Solving Real-World Inequalities: Graphing Solutions

A farmer has 100 meters of fencing to create a rectangular pen for his animals. 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. Graph the inequalities and identify the feasible region.

• x + y ≤ 75, x ≥ 0, y ≤ 0

• The system of inequalities is: { x + y ≤ 50, x ≥ 0, y ≥ 0 }.

• x + y ≤ 100, 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 a system of inequalities to represent the number of students that can attend the trip. Graph the inequalities and determine the maximum number of students that can go.

• 10

• 20

• 12

• 14

A company produces two types of gadgets, A and B. Each gadget A requires 2 hours of labor and each gadget B requires 3 hours. The company has a maximum of 12 hours of labor available. Write a system of inequalities to represent the production limits. Graph the inequalities and find the feasible production combinations.

• The feasible production combinations are (0, 4), (3, 2), (6, 0), (0, 0), (2, 2).

• (4, 1)

• (2, 4)

• (1, 3)

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 30 eggs available, write a system of inequalities to represent the number of cakes that can be made. Graph the inequalities and identify the possible combinations of cakes.

• The bakery can make a maximum of 10 chocolate cakes and 5 vanilla cakes.

• The possible combinations of cakes are represented by the area under the line 3x + 2y = 30 in the first quadrant.

• The system of inequalities is 3x + 2y ≤ 20 in the first quadrant.

• The possible combinations of cakes are represented by the area above the line 3x + 2y = 30.

A gym offers two types of memberships: basic and premium. The basic membership costs $25 per month and the premium costs $40 per month. If the gym wants to earn at least $1000 in a month, write a system of inequalities to represent the number of each type of membership sold. Graph the inequalities and find the combinations that meet the revenue goal.

• The system of inequalities is: 25x + 40y >= 1000, x >= 0, y >= 0.

• 25x + 40y <= 1000, x >= 0, y >= 0

• x + y >= 40, x >= 0, y >= 0

• 25x + 40y = 1000, x >= 0, y >= 0

A local theater has a seating capacity of 200. If tickets for adult seats cost $15 and tickets for child seats cost $10, and the theater wants to earn at least $1500 from ticket sales, write a system of inequalities to represent the ticket sales. Graph the inequalities and determine the possible combinations of adult and child tickets sold.

• The system of inequalities is: x + y ≤ 200 and 15x + 10y ≥ 1500.

• x + y = 200 and 15x + 10y = 1500

• x + y ≥ 200 and 15x + 10y ≤ 1500

• x + y ≤ 150 and 15x + 10y ≥ 2000

A charity event is selling two types of tickets: VIP and regular. The VIP ticket costs $50 and the regular ticket costs $30. If the charity wants to raise at least $2000, write a system of inequalities to represent the ticket sales. Graph the inequalities and find the combinations of tickets that meet the fundraising goal.

• The combinations of VIP and regular tickets that meet the fundraising goal are all points (x, y) in the region above the line 50x + 30y = 2000.

• The charity needs to sell at least 100 VIP tickets to meet the goal.

• The total revenue from ticket sales must be exactly $2000.

• The combinations of VIP and regular tickets are all points (x, y) below the line 50x + 30y = 2000.

A clothing store has a sale on two types of shirts: T-shirts and dress shirts. T-shirts are priced at $10 and dress shirts at $20. If the store wants to make at least $300 in sales, write a system of inequalities to represent the sales of each type of shirt. Graph the inequalities and identify the possible sales combinations.

• 5x + 15y >= 300

• 10x + 10y >= 300

• 15x + 25y >= 300

• 10x + 20y >= 300, x >= 0, y >= 0

A tech company produces two products: laptops and tablets. Each laptop requires 4 hours of assembly and each tablet requires 2 hours. If the company has 20 hours of assembly time available, write a system of inequalities to represent the production limits. Graph the inequalities and find the feasible production combinations.

• (6, 0)

• (0, 12)

• (3, 5)

• The feasible production combinations are (0, 10), (5, 0), (0, 0), (2, 6), (4, 2), and any combination along the line segment connecting (0, 10) to (5, 0) within the first quadrant.

A restaurant offers two types of meals: vegetarian and non-vegetarian. Each vegetarian meal costs $12 and each non-vegetarian meal costs $15. If the restaurant wants to earn at least $600 in a day, write a system of inequalities to represent the meal sales. Graph the inequalities and determine the possible combinations of meals sold.

• The system of inequalities is: 12x + 15y >= 600, x >= 0, y >= 0.

• x + y >= 50, x >= 0, y >= 0

• 12x + 15y = 600, x >= 0, y >= 0

• 12x + 15y <= 600, x >= 0, y >= 0