1. **State the problem:** We need to find and shade the region that satisfies the system of inequalities: $$y \geq x + 3$$ $$y < -x - 2$$ 2. **Understand the inequalities:** - The first inequality $y \geq x + 3$ means the region on or above the line $y = x + 3$. - The second inequality $y < -x - 2$ means the region strictly below the line $y = -x - 2$. 3. **Graph the boundary lines:** - For $y = x + 3$, the line passes through points $(0,3)$ and $(3,6)$. - For $y = -x - 2$, the line passes through points $(0,-2)$ and $(-2,0)$. 4. **Determine the shading:** - Shade the region above or on the line $y = x + 3$ (including the line). - Shade the region below the line $y = -x - 2$ (not including the line). 5. **Find the intersection points:** Solve for $x$ and $y$ where $y = x + 3$ and $y = -x - 2$: $$x + 3 = -x - 2$$ $$x + x = -2 - 3$$ $$2x = -5$$ $$x = \frac{-5}{2}$$ Substitute back to find $y$: $$y = x + 3 = \frac{-5}{2} + 3 = \frac{-5}{2} + \frac{6}{2} = \frac{1}{2}$$ So the lines intersect at $\left(\frac{-5}{2}, \frac{1}{2}\right)$. 6. **Shade the correct region:** The solution region is the set of points that satisfy both inequalities simultaneously, which is the area above or on $y = x + 3$ and below $y = -x - 2$. This region lies between the two lines and includes the boundary line $y = x + 3$ but excludes the line $y = -x - 2$. **Final answer:** The shaded region is the intersection of $y \geq x + 3$ and $y < -x - 2$.