1. **Stating the problem:** We need to find the system of inequalities that describes the shaded region in the given graph. 2. **Analyzing the graph:** - The shaded region is bounded by a line passing through points $(0,6)$ and $(6,0)$. - The region is above the line $y=4$. - The region is also in the first quadrant, so $x \geq 0$ and $y \geq 0$. 3. **Finding the equation of the line through $(0,6)$ and $(6,0)$:** The slope $m$ is given by $$m = \frac{0-6}{6-0} = \frac{-6}{6} = -1$$ Using point-slope form with point $(0,6)$: $$y - 6 = -1(x - 0) \implies y = -x + 6$$ 4. **Determining the inequality for the line:** Since the shaded region is above this line, the inequality is $$y \geq -x + 6$$ 5. **Inequality for the horizontal boundary:** The region is above the line $y=4$, so $$y \geq 4$$ 6. **Inequalities for the axes:** Since the region is in the first quadrant, $$x \geq 0, \quad y \geq 0$$ 7. **Final system of inequalities:** $$\begin{cases} y \geq -x + 6 \\ y \geq 4 \\ x \geq 0 \\ y \geq 0 \end{cases}$$ This system describes the shaded region in the graph.