Question

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

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

Explanation:

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

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

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

Rewrite it in slope - intercept form: \( 6y\leq - 5x-30\Rightarrow y\leq-\frac{5}{6}x - 5 \). 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 \( y=\frac{1}{3}x + 1 \) and below \( y =-\frac{5}{6}x - 5 \). By analyzing the graph (the line with positive slope is \( y=\frac{1}{3}x + 1 \) and the line with negative slope is \( y=-\frac{5}{6}x - 5 \)), the region that satisfies both inequalities is Region 4.

Answer:

D. Region 4