Inequalities in Action: Solving Real-World Scenarios

A farmer has 100 feet of fencing to create two adjacent rectangular pens for his animals. If the length of the first pen is represented by x and the length of the second pen by y, write the system of inequalities that represents the total area of the pens being less than or equal to 1000 square feet. What are the possible dimensions for the pens?

• The system of inequalities is: 2x + 2y <= 100 and xy <= 1000, with x >= 0, y >= 0, and x + y <= 50.

• x + y >= 50 and 2x + 2y <= 200

• x + y <= 100 and xy <= 500

• 2x + 2y >= 100 and xy >= 1000

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 (x) that can attend the trip. How many students can go if they want to stay within budget?

• 12

• 14

• 10

• 18

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 a month on memberships, write a system of inequalities to represent the number of basic (x) and premium (y) memberships they can purchase. What combinations of memberships are possible?

• (2, 4)

• The combinations of memberships (x, y) that satisfy the inequalities are: (0, 4), (1, 3), (2, 2), (3, 1), (4, 0), (0, 3), (0, 2), (0, 1), (0, 0) with x being the number of basic memberships and y being the number of premium memberships.

• (5, 0)

• (0, 5)

A bakery sells two types of cookies: chocolate chip and oatmeal raisin. Each chocolate chip cookie costs $2 and each oatmeal raisin cookie costs $3. If the bakery wants to make at least 50 cookies and spend no more than $120 on ingredients, write a system of inequalities to represent the situation. What are the possible numbers of each type of cookie they can make?

• The system of inequalities is: x + y >= 50 and 2x + 3y <= 120.

• x + y >= 50 and 2x + 3y = 120

• x + y <= 50 and 2x + 3y <= 100

• x + y <= 50 and 2x + 3y >= 120

A concert hall has a seating capacity of 500. Tickets for the front row cost $50 each, and tickets for the back row cost $30 each. If the concert hall wants to make at least $15,000 from ticket sales, write a system of inequalities to represent the number of front row (x) and back row (y) tickets sold. What combinations of ticket sales meet this requirement?

• x + y ≥ 500 and 50x + 30y ≤ 15000

• The system of inequalities is: x + y ≤ 500 and 50x + 30y ≥ 15000.

• x + y ≤ 400 and 50x + 30y ≥ 20000

• x + y = 500 and 50x + 30y = 15000

A charity organization is hosting a fundraiser and has a goal of raising at least $1,000. They sell tickets for $10 each and receive donations of $5 each. Write a system of inequalities to represent the number of tickets (x) sold and donations (y) received. How many tickets and donations are needed to meet their goal?

• x <= 0, y <= 0, 10x + 5y <= 1000

• x + y <= 100, 10x + 5y >= 500

• x >= 0, y >= 0, 10x + 5y >= 1000; e.g., (100, 0) or (50, 100) meet the goal.

• x >= 100, y >= 100, 10x + 5y >= 2000

A clothing store is having a sale on shirts and pants. Shirts cost $20 each and pants cost $30 each. If the store wants to sell at least 40 items and make no more than $800 in sales, write a system of inequalities to represent the number of shirts (x) and pants (y) sold. What combinations of shirts and pants can they sell?

• x + y <= 40 and 2x + 3y >= 80

• x + y >= 50 and 2x + 3y <= 100

• The combinations of shirts (x) and pants (y) that satisfy the inequalities are: (x, y) such that x + y >= 40 and 2x + 3y <= 80.

• x + y <= 30 and 2x + 3y >= 60

A pet shelter has a limit of 50 animals they can take care of. They have dogs and cats, with each dog requiring 3 square feet and each cat requiring 2 square feet. If they have a total of 120 square feet available, write a system of inequalities to represent the number of dogs (x) and cats (y) they can take in. What are the possible combinations of dogs and cats?

• (15, 35)

• The possible combinations of dogs and cats (x, y) are: (0, 50), (10, 40), (20, 30), (30, 20), (40, 10), (50, 0) and other combinations satisfying the inequalities.

• (25, 25)

• (5, 45)

A restaurant offers two types of meals: vegetarian and non-vegetarian. The vegetarian meal costs $12 and the non-vegetarian meal costs $15. If the restaurant wants to serve at least 100 meals and keep the total cost under $1,500, write a system of inequalities to represent the number of vegetarian (x) and non-vegetarian (y) meals served. What combinations of meals can they serve?

• x + y <= 100 and 12x + 15y > 1500

• x + y >= 100 and 12x + 15y = 1500

• x + y < 100 and 12x + 15y <= 1500

• The combinations of vegetarian (x) and non-vegetarian (y) meals must satisfy the inequalities: x + y >= 100 and 12x + 15y < 1500.

A tech company is producing two types of gadgets: A and B. Each gadget A costs $40 to produce and gadget B costs $60. If the company has a budget of $2,400 and wants to produce at least 30 gadgets in total, write a system of inequalities to represent the number of gadgets A (x) and B (y) they can produce. What are the possible production combinations?

• 40x + 60y ≤ 3000 and x + y ≥ 20

• The system of inequalities is: 40x + 60y ≤ 2400 and x + y ≥ 30. Possible combinations include (30, 0), (24, 6), (18, 12), (12, 18), (6, 24), and (0, 30) among others, as long as they satisfy both inequalities.

• x + y ≤ 40 and 40x + 60y ≥ 2400

• 40x + 60y ≤ 2400 and x + y ≤ 30