https://file.aiquickdraw.com/mathbot/file/1726019507387pk220p88.jpg

The given image shows a number line with a range marked between \(-12\) and \(-6\), both included. We can represent this range as an absolute value inequality.

To find the absolute value inequality, follow these steps:

1. The middle point between \(-12\) and \(-6\) is the average:
   \[
   \frac{-12 + (-6)}{2} = \frac{-18}{2} = -9
   \]
   So, the center of the interval is \(-9\).

2. Next, calculate the distance from \(-9\) to either endpoint (since the interval is symmetric):
   \[
   | -9 - (-12) | = 3 \quad \text{or} \quad | -9 - (-6) | = 3
   \]
   So, the distance from the center is 3.

Thus, the absolute value inequality can be written as:
\[
| x + 9 | \leq 3
\]

This inequality represents all the values of \(x\) that lie within 3 units of \(-9\), which is equivalent to the interval \([-12, -6]\).

Would you like further details on how to interpret this inequality or more examples? Here are five related questions to consider:

1. How do you solve an absolute value inequality like \( |x - a| \leq b \)?
2. What are the steps to graph an absolute value inequality?
3. How do you interpret an absolute value inequality involving "greater than" (e.g., \( |x - a| \geq b \))?
4. How would this inequality change if the interval were \((-12, -6]\) instead of \([-12, -6]\)?
5. How can you solve compound inequalities graphically?

**Tip:** When working with absolute value inequalities, always visualize the problem on a number line to better understand the range of solutions.

Explore how to solve the absolute value inequality | x + 9 | ≤ 3, based on the interval between -12 and -6 shown on a number line. This step-by-step solution covers key algebraic concepts and is suitable for students in Grades 7-9.

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