To determine which inequality best matches the shaded region in the graph, let's analyze the given options in relation to the boundary lines in the plot.

Step 1: Understanding the Boundaries

The graph shows two boundary lines:

1. Line 1: $y = 2x + 2$
2. Line 2: $y = x + 2$

The shaded area is above Line 1 and below Line 2, indicating that these lines form the limits of the region where the inequalities apply.

Step 2: Analyzing the Shaded Region

From observing the graph:

• The region is above the line $y = 2x + 2$, suggesting the inequality $y > 2x + 2$.
• The region is below the line $y = x + 2$, indicating the inequality $y < x + 2$.

Step 3: Verifying the Options

Let's compare these findings with the provided answer choices:

• Option A: $y > 2x + 2$ and $y > x + 2$ → Incorrect, as it requires the area to be above both lines.
• Option B: $y < 2x + 2$ and $y > x + 2$ → Incorrect, as it includes regions below $y = 2x + 2$.
• Option C: $y > 2x + 2$ and $y < x + 2$ → Correct, as this matches the shaded area (above $2x + 2$ and below $x + 2$).
• Option D: $y < 2x + 2$ and $y < x + 2$ → Incorrect, as it represents the area below both lines.

Conclusion

The correct answer is Option C: $y > 2x + 2$ and $y < x + 2$.

Would you like further details or have any questions about this solution?

Here are some related questions you might be interested in:

1. How do you determine which regions of the graph correspond to specific inequalities?
2. What techniques can you use to graph linear inequalities manually?
3. How do you interpret dashed vs. solid lines in graphing inequalities?
4. What are the possible applications of solving systems of inequalities in real life?
5. How do you check the solutions of a system of inequalities?

Tip: Always test a point in the shaded region (like the origin if it's not on a boundary) to verify which inequalities are satisfied by that region.