1. **State the problem:** We need to graph the system of linear inequalities: $$y \geq \frac{x}{3} - 1$$ $$y < x + 1$$ 2. **Understand the inequalities:** - The first inequality $y \geq \frac{x}{3} - 1$ means the region on or above the line $y = \frac{x}{3} - 1$. The line is solid because of the \geq (greater than or equal to) sign. - The second inequality $y < x + 1$ means the region strictly below the line $y = x + 1$. The line is dashed because the inequality is strict (<). 3. **Graph each boundary line:** - For $y = \frac{x}{3} - 1$, plot points and draw a solid line. - For $y = x + 1$, plot points and draw a dashed line. 4. **Shade the correct regions:** - Shade above the solid line $y = \frac{x}{3} - 1$. - Shade below the dashed line $y = x + 1$. 5. **Find the overlapping region:** - The solution to the system is where the shaded regions overlap. - This is the area between the two lines where $y$ is greater than or equal to $\frac{x}{3} - 1$ and less than $x + 1$. 6. **Summary:** - Solid line for $y \geq \frac{x}{3} - 1$ with shading above. - Dashed line for $y < x + 1$ with shading below. - Overlapping shaded region is the solution. This matches the first graph described, confirming the correct graphing of the system.