Graphing and Solving Systems of Inequalities in Real Scenarios

A farmer has 100 meters of fencing to create a rectangular pen for sheep. The length of the pen must be at least 10 meters. Set up a system of inequalities to represent the possible dimensions of the pen. What are the constraints on the length and width?

• L + W ≤ 100, L ≥ 10, W ≥ 5

• L + W < 50, L ≤ 10, W ≤ 0

• L + W ≤ 50, L ≥ 10, W ≥ 0

• L + W = 100, L ≥ 5, W ≥ 0

A school is planning a field trip and has a budget of $500. Each student ticket costs $15, and each adult ticket costs $20. Set up a system of inequalities to represent the number of student and adult tickets that can be purchased without exceeding the budget. What are the possible combinations?

• 15x + 20y >= 500

• The possible combinations of (x, y) are all pairs of non-negative integers that satisfy the inequality 15x + 20y <= 500.

• x + 2y <= 20

• x + y <= 25

A local gym offers two types of memberships: a basic membership for $30 per month and a premium membership for $50 per month. If a family wants to spend no more than $200 per month on memberships, set up a system of inequalities to represent the number of basic and premium memberships they can buy. What are the possible solutions?

• The possible solutions are the pairs (x, y) such that 30x + 50y ≤ 200, x ≥ 0, and y ≥ 0.

• 30x + 50y = 200, x < 0, y < 0

• 30x + 50y ≥ 200, x ≥ 0, y ≥ 0

• 30x + 50y ≤ 150, x ≥ 0, y ≥ 0

A bakery sells two types of cakes: chocolate and vanilla. Each chocolate cake requires 2 hours of baking time, and each vanilla cake requires 1 hour. If the bakery has a total of 10 hours available for baking, set up a system of inequalities to represent the number of each type of cake that can be baked. What are the constraints?

• The constraints are: 2x + y <= 10, x >= 0, y >= 0.

• x + 2y <= 10, x >= 0, y >= 0

• 2x + y >= 10, x >= 0, y >= 0

• 2x + 3y <= 10, x >= 0, y >= 0

A concert venue has a maximum capacity of 300 people. Tickets for adults cost $25, and tickets for children cost $15. If the venue wants to make at least $5000 from ticket sales, set up a system of inequalities to represent the number of adult and child tickets that can be sold. What are the possible combinations?

• x + y ≥ 300 and 25x + 15y ≤ 5000

• The possible combinations of adult and child tickets can be represented by the inequalities: x + y ≤ 300 and 25x + 15y ≥ 5000.

• x + y = 300 and 25x + 15y = 5000

• x + y ≤ 250 and 25x + 15y ≥ 6000

A clothing store is having a sale on shirts and pants. Each shirt costs $20, and each pair of pants costs $30. If a customer wants to spend no more than $200, set up a system of inequalities to represent the number of shirts and pants they can buy. What are the possible solutions?

• The possible solutions are the pairs (x, y) that satisfy the inequalities, such as (0, 0), (0, 6), (5, 0), (4, 2), (3, 4), (2, 6), (1, 8), and (0, 10).

• (10, 0)

• (0, 7)

• (6, 4)

A charity organization is organizing a fundraiser and needs to sell at least 200 tickets. Each adult ticket costs $10, and each child ticket costs $5. Set up a system of inequalities to represent the number of adult and child tickets that must be sold. What are the constraints?

• x + y <= 200, x >= 0, y >= 0

• x + y >= 250, x >= 0, y >= 0

• x + y = 200, x >= 0, y >= 0

• x + y >= 200, x >= 0, y >= 0

A pet store has a limit of 50 animals that can be adopted out in a month. If they have cats and dogs, and each cat counts as 1 animal while each dog counts as 2 animals, set up a system of inequalities to represent the number of cats and dogs that can be adopted. What are the possible solutions?

• x + y ≤ 50, x ≥ 0, y ≥ 0

• x + 3y ≤ 50, x ≥ 0, y ≥ 0

• 2x + y ≤ 50, x ≥ 0, y ≥ 0

• x + 2y ≤ 50, x ≥ 0, y ≥ 0

A restaurant offers two types of meal deals: a burger meal for $12 and a pasta meal for $15. If a family wants to spend no more than $100 on meals, set up a system of inequalities to represent the number of burger and pasta meals they can order. What are the possible combinations?

• The possible combinations of (x, y) are all pairs of non-negative integers satisfying 12x + 15y <= 100.

• The possible combinations of (x, y) are all pairs of integers satisfying 12x + 15y < 100.

• The possible combinations of (x, y) are all pairs of non-negative integers satisfying 12x + 15y >= 100.

• The possible combinations of (x, y) are all pairs of integers satisfying 12x + 15y = 100.

A school is organizing a sports day and needs to set up a system of inequalities to ensure that at least 50 students participate in either soccer or basketball. If each soccer team has 5 players and each basketball team has 7 players, set up the inequalities and interpret the solutions graphically. What are the constraints?

• 5x + 7y <= 50, x >= 0, y >= 0

• 5x + 7y = 50, x >= 0, y >= 0

• 5x + 7y >= 100, x >= 0, y >= 0

• 5x + 7y >= 50, x >= 0, y >= 0