To define the inequalities that bound the shaded region $R$, let's analyze the three lines shown in the graph.

Step 1: Determine the Equations of the Lines

1. Vertical Line at $x = -1$: This line runs vertically through $x = -1$, so its equation is: $x = -1$ Since the shaded region $R$ is to the right of this line, the inequality is: $x \geq -1$

2. Horizontal Line at $y = 1$: This line runs horizontally through $y = 1$, so its equation is: $y = 1$ The shaded region is below this line, so the inequality is: $y \leq 1$

3. Sloped Line Passing Through Points: The third line is slanted and appears to pass through points $(-3, -2)$ and $(1, 4)$. To find its equation, let's calculate the slope $m$ and then use the point-slope form of a line equation.

   • Slope $m$: $m = \frac{4 - (-2)}{1 - (-3)} = \frac{4 + 2}{1 + 3} = \frac{6}{4} = \frac{3}{2}$
   • Equation of the Line: Using the point-slope form $y - y_1 = m(x - x_1)$ with point $(1, 4)$, $y - 4 = \frac{3}{2}(x - 1)$ Expanding and simplifying, $y - 4 = \frac{3}{2}x - \frac{3}{2}$ $y = \frac{3}{2}x + \frac{5}{2}$

   Since the shaded region is below this line, the inequality is: $y \leq \frac{3}{2}x + \frac{5}{2}$

Final Inequalities Defining Region $R$

Combining these, the inequalities that define the shaded region $R$ are: