https://file.aiquickdraw.com/mathbot/file/1727935860620j7zthfja.jpg

The inequality displayed in the image is:

\[
|x - 9| \leq 2
\]

This is an absolute value inequality, which means we need to break it down into two separate inequalities, removing the absolute value. The rule for absolute values is:

\[
|a| \leq b \quad \text{implies} \quad -b \leq a \leq b
\]

Using this rule on the given inequality:

\[
-2 \leq x - 9 \leq 2
\]

### Step-by-step Solution:
1. Add 9 to all parts of the inequality to isolate \( x \):
   \[
   -2 + 9 \leq x - 9 + 9 \leq 2 + 9
   \]
   Simplifying the inequality:
   \[
   7 \leq x \leq 11
   \]

### Final Answer:
\[
x \in [7, 11]
\]

This means that \( x \) can be any value between 7 and 11, inclusive.

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

Here are 5 related questions:
1. How do we handle absolute value inequalities with "greater than" signs?
2. What happens if the inequality is strict, i.e., \( |x - 9| < 2 \)?
3. Can you solve absolute value inequalities that result in no solutions?
4. How do absolute value inequalities relate to intervals on a number line?
5. What would the solution look like if the inequality were \( |x - 9| \geq 2 \)?

**Tip**: Always remember to check if the inequality symbol is strict or not, as it determines whether you include endpoints in your solution!

Learn how to solve the absolute value inequality |x - 9| ≤ 2 with a step-by-step breakdown. This problem involves algebraic concepts such as absolute value inequalities and isolating variables. Suitable for students in Grades 8-10.

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