problem: What is the length of the segment of the number line consisting of the points that satisfy $(x-4)^2 \le 9$?
solution: Since the square of $x-4$ is at most 9, the value of $x-4$ must be between $-3$ and $3$ (or equal to either). So, we have $-3 \le x-4 \le 3$. Thus, $1 \le x \le 7$. Therefore, our answer is $\boxed{6}$.
  
  - OR -
  
  If $(x-4)^2 \le 9$, then $x$ can be no more than 3 away from 4.  Therefore, the values of $x$ from 1 to 7 satisfy the inequality, and our answer is $\boxed{6}$.