Question

1. which region represents the solution set to the following system of inequalities:

a. region 1 \\( y \geq \frac{2}{3}x + 1 \\)
b. region 2 \\( 5x + 6y \leq - 30 \\)
c. region 3
d. region 4

Explanation:

Step1: Analyze \( y \geq \frac{2}{3}x + 1 \)

The line \( y=\frac{2}{3}x + 1 \) has a slope of \( \frac{2}{3} \) and y - intercept 1. The inequality \( y\geq\frac{2}{3}x + 1 \) means we shade above this line.

Step2: Analyze \( 5x + 6y \leq - 30 \)

Rewrite it as \( y\leq-\frac{5}{6}x - 5 \). The line \( y =-\frac{5}{6}x-5 \) has a slope of \( -\frac{5}{6} \) and y - intercept - 5. The inequality \( y\leq-\frac{5}{6}x - 5 \) means we shade below this line.

Step3: Find the intersection region

We need to find the region that is above \( y=\frac{2}{3}x + 1 \) and below \( y =-\frac{5}{6}x-5 \). By looking at the graph, Region 3 is the region that satisfies both inequalities (above the first line and below the second line).

Answer:

C. Region 3