Solving Real-World Systems of Inequalities Challenge

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 inequalities that represent the constraints on the dimensions of the pen. Graph the inequalities and identify the feasible region.

• x + y ≤ 75, x ≥ 10, y ≥ 10

• The inequalities representing the constraints are: 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

• 14

• 12

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 of labor. The company has a maximum of 12 hours of labor available. Write the inequality for the labor constraint and graph it. What combinations of gadgets can the company produce?

• 2x + 3y ≥ 12

• x + y ≤ 4

• x + 2y ≤ 12

• The combinations of gadgets A and B that the company can produce are given by the inequality 2x + 3y ≤ 12, where x ≥ 0 and y ≥ 0.

A local gym offers two types of memberships: basic and premium. The basic membership costs $30 per month, while the premium membership costs $50 per month. If the gym wants to earn at least $1,200 in a month, write a system of inequalities to represent the number of each type of membership sold. Graph the inequalities and find the possible combinations of memberships.

• 30x + 50y = 1200, x >= 0, y >= 0

• The system of inequalities is: 30x + 50y >= 1200, x >= 0, y >= 0.

• 20x + 40y >= 1200, x >= 0, y >= 0

• 30x + 50y <= 1200, x >= 0, y >= 0

A bakery sells two types of cakes: chocolate and vanilla. Each chocolate cake requires 3 cups of flour and each vanilla cake requires 2 cups of flour. If the bakery has 18 cups of flour, write the inequality for the flour constraint and graph it. What is the maximum number of cakes that can be made?

• 5

• 8

• 7

• 6

A concert venue has a seating capacity of 500. If tickets for the concert are sold at $25 for general admission and $40 for VIP seating, write a system of inequalities to represent the total revenue generated if the venue wants to earn at least $10,000. Graph the inequalities and determine the possible ticket sales combinations.

• VIP tickets can only be sold if general admission tickets are sold out.

• The venue can accommodate 600 people for the concert.

• The total revenue must be exactly $10,000.

• The possible ticket sales combinations are represented by the shaded area in the graph of the inequalities.

A charity organization is organizing a fundraiser and has a goal of raising at least $2,000. They plan to sell tickets for $10 each and receive donations. Write a system of inequalities to represent the number of tickets sold and donations received. Graph the inequalities and find the combinations that meet the fundraising goal.

• 10x + y = 2000, x >= 0, y >= 0

• 10x + y <= 2000, x >= 0, y >= 0

• The system of inequalities is: 10x + y >= 2000, x >= 0, y >= 0.

• 10x + y >= 1000, x >= 0, y >= 0

A restaurant offers two types of meal plans: vegetarian and non-vegetarian. The vegetarian meal plan costs $15 and the non-vegetarian meal plan costs $20. If the restaurant wants to make at least $1,000 in a week, write a system of inequalities to represent the number of each meal plan sold. Graph the inequalities and determine the possible combinations of meal plans.

• 15x + 20y = 1000, x >= 0, y >= 0

• The system of inequalities is: 15x + 20y >= 1000, x >= 0, y >= 0.

• 15x + 20y <= 1000, x >= 0, y >= 0

• 10x + 25y >= 1000, x >= 0, y >= 0

A clothing store has a sale on shirts and pants. Shirts are priced at $20 each and pants at $30 each. If the store wants to make at least $600 in sales, write a system of inequalities to represent the number of shirts and pants sold. Graph the inequalities and find the combinations that meet the sales goal.

• Shirts must be sold at $30 each and pants at $20 each.

• The total sales must be exactly $600.

• The combinations of shirts and pants that meet the sales goal are all points (x, y) in the region defined by the inequalities.

• The store needs to sell exactly 20 shirts and 10 pants.

A tech company is developing two products, X and Y. Each product X requires 4 hours of development and each product Y requires 2 hours. If the company has a maximum of 20 hours available for development, write the inequality for the time constraint and graph it. What combinations of products can the company develop?

• 2x + 4y ≤ 20

• The combinations of products X and Y that can be developed are: (x, y) such that 4x + 2y ≤ 20, where x ≥ 0 and y ≥ 0.

• 4x + 3y ≤ 20

• 5x + y ≤ 20