The problem asks to find four inequalities that define the unshaded region $R$ in the given graph.

1. Problem Description

The problem asks to find four inequalities that define the unshaded region $R$ in the given graph.

2. Solution Steps

First, we identify the four lines that bound the region $R$.

Two are vertical lines and one is a horizontal line. One is a slanted line.

From the graph, we can see the two vertical lines are $x=-2$ and $x=4$. Since the unshaded region lies between these lines, we have $-2 \le x \le 4$.

The horizontal line appears to be $y=2$. Since the unshaded region lies above this line, we have $y \ge 2$.

The slanted line intersects the y-axis at $y=6$ and the x-axis at $x=8$. The equation of this line is found as follows:

The slope of the line is $m = \frac{6-0}{0-8} = -\frac{6}{8} = -\frac{3}{4}$.

Using the slope-intercept form $y = mx + b$, we have $y = -\frac{3}{4}x + 6$.

Since the unshaded region lies below this line, we have $y \le -\frac{3}{4}x + 6$.

Therefore, the four inequalities are:

$x \ge -2$

$x \le 4$

$y \ge 2$

$y \le -\frac{3}{4}x + 6$

3. Final Answer

$x \ge -2$

$x \le 4$

$y \ge 2$

$y \le -\frac{3}{4}x + 6$