The task here is to identify two inequalities whose intersection matches the shaded region on the graph. Let’s analyze the graph and the options:

1. Parabolic Boundary: The shaded region is bounded by a parabola that opens downward and is centered along the $y$-axis. It intersects the $y$-axis at $y = 3$. This parabola seems to represent the inequality $x^2 + y \leq 3$.

2. Linear Boundary: There's a line passing through the shaded region, approximately around $y = -\frac{1}{2}x$ or equivalently, $x + 2y = 0$. The inequality $x + 2y \leq 0$ appears to represent the boundary of the shaded region on the left side.

Conclusion

The correct pair of inequalities is:

• $x^2 + y \leq 3$ (parabolic boundary from above)
• $x + 2y \leq 0$ (linear boundary from the left)

So, the selected options should be:

• $x^2 + y \leq 3$
• $x + 2y \leq 0$

Would you like further clarification or details on how these inequalities are derived?

Here are five related questions to extend understanding:

1. What is the significance of each boundary in defining the region of intersection?
2. How can we determine if a given point is within the shaded region?
3. How do we derive the equation of a parabola from given points or constraints?
4. How does changing the inequality sign affect the solution region?
5. How can we verify graphically that a solution set belongs to an inequality?

Tip: When working with inequalities, testing a point inside and outside the region can help verify if an inequality correctly describes the shaded area.