Mastering Linear Inequalities: Writing & Graphing 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 a system of inequalities to represent the constraints on the dimensions of the pen. Graph the inequalities.

• x + y < 50, x > 0, y > 0

• x + y ≥ 100, x ≤ 0, y ≤ 0

• x + y = 50, x < 0, y < 0

• x + y ≤ 50, 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. Graph the inequalities.

• x ≤ 10, x ≥ -1

• x ≤ 20, x ≥ 0

• x ≤ 25, x ≥ 5

• x ≤ 14, x ≥ 0

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 a system of inequalities to represent the number of each type of cake that can be made. Graph the inequalities.

• 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 per month. If a customer wants to 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.

• 30x + 50y ≤ 150, x ≥ 0, y ≥ 0

• 20x + 40y ≤ 200, x ≥ 0, y ≥ 0

• 30x + 50y ≤ 250, x ≥ 0, y ≥ 0

• 30x + 50y ≤ 200, 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 a maximum of 24 hours of labor available, write a system of inequalities to represent the production limits. Graph the inequalities.

• 2x + 3y ≤ 24, x ≥ 0, y ≥ 0

• 3x + 2y ≤ 24, x ≤ 0, y ≥ 0

• 3x + 2y ≤ 30, x ≥ 0, y ≥ 0

• 3x + 2y ≤ 24, x ≥ 0, y ≥ 0

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 venue wants to make at least $10,000 in ticket sales, write a system of inequalities to represent the ticket sales. Graph the inequalities.

• x + y ≥ 500, 20x + 30y ≤ 10000

• x + y ≤ 500, 20x + 30y ≥ 10000

• x + y ≤ 400, 20x + 30y ≥ 12000

• x + y = 500, 20x + 30y = 10000

A charity organization is organizing a fundraiser and has a goal of raising at least $1,000. If they sell tickets for $15 each and receive donations of $50 each, write a system of inequalities to represent the number of tickets sold and donations received. Graph the inequalities.

• 15x + 50y >= 1000, x >= 0, y >= 0

• 10x + 30y >= 1000, x >= 0, y >= 0

• 20x + 40y >= 1000, x >= 0, y >= 0

• 15x + 50y <= 1000, x >= 0, y >= 0

A restaurant offers two types of meals: vegetarian and non-vegetarian. Each vegetarian meal costs $10 and each non-vegetarian meal costs $15. If the restaurant wants to make at least $300 in sales, write a system of inequalities to represent the number of each type of meal that can be sold. Graph the inequalities.

• 5x + 10y >= 300, x >= 0, y >= 0

• 10x + 15y >= 300, x >= 0, y >= 0

• 10x + 15y <= 300, x >= 0, y >= 0

• 10x + 20y >= 300, x >= 0, y >= 0

A clothing store has a sale on shirts and pants. Each shirt costs $25 and each pair of pants costs $40. If a customer wants to spend no more than $200, write a system of inequalities to represent the number of shirts and pants they can buy. Graph the inequalities.

• 30x + 50y ≤ 200, x ≥ 0, y ≥ 0

• 20x + 30y ≤ 200, x ≥ 1, y ≥ 1

• The system of inequalities is: 25x + 40y ≤ 200, x ≥ 0, y ≥ 0.

• 25x + 40y < 200, x > 0, y > 0

A tech company produces two types of gadgets: A and B. Each gadget A requires 4 hours of assembly and each gadget B requires 2 hours. If the company has a maximum of 32 hours available for assembly, write a system of inequalities to represent the production limits. Graph the inequalities.

• 4x + 2y ≤ 32, x ≥ 0, y ≥ 0

• 4x + 3y ≤ 32, x ≥ 0, y ≥ 0

• 2x + 4y ≤ 32, x ≥ 0, y ≥ 0

• 3x + 4y ≤ 32, x ≥ 0, y ≥ 0