To write the absolute value inequality for the graph on this number line, we need to examine the range of values shown. Here’s the breakdown:

1. The black dots at $-4$ and $4$ represent endpoints, indicating the interval between them.
2. The shading extends in both directions from $-10$ to $10$, which suggests that the values are between $-10$ and $10$.
3. Since the shading includes the range between these two points, this suggests an absolute value inequality.

Constructing the Inequality

The graph shows values that are less than or equal to $10$ in distance from $0$, which we can express with absolute value notation: $|x| \leq 10$

This inequality states that $x$ can range from $-10$ to $10$, including both endpoints.

Would you like more details or have any questions on this?

Here are 5 related questions for further exploration:

1. How would the inequality change if the shading went only up to $9$ instead of $10$?
2. What would the graph look like if the inequality were $|x| < 10$?
3. How would you write an absolute value inequality for the range $[-5, 5]$?
4. Can you explain the concept of absolute value in terms of distance on a number line?
5. What is the difference between $|x| \leq a$ and $|x| < a$ on a number line?

Tip: Absolute value inequalities like $|x| \leq a$ represent all points within a distance $a$ from $0$, which translates visually to an interval on the number line.