The problem asks us to find the three inequalities that define the unshaded region R in the given graph. The region is bounded by three lines. The coordinates of the intersection points are given as (1,1), (7,7), and (7,3).

1. Problem Description

The problem asks us to find the three inequalities that define the unshaded region R in the given graph. The region is bounded by three lines. The coordinates of the intersection points are given as (1,1), (7,7), and (7,3).

2. Solution Steps

First, let's find the equation of the line passing through (1,1) and (7,7). The slope of this line is:

$m_1 = \frac{7-1}{7-1} = \frac{6}{6} = 1$

The equation of the line is given by $y - y_1 = m_1(x - x_1)$. Using the point (1,1), we get:

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

$y - 1 = x - 1$

$y = x$

Since the unshaded region is above this line, the inequality is $y \ge x$.

Second, let's find the equation of the line passing through (1,1) and (7,3). The slope of this line is:

$m_2 = \frac{3-1}{7-1} = \frac{2}{6} = \frac{1}{3}$

The equation of the line is given by $y - y_1 = m_2(x - x_1)$. Using the point (1,1), we get:

$y - 1 = \frac{1}{3}(x - 1)$

$3(y - 1) = x - 1$

$3y - 3 = x - 1$

$3y = x + 2$

$y = \frac{1}{3}x + \frac{2}{3}$

Since the unshaded region is below this line, the inequality is $y \le \frac{1}{3}x + \frac{2}{3}$, which is equivalent to $3y \le x + 2$.

Third, the vertical line passes through (7,7) and (7,3), so the equation of this line is $x = 7$. Since the unshaded region is to the left of this line, the inequality is $x \le 7$.

3. Final Answer

The three inequalities that define the unshaded region R are:

$y \ge x$

$3y \le x + 2$

$x \le 7$