1. **State the problem:** We need to find the graph that best represents the solution set for the system of inequalities: $$y \leq -\frac{2}{3}x - 2$$ $$y \geq 2x + 2$$ 2. **Understand the inequalities:** - The first inequality, $y \leq -\frac{2}{3}x - 2$, means the solution includes all points on or below the line with slope $-\frac{2}{3}$ and y-intercept $-2$. - The second inequality, $y \geq 2x + 2$, means the solution includes all points on or above the line with slope $2$ and y-intercept $2$. 3. **Find the intersection point of the two lines:** Set the right sides equal to find where the lines cross: $$-\frac{2}{3}x - 2 = 2x + 2$$ Multiply both sides by 3 to clear the fraction: $$-2x - 6 = 6x + 6$$ Bring all terms to one side: $$-2x - 6 - 6x - 6 = 0$$ $$-8x - 12 = 0$$ Solve for $x$: $$-8x = 12$$ $$x = -\frac{12}{8} = -\frac{3}{2} = -1.5$$ Find $y$ by substituting $x = -1.5$ into one of the lines, e.g., $y = 2x + 2$: $$y = 2(-1.5) + 2 = -3 + 2 = -1$$ So the lines intersect at $(-1.5, -1)$. 4. **Analyze the solution region:** - For $y \leq -\frac{2}{3}x - 2$, the shaded region is below the line. - For $y \geq 2x + 2$, the shaded region is above the line. The solution set is the intersection of these two regions. 5. **Check the graphs:** - Graph a shows the correct shading: below the first line and above the second line, with an overlapping region. - Graphs b, c, and d have incorrect shading. **Final answer:** The graph that best represents the solution set is **Graph a**.