Mastering Inequalities: Feasible Regions & Substitution Skills

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 perimeter of the pen. Identify the feasible region for the dimensions of the pen.

• x + y < 100, x ≥ 0, y ≥ 0

• x + y = 50, x ≤ 100, y ≤ 100

• x + y ≤ 50, x ≥ 0, y ≥ 0

• 2x + 2y ≤ 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 (s) that can attend the trip. Use substitution to find the maximum number of students if 10 students are already signed up.

• 12

• 16

• 14

• 18

A bakery sells cookies for $2 each and muffins 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 (c) and muffins (m) sold. Identify the feasible region for the sales.

• c + m ≥ 50

• 2c + 3m ≤ 100

• 2c + 3m ≥ 100

• 4c + m ≥ 100

A concert hall has a seating capacity of 300. If tickets for adults are $15 and for children are $10, write an inequality to represent the total number of adult (a) and child (c) tickets that can be sold without exceeding the capacity. Use substitution to find the maximum number of adult tickets if 50 child tickets are sold.

• 250

• 200

• 300

• 150

A gym has a promotion where a membership costs $30 per month and personal training sessions cost $50 each. If a customer has a budget of $300 for the month, write an inequality to represent the number of training sessions (t) they can afford along with the membership. Identify the feasible region for their choices.

• t <= 5

• t <= 10

• t <= 7

• t <= 3

A local charity is organizing a fundraiser and needs to raise at least $2000. If they charge $25 per ticket for adults and $15 for children, write an inequality to represent the total number of adult (a) and child (c) tickets that need to be sold. Use substitution to find the number of adult tickets if 40 child tickets are sold.

• 40

• 56

• 30

• 72

A restaurant has a special where they offer a combo meal for $10. If they want to sell at least 50 combo meals to break even, write an inequality to represent the number of combo meals (m) they need to sell. Identify the feasible region for their sales strategy.

• m <= 50

• m < 50

• m = 50

• m >= 50

A clothing store has a sale where they offer a discount of 20% on all items. If the store wants to make at least $4000 in sales, write an inequality to represent the relationship between the number of items sold (x) and their price (p). Use substitution to find the number of items sold if the average price is $50.

• 150

• 120

• 80

• 100

A pet store has a limit of 200 animals they can house. If they have 50 dogs and each dog takes up space equivalent to 2 cats, write an inequality to represent the number of cats (c) they can have. Identify the feasible region for the number of cats they can adopt.

• c >= 150

• c <= 50

• c < 100

• c <= 100

A community center is planning a workshop and has a budget of $600. If they spend $100 on materials and $50 per participant, write an inequality to represent the maximum number of participants (p) they can accommodate. Use substitution to find the maximum number of participants if 5 participants are already registered.

• 10

• 5

• 15

• 20