Mastering Graphs: Analyzing Linear Inequalities in Context

A farmer has a rectangular field. The length of the field is at least 10 meters longer than its width. If the total area of the field must not exceed 600 square meters, graph the system of inequalities that represents this situation. What are the intersection points of the inequalities?

• The intersection points of the inequalities are (20, 30) and (0, 10).

• (15, 25) and (5, 5)

• (25, 25) and (0, 0)

• (30, 20) and (10, 0)

A school is planning to organize a sports event. They can accommodate at most 200 students and need at least 50 volunteers. If each student requires 2 square meters and each volunteer requires 1 square meter, graph the system of inequalities and find the intersection points that satisfy both conditions.

• (100, 100)

• (250, 50)

• (150, 30)

• The intersection points that satisfy both conditions are (200, 50) and (100, 200).

A company produces two types of gadgets, A and B. Each gadget A requires 3 hours of labor and each gadget B requires 2 hours. The company has a maximum of 30 hours of labor available. Additionally, they want to produce at least 5 gadgets of type A. Graph the inequalities and analyze the intersection points.

• The intersection points are (5, 7.5) and (10, 0).

• The intersection points are (0, 15) and (5, 10).

• The intersection points are (6, 5) and (12, 0).

• The intersection points are (3, 12) and (8, 2).

A restaurant offers two types of meals: vegetarian and non-vegetarian. They want to serve at least 40 meals but can serve no more than 100 meals in total. If each vegetarian meal costs $5 and each non-vegetarian meal costs $8, graph the system of inequalities and determine the feasible region's intersection points.

• The intersection points of the feasible region are (0, 40), (40, 0), (0, 100), and (100, 0).

• (20, 20)

• (50, 50)

• (0, 80)

A local gym has a maximum capacity of 150 members. They want to ensure that at least 30% of the members are enrolled in yoga classes. If each yoga class can accommodate 20 members, graph the inequalities and find the intersection points that meet these requirements.

• Only 2 yoga classes are needed.

• At least 3 yoga classes are needed.

• At least 5 yoga classes are needed.

• No yoga classes are required.

A charity organization is collecting donations. They need at least $500 to fund their project but cannot accept more than $2000 in total donations. If each individual can donate between $10 and $100, graph the system of inequalities and analyze the intersection points.

• The intersection points are (300, 70) and (1500, 30).

• The intersection points are (400, 60) and (1800, 25).

• The intersection points are (500, 50) and (2000, 20).

• The intersection points are (600, 40) and (2500, 10).

A factory produces two products, X and Y. Each product X requires 4 hours of machine time and each product Y requires 2 hours. The factory has a total of 40 hours available. Additionally, they want to produce at least 5 units of product Y. Graph the inequalities and find the intersection points.

• Intersection point is (5, 10).

• Intersection point is (8, 4).

• Intersection point is (6, 6).

• Intersection point is (7.5, 5).

A bookstore sells fiction and non-fiction books. They want to have at least 100 books in stock but can hold no more than 300 books. If each fiction book takes up 1 shelf space and each non-fiction book takes up 2 shelf spaces, graph the system of inequalities and analyze the intersection points.

• The feasible region is a line segment defined by the points (100, 50) and (200, 100).

• The intersection points are (100, 0), (0, 150), and (200, 0). The feasible region is a polygon defined by these points.

• The intersection points are (0, 100), (100, 100), and (300, 0).

• The intersection points are (50, 25), (150, 0), and (0, 200).

A tech company is developing two software applications. Each application requires a different amount of resources. Application A requires 3 units of resource 1 and 2 units of resource 2, while Application B requires 1 unit of resource 1 and 4 units of resource 2. If the company has a maximum of 12 units of resource 1 and 16 units of resource 2, graph the inequalities and find the intersection points.

• (3,2)

• (1,5)

• (5,1)

• The intersection points are (0,0), (0,4), (4,0), and (2,3).

A clothing store has a sale on shirts and pants. They want to sell at least 50 items but no more than 200 items in total. If each shirt costs $20 and each pair of pants costs $30, graph the system of inequalities and analyze the intersection points that satisfy these conditions.

• (0, 10)

• (15, 0)

• (5, 5)

• The intersection points are (0, 1.67), (2.5, 0), (0, 6.67), and (10, 0). The valid combinations of shirts and pants must lie within these points.