The problem asks to find the solution set of the following system of inequalities and graph it. $y \ge -2x$ $y \ge 3x+2$

1. Problem Description

The problem asks to find the solution set of the following system of inequalities and graph it.

$y \ge -2x$

$y \ge 3x+2$

2. Solution Steps

First, we graph the line $y = -2x$. This is a line with slope $-2$ and $y$-intercept $0$. Since the inequality is $y \ge -2x$, we shade the region above the line.

The line is included in the solution, so we draw a solid line.

Next, we graph the line $y = 3x + 2$. This is a line with slope $3$ and $y$-intercept $2$. Since the inequality is $y \ge 3x + 2$, we shade the region above the line.

The line is included in the solution, so we draw a solid line.

The solution to the system is the region where the shadings of both inequalities overlap. This is the region that is above both lines $y = -2x$ and $y = 3x + 2$.

To find the intersection point, we set the two equations equal to each other:

$-2x = 3x + 2$

$-5x = 2$

$x = -\frac{2}{5}$

Then, $y = -2(-\frac{2}{5}) = \frac{4}{5}$

So the intersection point is $(-\frac{2}{5}, \frac{4}{5})$.

3. Final Answer

The solution is the region where $y \ge -2x$ and $y \ge 3x + 2$. This is the area above both lines, and the lines are included in the solution set. The two lines intersect at the point $(-\frac{2}{5}, \frac{4}{5})$.