We are asked to find the four inequalities that define the unshaded region R on the XOY plane. The region is bounded by one line and three straight lines parallel to the axes.

1. Problem Description

We are asked to find the four inequalities that define the unshaded region R on the XOY plane. The region is bounded by one line and three straight lines parallel to the axes.

2. Solution Steps

First, we consider the vertical lines.

One line is $x = -1$. Since the unshaded region is to the right of the line, we have $x > -1$. Because the line is solid, we have $x \ge -1$.

The other vertical line is $x = 5$. Since the unshaded region is to the left of the line, we have $x < 5$. Because the line is solid, we have $x \le 5$.

So, $-1 \le x \le 5$.

Next, we consider the horizontal line.

The horizontal line is $y = 6$. Since the unshaded region is below the line, we have $y < 6$. Because the line is solid, we have $y \le 6$.

Finally, we consider the diagonal line. We need to find the equation of this line.

The line passes through $(0, 0)$ and $(4, 2)$.

The slope of the line is $m = \frac{2 - 0}{4 - 0} = \frac{2}{4} = \frac{1}{2}$.

The equation of the line is $y = \frac{1}{2}x$.

Since the unshaded region is above the line, we have $y > \frac{1}{2}x$. Because the line is solid, we have $y \ge \frac{1}{2}x$.

Therefore, the four inequalities are:

$x \ge -1$

$x \le 5$

$y \le 6$

$y \ge \frac{1}{2}x$

3. Final Answer

The four inequalities are:

$x \ge -1$

$x \le 5$

$y \le 6$

$y \ge \frac{1}{2}x$