The problem requires us to find the four inequalities that define the unshaded region R on the XOY plane in the given image.

1. Problem Description

The problem requires us to find the four inequalities that define the unshaded region R on the XOY plane in the given image.

2. Solution Steps

We can identify four lines that bound the region R. Two are vertical lines, one is a horizontal line, and one is a slanted line.

* Vertical Lines:

The vertical lines are at $x = -2$ and $x = 4$. Since the region R is between these lines, the inequalities are $x > -2$ and $x < 4$, which can be written as $-2 < x < 4$.

* Horizontal Line:

The horizontal line is at $y = 2$. Since the region R is above this line, the inequality is $y > 2$.

* Slanted Line:

The slanted line passes through the points $(0, 6)$ and $(8, 0)$. We can find the equation of this line using the slope-intercept form, $y = mx + b$, where $m$ is the slope and $b$ is the y-intercept.

The y-intercept is $b = 6$.

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

Thus, the equation of the line is $y = -\frac{3}{4}x + 6$.

Since the region R is below this line, the inequality is $y < -\frac{3}{4}x + 6$.

Therefore, the four inequalities that define the unshaded region R are:

$x > -2$

$x < 4$

$y > 2$

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

3. Final Answer

The four inequalities that define the unshaded region R are:

$x > -2$

$x < 4$

$y > 2$

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