Mastering Systems of Inequalities in Real-World Scenarios

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, write a system of inequalities to represent the situation and determine the maximum number of acres he can plant.

• 60 acres of corn and 0 acres of wheat

• 70 acres of corn and 10 acres of wheat

• 50 acres of corn and 25 acres of wheat

• 40 acres of corn and 20 acres of wheat

A school is organizing 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 maximum number of students that can attend the trip, and solve for the number of students if the budget is fully utilized.

• 20

• 14

• 10

• 12

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 $200 on memberships, write a system of inequalities to represent the situation and find the maximum number of each type of membership they can purchase.

• Maximum of 6 basic memberships and 0 premium memberships.

• Maximum of 4 basic memberships and 2 premium memberships.

• Maximum of 5 basic memberships and 1 premium membership.

• Maximum of 3 basic memberships and 3 premium memberships.

A company produces two types of gadgets: Type A and Type B. Each Type A gadget requires 3 hours of labor and each Type B gadget requires 2 hours. The company has a maximum of 60 hours of labor available. Write a system of inequalities to represent the production limits and determine how many of each type can be produced if they want to maximize production.

• 15 Type A gadgets and 5 Type B gadgets can be produced to maximize production.

• 30 Type A gadgets and 0 Type B gadgets can be produced to maximize production.

• 10 Type A gadgets and 20 Type B gadgets can be produced to maximize production.

• 20 Type B gadgets and 0 Type A gadgets can be produced to maximize production.

A restaurant has a seating capacity of 150 people. They serve two types of meals: vegetarian and non-vegetarian. Each vegetarian meal requires 1 seat and each non-vegetarian meal requires 2 seats. Write a system of inequalities to represent the seating arrangement and find the maximum number of each type of meal that can be served if the restaurant is at full capacity.

• Maximum vegetarian meals: 100, Maximum non-vegetarian meals: 50

• Maximum vegetarian meals: 150, Maximum non-vegetarian meals: 75

• Maximum vegetarian meals: 200, Maximum non-vegetarian meals: 100

• Maximum vegetarian meals: 75, Maximum non-vegetarian meals: 30

A local theater has 200 seats available for a show. Tickets for adults cost $15 and tickets for children cost $10. If the theater wants to make at least $1500 from ticket sales, write a system of inequalities to represent the ticket sales and determine the minimum number of adult and child tickets that need to be sold.

• At least 60 adult tickets and 140 child tickets need to be sold.

• 80 adult tickets and 120 child tickets

• 100 adult tickets and 100 child tickets

• 50 adult tickets and 150 child tickets

A clothing store sells shirts for $25 and pants for $40. The store has a budget of $600 for inventory. Write a system of inequalities to represent the maximum number of shirts and pants that can be purchased, and solve for the number of each type of clothing if the budget is fully utilized.

• 24 shirts and 0 pants

• 30 shirts and 2 pants

• 15 shirts and 10 pants

• 20 shirts and 5 pants

A charity event is planning to sell two types of tickets: VIP tickets for $100 and regular tickets for $50. If they want to raise at least $2000, write a system of inequalities to represent the ticket sales and determine the minimum number of each type of ticket that must be sold.

• At least 10 VIP tickets and 10 regular tickets must be sold.

• At least 20 VIP tickets (0 regular) or 0 VIP tickets and 40 regular tickets must be sold.

• At least 15 VIP tickets and 5 regular tickets must be sold.

• At least 30 regular tickets (0 VIP) must be sold.

A bakery makes two types of cakes: chocolate and vanilla. Each chocolate cake requires 3 eggs and each vanilla cake requires 2 eggs. If the bakery has 30 eggs available, write a system of inequalities to represent the cake production and find the maximum number of each type of cake that can be made.

• Maximum number of cakes: 15 (all vanilla) or 10 (all chocolate) or combinations thereof.

• Maximum number of cakes: 12 (6 chocolate and 6 vanilla)

• Maximum number of cakes: 5 (all vanilla)

• Maximum number of cakes: 20 (all chocolate)

A tech company is developing two products: a smartphone and a tablet. Each smartphone requires 4 hours of development and each tablet requires 3 hours. If the team has a total of 60 hours available, write a system of inequalities to represent the development time and determine how many of each product can be developed if they want to maximize their output.

• 20 smartphones and 0 tablets

• Maximum output is 15 smartphones or 12 tablets.

• 10 smartphones and 5 tablets

• 5 smartphones and 15 tablets