problem: Find the sum of all values of $x$ such that $|x-1| = 7$.
solution: We must have either $x-1 = 7$ or $x-1=-7$.  If $x-1=7$, we have $x=8$, and if $x-1 = -7$, we have $x= -6$, so the sum of the possible values of $x$ is $8+(-6) = \boxed{2}$.