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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?

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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.

Select the two inequalities whose intersection is the shaded region on the graph.

Learn how to determine the inequalities that define a shaded region on a graph. This problem explores concepts in algebra and coordinate geometry, using a parabola and a line to represent inequalities. Suitable for high school students in Grades 10-12.

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