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The problem involves solving a system of inequalities and identifying the correct graph that represents the solution set.

The system of inequalities is:
\[
\begin{cases}
y > -2x - 2 \\
y \leq x + 4
\end{cases}
\]

### Step-by-Step Solution:

1. **Graphing the first inequality**: 
   - \( y > -2x - 2 \)
   - The boundary line is \( y = -2x - 2 \). Since the inequality is strict (">"), the line should be dashed, and the region above the line should be shaded.
   - The line has a slope of \(-2\) and a y-intercept of \(-2\).

2. **Graphing the second inequality**: 
   - \( y \leq x + 4 \)
   - The boundary line is \( y = x + 4 \). Since the inequality is inclusive ("≤"), the line should be solid, and the region below the line should be shaded.
   - The line has a slope of \(1\) and a y-intercept of \(4\).

3. **Solution Region**:
   - The solution is the overlapping shaded region that satisfies both inequalities. This region is bounded by the dashed line of the first inequality and the solid line of the second inequality.

### Analyzing the Graphs:

- **Top left graph**: The shading is below the line \( y = -2x - 2 \), which is incorrect since the inequality requires shading **above**.
- **Top right graph**: The shading is on the correct side for \( y = -2x - 2 \) (above the dashed line), but the region for \( y \leq x + 4 \) (the solid line) is shaded incorrectly (above instead of below).
- **Bottom left graph**: The shading correctly represents both inequalities. The region is shaded **above** the dashed line \( y > -2x - 2 \) and **below** the solid line \( y \leq x + 4 \).
- **Bottom right graph**: The shaded region is too restricted, only capturing a small triangular area, which doesn't represent the entire solution set.

### Correct Answer:
The **bottom left graph** correctly represents the solution set for the given system of inequalities.

Would you like further clarification or additional details?

Here are five related questions to deepen your understanding:
1. How would the graph change if the inequality \( y > -2x - 2 \) were replaced with \( y \geq -2x - 2 \)?
2. What effect does changing the slope of the first inequality have on the solution set?
3. How do dashed lines and solid lines represent different types of inequalities in graphing?
4. What happens to the solution region if the signs of both inequalities are reversed?
5. Can you describe a real-world scenario where a system of inequalities like this might be used?

**Tip**: Always pay attention to the direction of the inequality symbol (">", "<", "≥", "≤") when shading regions on a graph.

Which graph represents the solution set of the system of inequalities? y > -2x - 2, y ≤ x + 4

Learn how to solve and graph the system of inequalities y > -2x - 2 and y ≤ x + 4. This guide provides a detailed step-by-step approach to graphing linear inequalities, understanding boundary lines, and shading the correct solution region. Suitable for Grades 9-12.

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