1. The problem asks to identify the two inequalities that form the system graphed. 2. The graph shows a dashed upward-opening parabola with vertex at (1,0) and a solid decreasing line crossing the y-axis at 4. 3. The parabola is dashed, indicating a strict inequality ($>$ or $<$), and the line is solid, indicating an inclusive inequality ($\geq$ or $\leq$). 4. The parabola opens upward and is centered at $x=1$, so it matches $y > 2(x - 1)^2$ or $y \geq (x - 1)^2$. 5. Since the parabola is dashed, the inequality must be strict: $y > 2(x - 1)^2$. 6. The line is decreasing and crosses the y-axis at 4, matching $y \leq \frac{1}{2}x + 4$ or $y \geq \frac{1}{3}x + 4$. 7. The shaded region is above the line, so the inequality must be $y \geq \frac{1}{3}x + 4$ or $y \leq \frac{1}{2}x + 4$. 8. Since the line is solid and the shading is above it, the correct inequality is $y \geq \frac{1}{3}x + 4$. 9. Therefore, the two inequalities making up the system are: $$y > 2(x - 1)^2$$ and $$y \geq \frac{1}{3}x + 4$$ Final answer: $y > 2(x - 1)^2$ and $y \geq \frac{1}{3}x + 4$.