The problem asks us to find four inequalities that define the unshaded region $R$ in the $xy$-plane shown in the image.

1. Problem Description

The problem asks us to find four inequalities that define the unshaded region $R$ in the $xy$-plane shown in the image.

2. Solution Steps

We need to identify the equations of the lines that bound the unshaded region and determine the appropriate inequality sign for each line.

* Vertical Line 1:

The vertical line on the left passes through $x = 1$. Since the unshaded region is to the right of this line, the inequality is $x > 1$. However, the line is solid so it means that the inequality is $x \ge 1$.

* Vertical Line 2:

The vertical line on the right passes through $x = 8$. Since the unshaded region is to the left of this line, the inequality is $x < 8$. However, the line is solid so it means that the inequality is $x \le 8$.

* Horizontal Line:

The horizontal line passes through $y = 7$. Since the unshaded region is below this line, the inequality is $y < 7$. However, the line is solid so it means that the inequality is $y \le 7$.

* Diagonal Line:

The diagonal line passes through the points $(0,0)$ and $(4,4)$. The slope is $\frac{4-0}{4-0} = 1$. The equation of the line is $y = x$. Since the unshaded region is above this line, the inequality is $y > x$. However, the line is solid so it means that the inequality is $y \ge x$.

3. Final Answer

The four inequalities that define the unshaded region R are:

$x \ge 1$

$x \le 8$

$y \le 7$

$y \ge x$