17. How many solutions does equation ||$|x-1|-1|-1|=1$ have?
The modulus function $|x|$ evaluates the absolute value of a number; for example $|6|=|-6|=6$.
Solution
004

Since ||$|x-1|-1|-1|=1$ we have ||$x-1|-1|-1= \pm 1$.
Therefore ||$x-1|-1|= \pm 1+1=0$ or 2 .
Continuing in this way, $|x-1|-1= \pm 0$ or $\pm 2$, so $|x-1|=3,1$ or -1 , (of which -1 is not possible). Therefore $x-1= \pm 3$ or $\pm 1$, so $x=-2,0,2$ or 4 .