Inequalities in Action: Interpreting Constraints & Regions

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. What is the feasible region for the dimensions?

• x + y ≤ 75, x ≥ 0, y ≥ 0

• x + y = 100, x ≥ 0, y ≥ 0

• x + y ≥ 100, x ≤ 0, y ≤ 0

• x + y ≤ 50, x ≥ 0, y ≥ 0; feasible region is in the first quadrant bounded by x + y = 50.

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 inequalities that represent the number of students that can attend the trip. Identify the feasible region for the number of students.

• x <= 12

• x <= 14, where x is the number of students.

• x <= 20

• x <= 10

A bakery sells two types of cakes: chocolate and vanilla. Each chocolate cake requires 2 hours of labor and each vanilla cake requires 1 hour. If the bakery has 10 hours of labor available, write the inequalities that represent the maximum number of cakes that can be made. What is the feasible region for the number of cakes?

• 2x + y ≤ 10, x ≥ 0, y ≥ 0

• 3x + y ≤ 10, x ≥ 0, y ≥ 0

• x + 2y ≤ 10, x ≥ 0, y ≥ 0

• 2x + y ≥ 10, x ≥ 0, y ≥ 0

A gym offers two types of memberships: basic and premium. The basic membership costs $30 per month and the premium costs $50. If a customer wants to spend no more than $200 a month, write the inequality that represents the number of each type of membership they can purchase. Identify the feasible region.

• 30x + 50y ≤ 200, x < 0, y ≥ 0

• 30x + 50y < 200, x ≥ 0, y ≥ 0

• 30x + 50y ≤ 200, x ≥ 0, y ≥ 0

• 30x + 50y ≤ 250, x ≥ 0, y ≥ 0

A company produces two products, A and B. Each product A requires 3 hours of labor and each product B requires 2 hours. If the company has 24 hours of labor available, write the inequalities that represent the production limits. What is the feasible region for the number of products produced?

• 2x + 3y ≤ 24, x ≥ 0, y ≥ 0

• 4x + 2y ≤ 24, x ≥ 0, y ≥ 0

• 3x + 2y ≤ 24, x ≥ 0, y ≥ 0

• 3x + y ≤ 24, x ≥ 0, y ≥ 0

A concert venue has a maximum capacity of 500 people. If tickets for adults cost $20 and tickets for children cost $10, and the venue wants to make at least $8000 from ticket sales, write the inequalities that represent the ticket sales. Identify the feasible region for the number of adult and child tickets sold.

• x + y ≤ 500, 20x + 10y ≥ 8000

• x + y = 500, 20x + 10y = 8000

• x + y ≥ 500, 20x + 10y ≤ 8000

• x + y ≤ 400, 20x + 10y ≥ 10000

A restaurant has a limited supply of 200 pounds of chicken and 150 pounds of beef. If each chicken dish requires 1 pound of chicken and each beef dish requires 2 pounds of beef, write the inequalities that represent the maximum number of dishes that can be prepared. What is the feasible region for the dishes?

• x <= 200, y <= 100

• x <= 250, y <= 50

• x <= 200, y <= 75; Feasible region: (0 <= x <= 200, 0 <= y <= 75)

• x <= 150, y <= 100

A clothing store sells shirts and pants. Each shirt costs $15 and each pair of pants costs $25. If a customer has $200 to spend, write the inequality that represents the maximum number of shirts and pants they can buy. Identify the feasible region for their purchases.

• 15x + 25y ≤ 200, x ≥ 0, y ≥ 0

• 20x + 25y ≤ 200, x ≥ 0, y ≥ 0

• 15x + 30y ≤ 200, x ≥ 0, y ≥ 0

• 10x + 20y ≤ 200, x ≥ 0, y ≥ 0

A charity event is selling tickets for a dinner. Each ticket costs $50, and they want to raise at least $3000. Write the inequality that represents the number of tickets they need to sell. What is the feasible region for ticket sales?

• t > 70

• t < 60

• t <= 50

• t >= 60

A student is saving money for a new laptop that costs $800. If they save $50 a week from their allowance and $20 a week from doing chores, write the inequality that represents the number of weeks it will take to save enough money. Identify the feasible region for their savings.

• w >= 12

• w > 12

• w <= 12

• w < 12