y>
2
3
​
x−6
y, is less than or equal to, minus, x, minus, 1
y≤−x−1

To analyze and graph the system of inequalities given by:

1. \( y > \frac{2}{3}x - 6 \)
2. \( y \leq -x - 1 \)

### Step-by-Step Analysis:
1. **Identify each inequality**:
   - The first inequality is \( y > \frac{2}{3}x - 6 \). This represents a line with a slope of \(\frac{2}{3}\) and a y-intercept at \(-6\). The inequality is strict (greater than), so the region above the line will be shaded with a dashed boundary (indicating the line itself is not included).
   - The second inequality is \( y \leq -x - 1 \). This represents a line with a slope of \(-1\) and a y-intercept at \(-1\). Since this is a "less than or equal to" inequality, the region below the line is shaded, and the boundary is solid.

2. **Graph the boundary lines**:
   - For \( y = \frac{2}{3}x - 6 \):
     - When \( x = 0 \), \( y = -6 \).
     - When \( y = 0 \), \( x = 9 \).
   - For \( y = -x - 1 \):
     - When \( x = 0 \), \( y = -1 \).
     - When \( y = 0 \), \( x = -1 \).

3. **Determine the solution region**:
   - The solution is the intersection of the regions that satisfy both inequalities. This will be the area above \( y = \frac{2}{3}x - 6 \) and below \( y = -x - 1 \).

Would you like a graph of this system, or a more detailed explanation? 

### Related Questions:
1. How do you determine the intersection point of two lines?
2. What does a dashed line represent in a graph of inequalities?
3. How can you convert a linear inequality to slope-intercept form?
4. What are the possible methods to solve a system of inequalities graphically?
5. How do you verify if a point satisfies a given inequality?

**Tip:** When graphing inequalities, always test a point (like \( (0, 0) \)) to determine which side of the boundary line satisfies the inequality.

The query submitted by the user: y > (2/3)x - 6, y ≤ -x - 1

Learn how to analyze and graph the system of inequalities y > (2/3)x - 6 and y ≤ -x - 1. This step-by-step guide covers identifying inequalities, graphing the boundary lines, and finding the solution region. Suitable for students in Grades 9-10.

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