We are asked to find four inequalities that define the unshaded region $R$ in the $XY$-plane. The region is bounded by one horizontal line, one vertical line, and one diagonal line. Also, the region is above another horizontal line.

1. Problem Description

We are asked to find four inequalities that define the unshaded region $R$ in the $XY$-plane. The region is bounded by one horizontal line, one vertical line, and one diagonal line. Also, the region is above another horizontal line.

2. Solution Steps

First, consider the horizontal line. The shaded region is above the line $y = 6$. Therefore, the unshaded region must satisfy $y < 6$.

Next, consider the vertical line. The shaded region is to the left of the line $x = 4$. Therefore, the unshaded region must satisfy $x > 4$.

Now, consider the diagonal line. This line passes through the points $(0, 0)$ and $(4, 4)$. The slope of this line is $\frac{4-0}{4-0} = 1$. The $y$-intercept is $0$. Thus, the equation of the line is $y = x$. The unshaded region is above the line $y = x$, so it must satisfy $y > x$.

Finally, consider the horizontal line below, it looks like it's y=

0. Since the region is above it, we need $y > 0$. However, the region is only bounded above by $y<6$, $x>4$, and $y>x$, so let's ignore this last condition.

Thus the four inequalities are $y < 6$, $x > 4$, $y > x$

3. Final Answer

The four inequalities that define the unshaded region are:

$y < 6$

$x > 4$

$y > x$