Graphing and Analyzing Systems of Inequalities in Real Life

A farmer has a rectangular field that is 100 meters long and 80 meters wide. He wants to plant two types of crops: corn and wheat. The corn requires at least 2 square meters per plant, and the wheat requires at least 3 square meters per plant. If he can plant a maximum of 40 corn plants and 30 wheat plants, graph the system of inequalities and identify the feasible region.

• x + y <= 300

• x <= 50

• y <= 25

• The feasible region is defined by the inequalities: x + y <= 400, x <= 40, y <= 30, x >= 0, y >= 0.

A school is organizing a fundraiser and has a goal of raising at least $500. They sell cookies for $5 each and brownies for $3 each. Write the inequalities representing the number of cookies (x) and brownies (y) they need to sell to meet their goal. Graph the system and analyze the intersection points.

• 5x + 4y <= 500

• 3x + 5y >= 500

• 5x + 3y >= 500

• 4x + 2y >= 500

A gym offers two types of memberships: a basic membership for $30 per month and a premium membership for $50 per month. If a customer can spend no more than $120 per month on memberships, graph the inequalities representing the number of basic (x) and premium (y) memberships they can purchase. What are the intersection points?

• The intersection points are (0, 4) and (6, 0)

• The intersection points are (2, 2) and (5, 0)

• The intersection points are (0, 2.4) and (4, 0).

• The intersection points are (1, 3) and (3, 1)

A local bakery sells two types of cakes: chocolate and vanilla. Each chocolate cake requires 2 hours to bake, and each vanilla cake requires 1 hour. If the bakery has a maximum of 10 hours available for baking, graph the system of inequalities and determine the feasible combinations of cakes they can bake.

• (6,0)

• Feasible combinations of cakes: (0,0), (0,10), (1,8), (2,6), (3,4), (4,2), (5,0)

• (2,5)

• (1,9)

A clothing store has a sale on shirts and pants. Each shirt costs $20 and each pair of pants costs $30. If a customer has a budget of $150, write the inequalities for the number of shirts (x) and pants (y) they can buy. Graph the inequalities and find the intersection points.

• 25x + 35y ≤ 150, x ≥ 0, y ≥ 0

• The inequalities are: 20x + 30y ≤ 150, x ≥ 0, y ≥ 0.

• 20x + 30y ≥ 150, x ≤ 0, y ≤ 0

• 10x + 15y ≤ 150, x ≥ 0, y ≥ 0

A charity event is selling tickets for $15 each and VIP tickets for $25 each. They want to raise at least $1,000. Write the inequalities for the number of regular tickets (x) and VIP tickets (y) sold. Graph the system and analyze the feasible region.

• 15x + 30y >= 1000

• x + y <= 100

• 10x + 20y >= 1000

• 15x + 25y >= 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 a day, graph the inequalities and find the intersection points that represent the number of meals sold.

• The intersection points are (0, 20) and (30, 0).

• (20, 10)

• (10, 15)

• (25, 5)

A tech company produces two types of gadgets: A and B. Each gadget A requires 3 hours of labor and gadget B requires 2 hours. If the company has 24 hours of labor available, graph the system of inequalities and determine the maximum number of each gadget they can produce.

• 6 gadgets A and 6 gadgets B

• 8 gadgets A and 4 gadgets B

• 12 gadgets A and 0 gadgets B

• 0 gadgets A and 12 gadgets B

A concert venue has a seating capacity of 500. They sell regular tickets for $20 and VIP tickets for $50. If they want to make at least $10,000 from ticket sales, write the inequalities for the number of regular (x) and VIP (y) tickets sold. Graph the system and analyze the intersection points.

• x + y ≥ 500; 20x + 50y ≤ 10,000

• x + y ≤ 400; 20x + 50y ≥ 12,000

• x + y = 500; 20x + 50y = 10,000

• x + y ≤ 500; 20x + 50y ≥ 10,000

A farmer has a limited amount of land and wants to plant two crops: tomatoes and peppers. Each tomato plant requires 4 square feet, and each pepper plant requires 2 square feet. If he has 200 square feet available, graph the system of inequalities and find the feasible region for planting.

• y <= 0

• x <= 0

• The feasible region is defined by the inequalities 4x + 2y <= 200, x >= 0, and y >= 0.

• 4x + 2y >= 200