Real-World Systems of Inequalities: Contextual Applications

A farmer has a total of 100 acres of land. He wants to plant corn and wheat. Each acre of corn requires 2 hours of labor, and each acre of wheat requires 3 hours of labor. If he has a maximum of 240 hours of labor available, what are the inequalities that represent the situation?

• x + y ≤ 100, 2x + 3y ≥ 240

• x + y ≥ 100, 2x + 3y ≤ 240

• x + y ≤ 100, 2x + 3y ≤ 240

• x + y ≤ 80, 2x + 3y ≤ 200

A school is organizing a field trip and has a budget of $500. The cost per student is $20 for transportation and $15 for food. If the number of students is represented by x, write the system of inequalities that represents the budget constraints.

• { "35x \u2264 500", "x \u2265 0" }

• $20x + $15x \leq 500

• x \leq 10

• x \leq 25

A gym offers two types of memberships: a basic membership for $30 per month and a premium membership for $50 per month. If a customer can spend no more than $300 in a year, what inequalities can be used to represent the number of basic (x) and premium (y) memberships they can purchase?

• 30x + 50y = 300

• 30x + 50y ≥ 300

• 20x + 40y ≤ 300

• 30x + 50y ≤ 300

A local bakery sells two types of cakes: chocolate and vanilla. Each chocolate cake requires 2 hours to bake, and each vanilla cake requires 1 hour. If the bakery can only bake for 10 hours a day, write the system of inequalities that represents the maximum number of cakes they can bake in a day.

• x + y ≤ 10, x ≥ 0, y ≥ 0

• x + 2y ≤ 10, x ≥ 0, y ≥ 0

• 2x + y ≥ 10, x ≥ 0, y ≥ 0

• 2x + y ≤ 10, x ≥ 0, y ≥ 0

A concert venue 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 venue wants to make at least $15,000 from ticket sales, what inequalities represent the ticket sales situation?

• x + y = 500, 50x + 30y = 15000

• x + y <= 500, 50x + 30y >= 15000

• x + y <= 400, 50x + 30y >= 20000

• x + y >= 500, 50x + 30y <= 15000

A clothing store sells shirts for $25 and pants for $40. If the store wants to sell at least 100 items and make at least $3,000 in sales, write the system of inequalities that represents this scenario.

• x + y >= 150, 25x + 40y >= 2500

• x + y = 100, 25x + 40y = 3000

• x + y >= 100, 25x + 40y >= 3000

• x + y <= 100, 25x + 40y <= 3000

A charity event is selling two types of tickets: VIP tickets for $100 and regular tickets for $50. If they want to raise at least $5,000 and sell no more than 100 tickets in total, what inequalities can be used to represent this situation?

• 100x + 50y >= 5000, x + y <= 100, x >= 0, y >= 0

• x + y >= 50

• x + y >= 100

• 100x + 50y <= 5000

A pet shelter has a limit of 50 animals they can care for. They have dogs and cats, with each dog taking up 2 spaces and each cat taking up 1 space. If they want to take in as many animals as possible, what system of inequalities represents this situation?

• 2x + y <= 50, x >= 0, y >= 0

• x + y <= 50, x >= 0, y >= 0

• 2x + 3y <= 50, x >= 0, y >= 0

• x + 2y <= 50, x >= 0, y >= 0

A restaurant offers two types of meals: vegetarian and non-vegetarian. Each vegetarian meal costs $15 and each non-vegetarian meal costs $25. If the restaurant wants to earn at least $1,000 in a week and sell at least 50 meals, what inequalities represent this scenario?

• 15x + 25y ≥ 1000, x + y ≥ 50

• 10x + 20y ≥ 1000, x + y ≥ 50

• 15x + 25y ≤ 1000, x + y ≤ 50

• 15x + 25y = 1000, x + y = 50

A tech company produces two types of gadgets: A and B. Each gadget A requires 3 hours of assembly and gadget B requires 2 hours. If the company has 60 hours available for assembly, what system of inequalities can be used to represent the production limits?

• 3x + 2y ≤ 60, x ≥ 0, y ≥ 0

• 2x + 3y ≤ 60, x ≥ 0, y ≥ 0

• 3x + 2y ≤ 50, x ≥ 0, y ≥ 0

• 3x + 2y ≥ 60, x ≥ 0, y ≥ 0