1. Consider the domain x >= 1.
   In this domain |x-1| = x-1,
   and yours equation (1) becomes

    = ,  or

    = ,

   (x-2)*(x+1) = 0

   with the roots x = 2 and x = -1.

   Of them, only x = 2 belongs to the domain x >= 1.
   So, x = 2 is the ONLY solution of the equation (1) in this domain.


2. Next, consider the domain x < 1.
   In this domain |x-1| = -(x-1). 
   and yours equation (1) becomes

    = ,  or

    = .

   Find the roots using the quadratic formula.

    = ,   = .

   Of them, only  =  belongs to the domain x < 1.

   So,  =   is the ONLY solution of the equation (1) in this domain.

Answer.  The solutions are  x = 2  and  x = .