The inequality you provided is:

\[
|x - 7| \leq 9
\]

To solve this absolute value inequality, we break it into two cases, as follows:

### 1. Case 1: 
\[
x - 7 \leq 9
\]
Solving for \( x \):
\[
x \leq 16
\]

### 2. Case 2:
\[
-(x - 7) \leq 9 \quad \text{or equivalently} \quad x - 7 \geq -9
\]
Solving for \( x \):
\[
x \geq -2
\]

### Final Solution:
The solution to \( |x - 7| \leq 9 \) is the intersection of both cases, which gives:
\[
-2 \leq x \leq 16
\]

Thus, the solution is:

\[
x \in [-2, 16]
\]

Would you like further details or have any questions?

Here are some related questions to expand on this:

1. How do we solve strict inequalities like \( |x - 7| < 9 \)?
2. What changes when the inequality is \( |x - 7| \geq 9 \)?
3. How do you graph the solution \( -2 \leq x \leq 16 \) on a number line?
4. How does the solution change if we have \( |x - 7| > 9 \)?
5. Can this technique be applied to equations like \( |2x - 5| \leq 7 \)?

**Tip:** When dealing with absolute value inequalities, always split them into two separate inequalities based on the definition of absolute value.

Learn how to solve the absolute value inequality |x - 7| ≤ 9 with a step-by-step solution. This problem covers key algebraic concepts such as absolute value inequalities and their properties. Suitable for students in Grades 7-9.

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