By definition |z-a| is the 'distance' of the point z from the point a...
... and what is the z-plane region the points of which are less distant from z=1 than from z=2?...
Kind regards
Read this thread: http://www.mathhelpforum.com/math-he...-equation.html
Draw the locus defined by |z-1| = |z-2|. It divides the complex plane into two regions. Now choose a simple and convenient point from one of those regions and test whether it satisfies |z-1| < |z-2|. If it does, then the region the point belongs to is the solution. If it doesn't, then the other region is the solution.