Graphing and Interpreting Linear Inequalities in Real Life

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 and identify the feasible region.

• x + y < 100

• 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 and interpret the solution.

• x ≤ 14, where x is the number of students.

• x ≤ 10

• x ≤ 20

• x ≤ 12

A factory produces two types of toys: type A and type B. Each type A toy requires 2 hours of labor and each type B toy requires 3 hours. If the factory has 60 hours of labor available, write a system of inequalities to represent the production limits. Graph the inequalities and describe the feasible production combinations.

• 3x + 2y ≤ 60, x ≥ 0, y ≤ 0

• x + y ≤ 20, x ≥ 0, y ≥ 0

• The system of inequalities is: 2x + 3y ≤ 60, x ≥ 0, y ≥ 0.

• 2x + 3y ≥ 60, x ≤ 0, y ≤ 0

A restaurant offers two types of meals: vegetarian and non-vegetarian. The vegetarian meal costs $10 and the non-vegetarian meal costs $15. If the restaurant wants to make at least $300 in a day, write a system of inequalities to represent the number of each type of meal they need to sell. Graph the inequalities and interpret the solution.

• 5x + 10y ≥ 300, x ≥ 0, y ≥ 0

• The system of inequalities is: 10x + 15y ≥ 300, x ≥ 0, y ≥ 0.

• 10x + 15y ≤ 300, x ≥ 0, y ≥ 0

• 10x + 15y = 300, x ≥ 0, y ≥ 0

A clothing store sells shirts and pants. Each shirt costs $25 and each pair of pants costs $40. The store wants to make at least $1000 in sales. Write a system of inequalities to represent the sales goals. Graph the inequalities and identify the combinations of shirts and pants that meet the goal.

• The store needs to sell exactly 40 shirts and 25 pairs of pants to meet the sales goal.

• The combinations of (x, y) that satisfy the inequalities can be found by graphing the line 25x + 40y = 1000 and identifying the region above this line in the first quadrant.

• The combinations of (x, y) that satisfy the inequalities can be found by graphing the line 25x + 40y = 500 and identifying the region below this line.

• The combinations of (x, y) that satisfy the inequalities can be found by graphing the line 25x + 40y = 1500 and identifying the region to the left of this line.

A charity event is selling tickets for a concert. Each ticket costs $50, and they want to raise at least $2000. Write a system of inequalities to represent the number of tickets they need to sell. Graph the inequalities and interpret the solution regarding ticket sales.

• x <= 40, x >= 0

• x >= 50, x >= 0

• x >= 40, x <= 0

• x >= 40, x >= 0

A gym has a maximum capacity of 150 members. If they currently have x members enrolled in yoga classes and y members in pilates classes, write a system of inequalities to represent the membership limits. Graph the inequalities and describe the feasible membership combinations.

• The system of inequalities is: x + y ≤ 150, x ≥ 0, y ≥ 0.

• x + y ≥ 150, x ≤ 0, y ≤ 0

• x + y < 150, x > 0, y > 0

• x + y = 150, x < 0, y < 0

A local bakery sells two types of cakes: chocolate and vanilla. Each chocolate cake requires 3 eggs and each vanilla cake requires 2 eggs. If the bakery has 30 eggs available, write a system of inequalities to represent the cake production limits. Graph the inequalities and interpret the solution regarding cake production.

• The system of inequalities is: 3x + 2y ≤ 30, x ≥ 0, y ≥ 0.

• x + y ≤ 15, x ≥ 0, y ≥ 0

• 2x + 3y ≤ 30, x ≥ 0, y ≥ 0

• 3x + 2y ≥ 30, x ≥ 0, y ≥ 0

A tech company produces laptops and tablets. Each laptop requires 4 hours of assembly and each tablet requires 2 hours. If the company has 40 hours of assembly time available, write a system of inequalities to represent the production limits. Graph the inequalities and describe the feasible production combinations.

• 4x + 3y ≤ 40, x ≥ 0, y ≥ 0

• 4x + 2y ≥ 40, x ≥ 0, y ≥ 0

• The system of inequalities is: 4x + 2y ≤ 40, x ≥ 0, y ≥ 0.

• 3x + 2y ≤ 40, x ≥ 0, y ≥ 0

A concert venue has a seating capacity of 500. If x represents the number of VIP tickets sold and y represents the number of general admission tickets sold, write a system of inequalities to represent the seating constraints. Graph the inequalities and interpret the solution regarding ticket sales.

• The system of inequalities is: x + y ≤ 500, x ≥ 0, y ≥ 0.

• x + y = 500, x ≤ 0, y ≤ 0

• x + y < 500, x > 0, y > 0

• x + y ≥ 500, x < 0, y < 0