The problem asks us to graph and shade the solution region for the following system of inequalities: $4x - 5y \ge 20$ $y < \frac{3}{2}x - 2$

1. Problem Description

The problem asks us to graph and shade the solution region for the following system of inequalities:

$4x - 5y \ge 20$

$y < \frac{3}{2}x - 2$

2. Solution Steps

First, let's consider the first inequality: $4x - 5y \ge 20$.

To graph this, we first treat it as an equation: $4x - 5y = 20$.

We can find the x-intercept by setting $y=0$: $4x = 20$, so $x = 5$.

We can find the y-intercept by setting $x=0$: $-5y = 20$, so $y = -4$.

Thus, the line passes through $(5, 0)$ and $(0, -4)$.

Since the inequality is $\ge$, we draw a solid line.

To determine which side to shade, we test the point $(0, 0)$: $4(0) - 5(0) \ge 20$ simplifies to $0 \ge 20$, which is false. Therefore, we shade the region that does *not* include $(0, 0)$.

Now, let's consider the second inequality: $y < \frac{3}{2}x - 2$.

To graph this, we first treat it as an equation: $y = \frac{3}{2}x - 2$.

The y-intercept is $-2$. The slope is $\frac{3}{2}$, so for every 2 units we move to the right, we move 3 units up.

Since the inequality is $<$, we draw a dashed line.

To determine which side to shade, we test the point $(0, 0)$: $0 < \frac{3}{2}(0) - 2$ simplifies to $0 < -2$, which is false. Therefore, we shade the region that does *not* include $(0, 0)$.

The solution region is the intersection of the shaded regions for both inequalities.

3. Final Answer

The solution is the graph with a solid line through (5,0) and (0,-4) shaded to the side that does not contain the origin and a dashed line for $y = \frac{3}{2}x - 2$ with a y intercept of -2 shaded above the line. The solution is the overlapping shaded region.