The problem is to solve the following system of inequalities: $4x - 5y \ge 20$ $y < \frac{3}{2}x - 2$

1. Problem Description

The problem is to solve the following system of inequalities:

$4x - 5y \ge 20$

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

2. Solution Steps

To solve a system of inequalities, we need to graph each inequality on the same coordinate plane and find the region where the solutions overlap.

Step 1: Graph the first inequality, $4x - 5y \ge 20$.

First, treat the inequality as an equation: $4x - 5y = 20$.

To find the intercepts, set $x=0$ and solve for $y$: $4(0) - 5y = 20$, which gives $-5y = 20$, so $y = -4$. The y-intercept is $(0, -4)$.

Set $y=0$ and solve for $x$: $4x - 5(0) = 20$, which gives $4x = 20$, so $x = 5$. The x-intercept is $(5, 0)$.

Plot these two points $(0, -4)$ and $(5, 0)$ and draw a straight line through them. Since the inequality is $\ge$, the line will be solid.

Next, test a point to determine which side of the line to shade. A common point to test is $(0,0)$. Plug $(0,0)$ into the inequality: $4(0) - 5(0) \ge 20$, which simplifies to $0 \ge 20$. This is false, so we shade the region that does not contain $(0,0)$. In other words, shade below and to the right of the line.

Step 2: Graph the second inequality, $y < \frac{3}{2}x - 2$.

Treat the inequality as an equation: $y = \frac{3}{2}x - 2$. This is in slope-intercept form ($y = mx + b$), where $m = \frac{3}{2}$ is the slope and $b = -2$ is the y-intercept. The y-intercept is $(0, -2)$.

Since the inequality is $<$, the line will be dashed (or dotted).

From the y-intercept $(0, -2)$, use the slope $\frac{3}{2}$ to find another point. Go up 3 units and right 2 units, which gives the point $(2, 1)$.

Draw a dashed line through the points $(0, -2)$ and $(2, 1)$.

Now, test a point to determine which side of the line to shade. Again, use $(0,0)$. Plug $(0,0)$ into the inequality: $0 < \frac{3}{2}(0) - 2$, which simplifies to $0 < -2$. This is false, so we shade the region that does not contain $(0,0)$. In other words, shade below the line.

Step 3: Find the overlapping region.

The solution to the system of inequalities is the region where the shaded areas from both inequalities overlap. Any point in this region is a solution to the system.

3. Final Answer

The solution to the system of inequalities is the region where the graphs of $4x - 5y \ge 20$ and $y < \frac{3}{2}x - 2$ overlap. The graph of $4x - 5y \ge 20$ is a solid line with x-intercept (5,0) and y-intercept (0,-4), shaded below and to the right. The graph of $y < \frac{3}{2}x - 2$ is a dashed line with y-intercept (0,-2) and slope $\frac{3}{2}$, shaded below. The overlapping region is the solution.