Question

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

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

Explanation:

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

The inequality \( y \geq \frac{2}{3}x + 1 \) means we shade above the line \( y=\frac{2}{3}x + 1 \) (since the inequality is "greater than or equal to"). The line has a slope of \( \frac{2}{3} \) and a y - intercept of 1.

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

First, rewrite \( 5x + 6y \leq - 30 \) in slope - intercept form (\( y=mx + b \)):

$$
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$$

The inequality \( y\leq-\frac{5}{6}x - 5 \) means we shade below the line \( y =-\frac{5}{6}x - 5 \) (since the inequality is "less than or equal to"). The slope of this line is \( -\frac{5}{6} \) and the y - intercept is - 5.

Step3: Find the overlapping 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 (and analyzing the two inequalities), the region that satisfies both inequalities is Region 3.

Answer:

C. Region 3