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.