Solving Inequalities: Feasible Regions & Variable Values

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 an inequality that represents the maximum area of the pen. Identify the feasible region for the dimensions of the pen.

• x + y = 50, x ≥ 0, y ≥ 0

• 2x + 2y ≤ 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 an inequality to represent the maximum number of students that can attend the trip. Solve for the number of students.

• 14

• 20

• 10

• 12

A bakery sells cookies for $2 each and brownies for $3 each. If the bakery wants to make at least $100 in one day, write an inequality to represent the relationship between the number of cookies (x) and brownies (y) sold. Identify the feasible region for the sales.

• x + y >= 50

• 2x + 3y >= 100

• 2x + 3y <= 100

• 2x + y >= 100

A gym has a membership limit of 200 members. If x represents the number of adult members and y represents the number of youth members, write an inequality to represent the maximum number of members allowed. Solve for the possible values of x and y.

• x + y = 200; 0 ≤ x ≤ 100, 0 ≤ y ≤ 100

• x + y ≤ 200; 0 ≤ x ≤ 200, 0 ≤ y ≤ 200 - x

• x + y < 200; 0 ≤ x ≤ 150, 0 ≤ y ≤ 50

• x + y ≥ 200; 0 ≤ x ≤ 250, 0 ≤ y ≤ 250

A concert venue has a seating capacity of 300. If tickets for the front row cost $50 each and tickets for the back row cost $30 each, write an inequality to represent the total revenue needed to cover costs of $10,000. Identify the feasible region for ticket sales.

• 50x + 30y = 10000, x + y = 300

• 50x + 30y >= 5000, x + y <= 200

• 50x + 30y <= 10000, x + y >= 300

• 50x + 30y >= 10000, x + y <= 300

A company produces two types of gadgets: Type A and Type B. Each Type A gadget requires 2 hours of labor and each Type B gadget requires 3 hours. If the company has 60 hours of labor available, write an inequality to represent the production limits. Solve for the possible combinations of Type A and Type B gadgets.

• x + y ≤ 20

• The possible combinations of Type A and Type B gadgets are given by the inequality 2x + 3y ≤ 60, where x ≥ 0 and y ≥ 0.

• 2x + 3y < 60

• 4x + 2y ≤ 60

A charity event aims to raise at least $1,000. If each ticket sold for the event costs $25 and each donation is $50, write an inequality to represent the relationship between tickets sold (x) and donations (y). Identify the feasible region for fundraising.

• 25x + 50y <= 1000

• 25x + 50y >= 1000

• 25x + 50y = 1000

• 50x + 25y >= 1000

A restaurant has a special offer where a meal costs $15 and a drink costs $5. If the restaurant wants to make at least $200 in one evening, write an inequality to represent the number of meals (x) and drinks (y) sold. Solve for the possible values of x and y.

• 20x + 10y ≥ 200; y ≤ 10 - x

• 15x + 10y ≥ 200; y ≥ 30 - 2x

• 10x + 2y ≥ 200; y ≤ 20 - 2x

• 15x + 5y ≥ 200; y ≥ 40 - 3x

A local park has a limit of 150 visitors at any given time. If x represents adults and y represents children, write an inequality to represent the maximum number of visitors allowed. Identify the feasible region for the park's capacity.

• x + y ≤ 150

• x + y < 150

• x + y ≥ 150

• x + y = 150

A clothing store has a sale where shirts are $20 and pants are $30. If the store wants to make at least $500 in sales, write an inequality to represent the relationship between the number of shirts (x) and pants (y) sold. Solve for the possible combinations of shirts and pants.

• 25x + 20y = 500

• 10x + 40y ≤ 500

• 15x + 25y ≥ 500

• 20x + 30y ≥ 500