https://file.aiquickdraw.com/mathbot/file/1726758894418tmph1y6l.jpg

We are asked to identify the correct graph representing the solution set of the inequality:

\[
y \leq -\frac{2}{3}x - 1
\]

### Step-by-step Analysis:

1. **Equation of the boundary line:**
   The boundary line corresponds to the equation:
   \[
   y = -\frac{2}{3}x - 1
   \]
   This is a straight line with:
   - Slope: \(-\frac{2}{3}\)
   - y-intercept: \(-1\) (i.e., the line crosses the y-axis at \(y = -1\)).

2. **Graphing the boundary line:**
   The slope of \(-\frac{2}{3}\) means that for every increase of 3 units in \(x\), \(y\) decreases by 2 units. This gives us the shape and tilt of the line.

3. **Solid or dashed line:**
   The inequality is \(y \leq -\frac{2}{3}x - 1\), which includes equality (i.e., \(y\) can be exactly on the line). Therefore, the boundary line should be solid, not dashed.

4. **Shading region:**
   Since the inequality is \(y \leq -\frac{2}{3}x - 1\), the region below the line should be shaded. This means we are looking for the area where the \(y\)-coordinates are less than or equal to the line's \(y\)-coordinates.

### Inspection of the graphs:

- **Graph W:** The line is solid, and the region above the line is shaded.
- **Graph X:** The line is solid, and the region below the line is shaded.
- **Graph Y:** The line is dashed, and the region below the line is shaded.
- **Graph Z:** The line is dashed, and the region above the line is shaded.

### Conclusion:
Graph **X** is the correct representation of the inequality \(y \leq -\frac{2}{3}x - 1\). The line is solid, and the region below the line is shaded.

Would you like more details on any part of this solution? 

Here are 5 related questions to expand on this topic:
1. How do you determine which side of the boundary line to shade in an inequality?
2. What changes in the graph if the inequality is \(y > -\frac{2}{3}x - 1\)?
3. How do you graph inequalities involving vertical or horizontal lines?
4. How can you use a test point to verify the shading for an inequality?
5. How does changing the slope affect the graph of a linear inequality?

**Tip:** Always check whether the inequality includes equality (e.g., \(\leq\) or \(\geq\)) to decide whether the boundary line should be solid or dashed.

Which of the following graphs represents the solution set of the inequality y ≤ -2/3x - 1?

Learn how to solve and graph the linear inequality y ≤ -2/3x - 1 by identifying the correct graph representation. This problem involves key concepts of linear inequalities, graphing lines, and understanding slope-intercept form. Suitable for students in Grades 9-10.

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