The problem asks us to find four inequalities that define the unshaded region R in the given XOY plane. The region is bounded by one line with a positive slope and three lines that are either horizontal or vertical.

1. Problem Description

The problem asks us to find four inequalities that define the unshaded region R in the given XOY plane. The region is bounded by one line with a positive slope and three lines that are either horizontal or vertical.

2. Solution Steps

First, we identify the four lines that define the boundaries of the unshaded region.

* Line 1: Vertical line at $x = -1$. The region is to the right of this line, so the inequality is $x > -1$.

* Line 2: Vertical line at $x = 6$. The region is to the left of this line, so the inequality is $x < 6$.

* Line 3: Horizontal line at $y = 6$. The region is below this line, so the inequality is $y < 6$.

* Line 4: A line with a positive slope. This line appears to pass through the points $(0, -1)$ and $(1, 0)$.

The slope, $m$, of the line is calculated as:

$m = \frac{y_2 - y_1}{x_2 - x_1} = \frac{0 - (-1)}{1 - 0} = \frac{1}{1} = 1$

The equation of the line is given by $y = mx + c$, where $c$ is the y-intercept. Since the line passes through $(0, -1)$, $c = -1$.

Therefore, the equation of the line is $y = x - 1$.

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

3. Final Answer

The four inequalities that define the unshaded region R are:

$x > -1$

$x < 6$

$y < 6$

$y > x - 1$