The problem asks us to determine the four inequalities that define the unshaded region $R$ on the $XOY$ plane shown in the image.

1. Problem Description

The problem asks us to determine the four inequalities that define the unshaded region $R$ on the $XOY$ plane shown in the image.

2. Solution Steps

From the image, we can identify four lines that bound the unshaded region.

First, there is a vertical line at $x = 1$. The unshaded region is to the right of this line, so $x > 1$.

Second, there is another vertical line at $x = 7$. The unshaded region is to the left of this line, so $x < 7$. Thus we have $1 < x < 7$ or $x > 1$ and $x < 7$.

Third, there is a horizontal line at $y = 8$. The unshaded region is below this line, so $y < 8$.

Fourth, there is a diagonal line that appears to pass through $(1, 0)$ and $(7, 6)$.

Let's find the equation of the line. The slope $m$ is given by:

$m = \frac{6 - 0}{7 - 1} = \frac{6}{6} = 1$

Using the point-slope form $y - y_1 = m(x - x_1)$ with point $(1, 0)$:

$y - 0 = 1(x - 1)$

$y = x - 1$

The unshaded region is above this line, so $y > x - 1$.

Combining the inequalities, we have:

$x > 1$

$x < 7$

$y < 8$

$y > x - 1$

3. Final Answer

The four inequalities are:

$x > 1$

$x < 7$

$y < 8$

$y > x - 1$