Question

Suppose you are buying two kinds of notebooks for school. A spiral notebook costs $2,

and a three-ring notebook costs $5. You must have at least 6 notebooks. The cost of the
notebooks can be no more than $20.
a. Write a system of inequalities to model the situation.
b. Graph and solve the system.

Answer 1

The situation is modeled with the system of inequalities: x + y ≥ 6 and 2x + 5y ≤ 20. To graph the inequalities, each inequality is turned into an equation and the shading represents the solution set. The solution is found where the shaded regions overlap on the graph.

Step-by-step explanation:

To model the student's notebook purchasing situation with inequalities, let x be the number of spiral notebooks and y be the number of three-ring notebooks. The system of inequalities representing the situation is given by:

• x + y ≥ 6 (At least 6 notebooks in total)
• 2x + 5y ≤ 20 (Total cost no more than $20)

To graph the system, follow these steps:

1. Draw the y-axis and x-axis on a coordinate plane.
2. For each inequality, turn the inequality into an equation (e.g., x + y = 6) and graph the line.
3. Since they are inequalities, shade the region that satisfies each inequality. In this case, the region above the line x + y = 6 and the region below the line 2x + 5y = 20.
4. The feasible region where the two shaded regions overlap is the solution to the system.

Solve the system by finding the area of overlap on the graph. The coordinates of the intersection points represent the possible numbers of each type of notebook that you can buy.

For example, a point (3,3) on the graph means you can buy 3 spiral notebooks and 3 three-ring notebooks, staying within the budget and the required minimum amount.