and each cookie requires 1 cup of flour. If the bakery has 12 cups of flour available, write an inequality to represent the baking constraints. Graph the inequality and discuss the possible combinations of cakes and cookies that can be made.

• 2x + y ≤ 12

• 3x + y ≤ 12

• x + y ≤ 12

• 3x + 2y ≤ 12

A gym has a maximum capacity of 50 members. If each member pays a monthly fee of $30, write an inequality to represent the total income from memberships. Graph the inequality and interpret the financial implications of reaching full capacity.

• 30x = 1500

• 30x <= 1500

• 30x >= 1500

• 30x < 1500

A restaurant has a seating capacity of 80 people. If each table seats 4 people, write an inequality to represent the maximum number of tables that can be set up. Graph the inequality and discuss how this affects the restaurant's service capacity.

• t ≥ 25

• t = 20

• t < 15

• t ≤ 20

A clothing store has a sale on shirts and pants. If shirts are $20 each and pants are $30 each, and the store wants to keep total sales under $600, write an inequality to represent the sales constraints. Graph the inequality and analyze the potential sales combinations.

• 20x + 30y = 600

• 10x + 15y < 300

• 20x + 30y < 600

• 20x + 30y > 600