Exploring Feasible Regions: Real-Life Linear Inequalities

A farmer has 100 meters of fencing to create a rectangular pen for sheep. If the length of the pen is represented by x and the width by y, write the inequalities that represent the constraints on the dimensions of the pen. Graph the inequalities and identify the feasible region.

• The inequalities representing the constraints are: x + y ≤ 50, x ≥ 0, y ≥ 0.

• x + y = 100, 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 the system of inequalities that represents the number of students that can attend the trip. Graph the inequalities and identify the feasible region.

• 0 ≤ x ≤ 5

• 0 ≤ x ≤ 10

• 0 ≤ x ≤ 14

• 0 ≤ x ≤ 20

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. The company has a maximum of 12 hours of labor available. Write the inequalities for the production limits and graph them to find the feasible region.

• 2x + 3y ≥ 12, x ≥ 0, y ≥ 0

• x + y ≤ 12, x ≥ 0, y ≥ 0

• 2x + 3y ≤ 12, x ≥ 0, y ≥ 0

• 2x + y ≤ 12, 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 30 eggs available, write the system of inequalities and graph them to identify the feasible region for the number of cakes that can be made.

• 2x + 3y ≤ 30, x ≥ 0, y ≥ 0

• 3x + 2y ≤ 30, x ≥ 0, y ≥ 0

• 4x + y ≤ 30, x ≥ 0, y ≥ 0

• 3x + 3y ≤ 30, x ≥ 0, y ≥ 0

A gym has a maximum capacity of 50 members. If each membership costs $30 per month and the gym wants to earn at least $1200 in a month, write the inequalities that represent the situation. Graph the inequalities and identify the feasible region for the number of members.

• x >= 35 and x <= 45

• x >= 30 and x <= 50

• x >= 45 and x <= 55

• The inequalities are x >= 40 and x <= 50.

A local theater has 200 seats and charges $10 for adult tickets and $5 for children's tickets. If the theater wants to earn at least $1000 from ticket sales, write the system of inequalities and graph them to find the feasible region for the number of adult and children's tickets sold.

• x + y >= 200 and 10x + 5y <= 1000

• The system of inequalities is: x + y <= 200 and 10x + 5y >= 1000.

• x + y <= 150 and 10x + 5y >= 800

• x + y <= 200 and 10x + 5y <= 1000

A delivery service can carry a maximum of 100 kg of packages. If package A weighs 5 kg and package B weighs 10 kg, write the inequalities that represent the weight limits for the packages. Graph the inequalities and identify the feasible region for the number of each type of package that can be delivered.

• 5x + 10y >= 100, x >= 0, y >= 0

• 5x + 10y <= 100, x >= 0, y >= 0

• 5x + 10y = 100, x < 0, y < 0

• 5x + 10y <= 50, x >= 0, y >= 0

A clothing store sells shirts for $15 and pants for $25. The store wants to make at least $600 in sales. Write the system of inequalities that represents the sales goals and graph them to identify the feasible region for the number of shirts and pants sold.

• 15x + 25y >= 600, x >= 0, y >= 0

• 15x + 25y <= 600, x >= 0, y >= 0

• 15x + 25y = 600, x >= 0, y >= 0

• 15x + 25y >= 600, x < 0, y >= 0

A charity event has a goal of raising at least $2000. If each ticket sold is $50 and each donation is $100, write the inequalities that represent the fundraising goals. Graph the inequalities and identify the feasible region for the number of tickets sold and donations received.

• 50x + 100y >= 1000; x >= 0; y >= 0

• 50x + 100y = 2000; x >= 0; y >= 0

• 50x + 100y <= 2000; x >= 0; y >= 0

• 50x + 100y >= 2000; x >= 0; y >= 0

A tech company can produce a maximum of 300 devices. Each device A takes 2 hours to produce and each device B takes 1 hour. If the company has 400 hours available for production, write the system of inequalities and graph them to find the feasible region for the number of devices A and B that can be produced.

• x + y <= 300, 2x + y <= 500, x >= 0, y >= 0

• x + y <= 400, 2x + y <= 300, x >= 0, y >= 0

• The system of inequalities is: x + y <= 300, 2x + y <= 400, x >= 0, y >= 0.

• x + y >= 300, 2x + y >= 400, x <= 0, y <= 0