The problem asks to graph the solution of the following system of inequalities: $y \ge -2x$ $y \ge 4x + 3$

1. Problem Description

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

$y \ge -2x$

$y \ge 4x + 3$

2. Solution Steps

To graph the solution to this system of inequalities, we first consider each inequality separately.

For the first inequality, $y \ge -2x$, we graph the line $y = -2x$. This line passes through the origin $(0,0)$ and has a slope of $-2$. Since the inequality is $y \ge -2x$, we shade the region above the line.

For the second inequality, $y \ge 4x + 3$, we graph the line $y = 4x + 3$. This line has a $y$-intercept of 3 and a slope of

4. Since the inequality is $y \ge 4x + 3$, we shade the region above the line.

The solution to the system is the region where the shaded regions of both inequalities overlap. The lines $y=-2x$ and $y=4x+3$ intersect where $-2x=4x+3$, which implies $6x=-3$, so $x=-\frac{1}{2}$. When $x = -\frac{1}{2}$, $y = -2(-\frac{1}{2}) = 1$. Therefore the two lines intersect at the point $(-\frac{1}{2}, 1)$.

3. Final Answer

The solution to the system of inequalities is the region above both lines $y = -2x$ and $y = 4x + 3$. The intersection point of the two lines is $(-\frac{1}{2}, 1)$.