Q.Solve the following for $x$, where $|x|$ is the modulus function, $[x]$ is the greatest integer function, $\{x\}$ is the fractional part function: $1<|x-1|<4$.

A piecewise function is defined by two or more separate formulas, each valid only on its own specified part of the domain; evaluating it always starts by checking which piece the input falls into (paying close attention to which endpoint each inequality includes — a filled dot on a graph marks an included point, a hollow dot an excluded one). Two named piecewise functions recur constantly.

The signum function, $\text{sgn}(x)=-1$ for $x<0$, $0$ at $x=0$, $+1$ for $x>0$, reduces any real number to just its sign; domain $R$, range the three-point set $\{-1,0,1\}$. …