https://file.aiquickdraw.com/mathbot/file/1726756975751bs3k5hv7.png

The graph shows a number line with the shaded region between -6 and 5, inclusive of the endpoints. To express this graph as an absolute value inequality, we need to define an inequality that describes the distance of the variable \( x \) from the midpoint of the interval.

### Steps:

1. The endpoints are at \( -6 \) and \( 5 \), and the midpoint between them is:
   \[
   \text{Midpoint} = \frac{-6 + 5}{2} = -\frac{1}{2}
   \]

2. The distance from the midpoint \( -\frac{1}{2} \) to either endpoint (which is the same for both sides) is:
   \[
   \text{Distance} = 5 - \left( -\frac{1}{2} \right) = 5 + \frac{1}{2} = \frac{11}{2}
   \]

3. The absolute value inequality for this situation describes all values of \( x \) that are within \( \frac{11}{2} \) units of \( -\frac{1}{2} \):
   \[
   |x + \frac{1}{2}| \leq \frac{11}{2}
   \]

### Final Answer:
The absolute value inequality for the graph is:
\[
|x + \frac{1}{2}| \leq \frac{11}{2}
\]

This inequality represents all values of \( x \) between \( -6 \) and \( 5 \).

Do you want more details or have any questions?

Here are 5 related questions to think about:
1. How would the inequality change if the endpoints were not included?
2. Can you express this inequality as a compound inequality instead of using absolute value?
3. What happens to the inequality if you shift the graph by 2 units to the right?
4. How would you solve this inequality algebraically to get the interval notation?
5. Can this inequality be applied to real-world scenarios involving distances?

**Tip:** When working with absolute value inequalities, remember that the inequality symbol (less than or greater than) tells you whether you're dealing with a range of values within a specific distance (for \( \leq \)) or beyond a certain distance (for \( \geq \)).

Write an absolute value inequality for the graph below. Use x for your variable.

Learn how to express a graph of numbers between -6 and 5 as an absolute value inequality. This example covers key algebraic concepts, including the midpoint formula and the absolute value inequality theorem. Suitable for students in Grades 8-10.

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