Graphing and Solving Systems of Inequalities in Real Life

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 a system of inequalities to represent the constraints on the dimensions of the pen. Graph the inequalities.

• 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 for transportation and $15 for admission. Write a system of inequalities to represent the maximum number of students that can attend the trip. Solve the system and graph the inequalities.

• The maximum number of students that can attend the trip is 14.

• The maximum number of students that can attend the trip is 20.

• The maximum number of students that can attend the trip is 10.

• The maximum number of students that can attend the trip is 5.

A company produces two types of gadgets, A and B. Each gadget A requires 2 hours of labor and each gadget B requires 3 hours of labor. If the company has 12 hours of labor available, write a system of inequalities to represent the production limits. Graph the inequalities and identify feasible production combinations.

• 2x + 3y >= 12, x >= 0, y >= 0

• x + y <= 12, x >= 0, y >= 0

• The system of inequalities is: 2x + 3y <= 12, x >= 0, y >= 0.

• 2x + y <= 12, x >= 0, y >= 0

A local 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 number of each type of membership they can purchase. Graph the inequalities.

• 20x + 40y ≤ 200, x ≥ 0, y ≤ 0

• 30x + 50y = 200, x < 0, y < 0

• 30x + 50y ≥ 200, x ≥ 0, y ≥ 0

• 30x + 50y ≤ 200, x ≥ 0, y ≥ 0

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 24 eggs available, write a system of inequalities to represent the maximum number of cakes that can be made. Solve the system and graph the inequalities.

• The maximum number of cakes that can be made is 12.

• The maximum number of cakes that can be made is 10.

• The maximum number of cakes that can be made is 16.

• The maximum number of cakes that can be made is 8.

A charity event is selling tickets for adults and children. Adult tickets cost $15 and children tickets cost $10. If the total revenue from ticket sales cannot exceed $600, write a system of inequalities to represent the number of adult and child tickets that can be sold. Graph the inequalities and find the feasible region.

• The feasible region is defined by the intersection of the inequalities in the first quadrant of the coordinate plane.

• The feasible region is a single point at the origin of the coordinate plane.

• The feasible region is defined by the area outside the inequalities in the first quadrant of the coordinate plane.

• The feasible region is defined by the intersection of the inequalities in the second quadrant of the coordinate plane.

A concert venue has a seating capacity of 500. If tickets for the concert are sold at $20 for general admission and $30 for VIP seating, and the total revenue cannot exceed $10,000, write a system of inequalities to represent the ticket sales. Graph the inequalities and find the feasible solutions.

• VIP tickets can only be sold if general admission tickets are sold out.

• The maximum number of tickets sold is 300.

• The feasible solutions are the points (x, y) that satisfy the inequalities, which can be graphed to find the region of feasible solutions.

• The total revenue must be exactly $10,000.

A delivery service can handle a maximum of 200 packages per day. If each standard package takes 1 hour to deliver and each express package takes 2 hours, write a system of inequalities to represent the delivery limits. Graph the inequalities and identify the possible combinations of standard and express packages.

• x + y ≤ 200 and x + 2y ≥ 200

• The system of inequalities is: x + y ≤ 200 and x + 2y ≤ 200.

• x + y ≤ 150 and x + 2y ≤ 100

• x + y ≥ 200 and x + 2y ≥ 200

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 team has 20 hours available for development, write a system of inequalities to represent the production limits. Solve the system and graph the inequalities.

• 2x + 4y <= 20, x >= 0, y >= 0

• 4x + 3y <= 20, x >= 0, y >= 0

• 3x + 2y <= 20, x >= 0, y >= 0

• The system of inequalities is: 4x + 2y <= 20, x >= 0, y >= 0.