The problem asks us to graph the solution to the following system of inequalities: $y \ge 4x - 5$ $y \le \frac{2}{5}x$

1. Problem Description

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

$y \ge 4x - 5$

$y \le \frac{2}{5}x$

2. Solution Steps

First, we consider the inequality $y \ge 4x - 5$. To graph the line $y = 4x - 5$, we can find two points on the line. When $x = 0$, $y = 4(0) - 5 = -5$, so $(0, -5)$ is a point. When $y = 0$, $0 = 4x - 5$, so $4x = 5$ and $x = \frac{5}{4} = 1.25$. Thus, $(1.25, 0)$ is another point. Since the inequality is $y \ge 4x - 5$, the line is solid, and the region above the line is shaded.

Next, we consider the inequality $y \le \frac{2}{5}x$. To graph the line $y = \frac{2}{5}x$, we can find two points on the line. When $x = 0$, $y = \frac{2}{5}(0) = 0$, so $(0, 0)$ is a point. When $x = 5$, $y = \frac{2}{5}(5) = 2$, so $(5, 2)$ is another point. Since the inequality is $y \le \frac{2}{5}x$, the line is solid, and the region below the line is shaded.

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

3. Final Answer

The graph of the solution is the region where $y \ge 4x - 5$ and $y \le \frac{2}{5}x$. This area is bounded by the solid lines $y = 4x - 5$ and $y = \frac{2}{5}x$.

The line $y=4x-5$ passes through $(0, -5)$ and $(1.25, 0)$. The line $y=\frac{2}{5}x$ passes through $(0, 0)$ and $(5, 2)$.

The solution includes the region above the line $y = 4x - 5$ and below the line $y = \frac{2}{5}x$.