Inequalities in Action: Writing and Solving Systems

A farmer has 100 acres of land 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 the farmer has a total of 240 hours of labor available, write a system of inequalities to represent the situation and find the possible combinations of corn and wheat that can be planted.

• x + y >= 100, 2x + 3y >= 240, x <= 0, y <= 0

• x + y <= 50, 2x + 3y <= 120, x >= 0, y >= 0

• The system of inequalities is: x + y <= 100, 2x + 3y <= 240, x >= 0, y >= 0.

• x + y <= 100, 2x + 3y <= 300, x >= 0, y >= 0

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 determine the feasible number of students that can go.

• 10

• 18

• 12

• 14

A bakery sells 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, write a system of inequalities to represent the number of each type of cake that can be made, and find the possible combinations of cakes that can be baked.

• Possible combinations of cakes: (0, 15), (1, 13), (2, 12), (3, 10), (4, 9), (5, 7), (6, 6), (7, 4), (8, 3), (9, 1), (10, 0).

• (5, 10)

• (0, 20)

• (10, 5)

A gym offers two types of memberships: basic and premium. The basic membership costs $30 per month, while the premium membership costs $50 per month. If the gym wants to earn at least $1,200 in a month, write a system of inequalities to represent the number of each type of membership sold, and determine the combinations that meet the goal.

• The system of inequalities is: 30x + 50y >= 1200, x >= 0, y >= 0.

• 30x + 50y = 1200, x >= 0, y >= 0

• 30x + 50y >= 1000, x >= 0, y >= 0

• 30x + 50y <= 1200, x >= 0, y >= 0

A clothing store sells shirts and pants. Each shirt costs $15 and each pair of pants costs $25. The store wants to make at least $1,000 in sales. Write a system of inequalities to represent the sales goal, and find the combinations of shirts and pants that can achieve this target.

• x >= 100, y >= 50

• x + y <= 1000

• The combinations of shirts (x) and pants (y) that satisfy the inequality are given by y >= (1000 - 15x) / 25, where x >= 0 and y >= 0.

• y <= (1000 - 25x) / 15

A concert venue has a seating capacity of 500. Tickets for the front row cost $100 each, and tickets for the back row cost $50 each. If the venue wants to make at least $30,000 from ticket sales, write a system of inequalities to represent the ticket sales, and determine the possible combinations of front and back row tickets that can be sold.

• x + y ≤ 500, 100x + 50y ≥ 30000

• x + y = 500, 100x + 50y = 30000

• x + y ≤ 400, 100x + 50y ≥ 25000

• x + y ≥ 500, 100x + 50y ≤ 30000

A charity event is selling two types of tickets: regular and VIP. Regular tickets cost $25, and VIP tickets cost $50. If the charity wants to raise at least $2,000, write a system of inequalities to represent the ticket sales, and find the combinations of regular and VIP tickets that can be sold to meet the goal.

• x + y >= 80, with x >= 0 and y >= 0

• The combinations of regular and VIP tickets that can be sold to meet the goal are given by the inequality 25x + 50y >= 2000, with x >= 0 and y >= 0.

• 25x + 50y = 2000, with x > 0 and y > 0

• 25x + 50y <= 2000, with x >= 0 and y >= 0

A restaurant offers two types of meals: vegetarian and non-vegetarian. Each vegetarian meal costs $10 and each non-vegetarian meal costs $15. If the restaurant wants to earn at least $500 in one day, write a system of inequalities to represent the sales, and determine the combinations of meals that can achieve this target.

• 10x + 15y = 500, x >= 0, y >= 0

• The system of inequalities is: 10x + 15y >= 500, x >= 0, y >= 0.

• 5x + 10y >= 500, x >= 0, y >= 0

• 10x + 15y <= 500, x >= 0, y >= 0

A local gym has a maximum capacity of 200 members. Each membership costs $40 per month. If the gym wants to earn at least $6,000 in monthly revenue, write a system of inequalities to represent the membership sales, and find the combinations of memberships that can meet this goal.

• 150 <= x <= 200

• x < 150

• 200 < x <= 250

• 100 <= x <= 150

A farmer has a total of 150 animals consisting of cows and chickens. Each cow has 4 legs and each chicken has 2 legs. If the total number of legs is 500, write a system of inequalities to represent the number of cows and chickens, and determine the possible combinations of animals that satisfy these conditions.

• 60 cows and 90 chickens

• 100 cows and 50 chickens

• 120 cows and 30 chickens

• 80 cows and 70 chickens