Exploring Linear Inequalities: Real-Life Applications

A farmer has 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 1 hour of labor. If he has a total of 120 hours of labor available, what is the feasible region for the number of acres of corn (x) and wheat (y) he can plant?

• The feasible region is defined by the inequalities x + y <= 100 and 2x + y <= 120, with x, y >= 0.

• x + y <= 100 and 2x + y >= 120

• x + y <= 120 and 2x + y <= 100

• x + y >= 100 and 2x + y >= 120

A school is organizing a field trip and has a budget of $500. The cost per student is $10 for transportation and $15 for admission. If the number of students is represented by x, how can you express the inequalities that represent the maximum number of students that can attend the trip?

• 25x <= 500

• 20x <= 500

• 30x <= 500

• 15x <= 500

A company produces two types of gadgets, A and B. Each gadget A requires 3 hours of assembly and each gadget B requires 2 hours. The company has 30 hours available for assembly. If they want to produce at least 5 gadgets A, what are the inequalities that represent the production limits?

• 3x + 2y <= 30, x >= 5

• 3x + 2y <= 20, x >= 5

• 3x + 2y >= 30, x <= 5

• x + y <= 30, x >= 5

A gym offers two types of memberships: basic and premium. The basic membership costs $20 per month, and the premium membership costs $40 per month. If the gym wants to earn at least $1,000 in a month, how can you express the inequalities for the number of basic (x) and premium (y) memberships sold?

• 20x + 40y ≤ 1000

• 30x + 20y ≥ 1000

• 10x + 50y ≥ 1000

• 20x + 40y ≥ 1000

A 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 has 10 hours available for baking, what inequalities represent the maximum number of cakes they can bake?

• x + y ≤ 10

• 3x + y ≤ 10

• x + 2y ≤ 10

• 2x + y ≤ 10

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

• x + y = 500, 50x + 30y = 15000

• x + y <= 400, 50x + 30y >= 20000

• x + y >= 500, 50x + 30y <= 15000

• x + y <= 500, 50x + 30y >= 15000

A clothing store sells shirts and pants. Each shirt costs $15 and each pair of pants costs $25. If the store wants to make at least $1,200 in sales, what inequalities can represent the number of shirts (x) and pants (y) sold?

• 15x + 25y <= 1200

• 20x + 30y = 1200

• 10x + 20y >= 1200

• 15x + 25y >= 1200

A charity organization is planning a fundraiser. They can sell tickets for $10 each or offer a dinner for $50 per person. If they want to raise at least $2,000, what inequalities can represent the number of tickets (x) and dinners (y) sold?

• 20x + 30y > 2000

• 5x + 25y ≥ 2000

• 15x + 40y ≤ 2000

• 10x + 50y ≥ 2000

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 40 hours available for development, what inequalities represent the maximum number of products they can develop?

• 2x + 3y ≤ 40, x ≥ 0, y ≥ 0

• 5x + y ≤ 40, x ≥ 0, y ≥ 0

• 3x + 4y ≤ 40, x ≥ 0, y ≥ 0

• 4x + 2y ≤ 40, x ≥ 0, y ≥ 0

A local farmer's market sells fruits and vegetables. Each fruit basket costs $5 and each vegetable basket costs $3. If the market wants to earn at least $600, what inequalities can represent the number of fruit (x) and vegetable (y) baskets sold?

• 5x + 3y >= 600, x >= 0, y >= 0

• 5x + 3y <= 600, x >= 0, y >= 0

• 5x + 3y = 600, x >= 0, y >= 0

• 2x + 4y >= 600, x >= 0, y >= 0