https://file.aiquickdraw.com/mathbot/file/17270937279599213hg54.jpg

The graph in the image shows a number line with points marked at -8, -2, and a line segment connecting these points. This suggests the inequality involves absolute value.

The given options are:

1. \(|x + 5| < 3\)
2. \(|x + 5| \leq 3\)
3. \(|x + 5| > 3\)
4. \(|x + 5| \geq 3\)

### Analyzing the Inequality

To determine the correct inequality, let's consider each option and what it represents on a number line:

1. \(|x + 5| < 3\):
   - This means the distance from \(-5\) is less than \(3\).
   - The solution would be: \(-8 < x + 5 < -2\).
   - Subtract \(5\) from all parts: \(-13 < x < -7\).
   - This doesn't match the segment between \(-8\) and \(-2\).

2. \(|x + 5| \leq 3\):
   - This means the distance from \(-5\) is less than or equal to \(3\).
   - The solution would be: \(-8 \leq x + 5 \leq -2\).
   - Subtract \(5\) from all parts: \(-13 \leq x \leq -7\).
   - This also doesn't match the segment between \(-8\) and \(-2\).

3. \(|x + 5| > 3\):
   - This means the distance from \(-5\) is more than \(3\).
   - The solution would be: \(x + 5 > -2\) or \(x + 5 < -8\).
   - Subtract \(5\) from all parts: \(x > -7\) or \(x < -13\).
   - This doesn't match the segment between \(-8\) and \(-2\).

4. \(|x + 5| \geq 3\):
   - This means the distance from \(-5\) is more than or equal to \(3\).
   - The solution would be: \(x + 5 \geq -2\) or \(x + 5 \leq -8\).
   - Subtract \(5\) from all parts: \(x \geq -7\) or \(x \leq -13\).
   - This also doesn't match the segment between \(-8\) and \(-2\).

### Correct Inequality

The inequality that matches the given graph is:

\[ |x + 5| \leq 3 \]

**Explanation**:
- The graph shows a closed interval \([-8, -2]\).
- This represents all \(x\) values such that the distance from \(-5\) is less than or equal to \(3\).
- The correct inequality is \(|x + 5| \leq 3\).

Would you like further details or have any questions?

### Relative Questions:
1. What is the general solution for the inequality \(|x + a| \leq b\)?
2. How do you graph \(|x - a| \leq b\) on a number line?
3. Can you explain why \(|x + 5| \leq 3\) and \(|x + 5| < 3\) have different solution sets?
4. How would the graph change if the inequality were \(|x + 5| > 3\)?
5. What are the steps to solve absolute value inequalities?

### Tip:
When solving absolute value inequalities, remember to consider both the positive and negative scenarios to find the complete solution set.

Which inequality is represented by the graph shown?

Explain why your choice is correct, compared to the other choices.

Learn how to determine which absolute value inequality corresponds to a given graph. This problem covers key algebraic concepts, including absolute value inequalities and their graphical representations. Suitable for students in Grades 7-9.

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