Solve $|x-8| \leq 4$

Step 1 ：The absolute value of a number is its distance from zero on the number line. Therefore, the inequality \(|x-8| \leq 4\) means that the distance between \(x\) and \(8\) on the number line is less than or equal to \(4\). This can be solved by setting up two inequalities: \(x-8 \leq 4\) and \(- (x-8) \leq 4\).

Step 2 ：Solving these inequalities gives us the solutions \(x = 12\) and \(x = 4\). However, these are the endpoints of the interval of solutions.

Step 3 ：The solution to the original inequality \(|x-8| \leq 4\) is the interval \([4, 12]\).

Step 4 ：Final Answer: The solution to the inequality \(|x-8| \leq 4\) is \(\boxed{[4, 12]}\).