https://file.aiquickdraw.com/mathbot/file/1728428495159mzv4voev.jpg

The graph shows a number line with solid dots at -6 and 4, and a shaded region between them. This suggests that the solution includes all values of \( x \) between -6 and 4, inclusive.

To express this as an absolute value inequality, we want to rewrite the interval \( -6 \leq x \leq 4 \) in terms of an absolute value. The midpoint between -6 and 4 is:

\[
\frac{-6 + 4}{2} = -1
\]

This midpoint, -1, will be the center of our absolute value expression. The distance from -1 to either endpoint (4 or -6) is:

\[
|4 - (-1)| = 5 \quad \text{or} \quad |-6 - (-1)| = 5
\]

Thus, the inequality can be written as:

\[
|x + 1| \leq 5
\]

This is the absolute value inequality that represents the shaded region between -6 and 4.

Would you like more details or have any other questions?

### Related Questions:
1. How do we derive absolute value inequalities from graphs?
2. What is the significance of the midpoint in absolute value inequalities?
3. How do we handle open and closed dots in inequalities?
4. Can this inequality be written in interval notation?
5. How would the inequality change if the dots were open (not solid)?

### Tip:
Always check the endpoints on the number line to determine whether to use strict inequality (<) or inclusive inequality (≤) based on whether the dots are open or closed.

Write the absolute value inequality represented by this graph in terms of x.

Learn how to interpret a number line graph to form an absolute value inequality in terms of x. This problem involves finding the midpoint and the distance from endpoints to represent the inequality |x + 1| ≤ 5. Suitable for students in Grades 8-10.

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