JOURNAL

Describe how you know which region, if any, represents the solution to a system of linear inequalities.

REMEMBER

The solution of a system of linear inequalities is the intersection of the solutions to each inequality. Every point in the intersection region satisfies all inequalities in the system.

PRACTICE

1. Samuel is remodeling his basement. One part of the planning involves the flooring. He knows that he would like both carpet and hardwood but isn't sure how much of each he will use. The maximum amount of flooring area he can cover is 2000 square feet. The carpet costs [tex]$4.50[/tex] per square foot, and the hardwood costs [tex]$8.25[/tex] per square foot. Both prices include labor costs. Samuel has budgeted $10,000 for the flooring.

- Write a system of inequalities to represent the maximum amount of flooring needed and the maximum amount of money Samuel wants to spend.

- One idea Samuel has is to make two rooms—one having 400 square feet of carpeting and the other having 1200 square feet of hardwood. Determine whether this amount of carpeting and hardwood are solutions to the system of inequalities. Explain your reasoning in terms of the problem situation.

To determine which region represents the solution to a system of linear inequalities, you must analyze the inequalities individually and then find the intersection of their solutions. The solution to a system of linear inequalities is the set of all points that simultaneously satisfy every inequality in the system.

Let's break it down using Samuel's flooring situation:

Define Variables:
- Let [tex]x[/tex] represent the area in square feet of carpet.
- Let [tex]y[/tex] represent the area in square feet of hardwood.
Write the Inequalities:
- The maximum area of flooring: [tex]x + y \leq 2000[/tex].
- The budget constraint: [tex]4.5x + 8.25y \leq 10000[/tex].
Test the Solution (400 square feet of carpet and 1200 square feet of hardwood):
- Substitute [tex]x = 400[/tex] and [tex]y = 1200[/tex] into each inequality:
  - For the area: [tex]400 + 1200 = 1600[/tex]. This satisfies the inequality [tex]1600 \leq 2000[/tex].
  - For the cost: [tex]4.5(400) + 8.25(1200) = 1800 + 9900 = 11700[/tex]. This does not satisfy the inequality [tex]11700 \leq 10000[/tex].

Since the test solution does not satisfy the budget inequality, [tex]x = 400[/tex] and [tex]y = 1200[/tex] are not within the region representing the solution to the system of inequalities.

To visualize which region represents the solution, you would graph these inequalities on a coordinate plane. The overlapping (or intersecting) region of the shaded areas corresponding to each inequality is the solution region. Every point within this overlap satisfies all constraints.