The problem asks to graph the solution of the following system of inequalities: $y \ge -3x$ $y \ge 2x + 1$

1. Problem Description

The problem asks to graph the solution of the following system of inequalities:

$y \ge -3x$

$y \ge 2x + 1$

2. Solution Steps

First, we graph the line $y = -3x$. This line passes through the origin (0, 0). When $x = 1$, $y = -3$. So, another point on the line is (1, -3). Since the inequality is $y \ge -3x$, we shade the region above the line $y = -3x$.

Next, we graph the line $y = 2x + 1$. When $x = 0$, $y = 1$. So, the line passes through (0, 1). When $x = 1$, $y = 2(1) + 1 = 3$. So, another point on the line is (1, 3). Since the inequality is $y \ge 2x + 1$, we shade the region above the line $y = 2x + 1$.

The solution to the system of inequalities is the region where the shaded regions of both inequalities overlap.

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

$-3x = 2x + 1$

$-5x = 1$

$x = -\frac{1}{5}$

Then, $y = -3(-\frac{1}{5}) = \frac{3}{5}$

The point of intersection is $(-\frac{1}{5}, \frac{3}{5})$.

The graph should show two solid lines: $y = -3x$ and $y = 2x + 1$. The region above both lines is the solution.

3. Final Answer

The solution is the region above both lines $y=-3x$ and $y=2x+1$, including the lines themselves. The intersection point of these lines is $(-\frac{1}{5}, \frac{3}{5})$.