Mastering Linear Inequalities: Real-World Applications

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 an inequality to represent the maximum area of the pen. What are the possible values for x and y?

• x + y = 100, x ≤ 0, y ≤ 0

• x + y ≤ 50, x ≥ 0, y ≥ 0

• x + y ≥ 50, x < 0, y < 0

• x + y < 100, x > 0, y > 0

A school is planning a field trip and has a budget of $500. The cost per student is $20, and the bus rental is a fixed cost of $200. Write an inequality to represent the maximum number of students that can attend the trip. How many students can they take?

• 20

• 25

• 15

• 10

A local gym offers two types of memberships: a monthly membership for $30 and an annual membership for $300. If a person can spend no more than $600 on memberships, write an inequality to represent the number of monthly memberships (x) and annual memberships (y) they can purchase. What are the possible combinations?

• 30x + 300y ≥ 600

• 30x + 300y = 600

• 30x + 300y ≤ 600

• 15x + 150y ≤ 600

A concert venue has a seating capacity of 2000. If tickets are sold for $50 each and the venue wants to make at least $80,000, write an inequality to represent the number of tickets (x) that must be sold. How many tickets need to be sold to meet the goal?

• 1800

• 1400

• 2000

• 1600

A bakery sells cupcakes for $2 each and cookies for $1 each. If the bakery wants to make at least $100 in one day, write an inequality to represent the number of cupcakes (x) and cookies (y) they need to sell. What are the possible combinations of cupcakes and cookies?

• 2x + y >= 100

• 3x + y <= 100

• x + y >= 50

• x + 2y >= 100

A company produces two types of gadgets, A and B. Each gadget A requires 2 hours of labor and gadget B requires 3 hours. If the company has 30 hours of labor available, write a system of inequalities to represent the production limits of gadgets A (x) and B (y). What are the possible production combinations?

• 3x + 2y ≤ 30, x ≥ 0, y ≤ 0

• 2x + 3y ≥ 30, x ≤ 0, y ≤ 0

• x + y ≤ 10, x ≥ 0, y ≥ 0

• The possible production combinations are represented by the region defined by the inequalities: 2x + 3y ≤ 30, x ≥ 0, y ≥ 0.

A charity event aims to raise at least $2000. If each ticket sold is $25 and each donation is $50, write an inequality to represent the relationship between the number of tickets (x) sold and donations (y) received. How many tickets and donations are needed to meet the goal?

• At least 100 tickets with no donations.

• At least 20 tickets and 10 donations.

• At least 60 tickets or 30 donations.

• At least 80 tickets or combinations of tickets and donations such as 40 tickets and 20 donations.

A restaurant has a maximum capacity of 150 customers. If they want to maintain a ratio of at least 2:1 between diners and takeout orders, write a system of inequalities to represent the number of dine-in customers (x) and takeout orders (y). What are the possible values for x and y?

• x + y <= 150, x >= 2y

• x + y >= 150, x <= 2y

• x + y = 150, x > 2y

• x + y < 150, x = 2y

A clothing store has a sale on shirts and pants. Shirts are $15 each and pants are $25 each. If a customer wants to spend no more than $100, write an inequality to represent the number of shirts (x) and pants (y) they can buy. What are the possible combinations?

• 15x + 25y ≤ 100

• 20x + 15y ≤ 100

• 5x + 30y ≤ 100

• 10x + 20y ≤ 100

A student is saving money for a new laptop that costs $800. If they save $50 a week from their allowance and also plan to earn $100 from a part-time job, write an inequality to represent the number of weeks (x) they need to save to afford the laptop. How many weeks will it take?

• 20 weeks

• 12 weeks

• 10 weeks

• 14 weeks