Question

State True and False

The inequality $|z-4|<|z-2|$ represents the region given by Re(z) > 3.

Answer 1

$|z-4|<|z-2|\Rightarrow {|z-4|}^2<{|z-2|}^2$
$\Rightarrow {|x+iy-4|}^2<{|x+iy-2|}^2$
$\Rightarrow {(x-4)}^2+y^2<{(x-2)}^2+y^2$
$\Rightarrow x^2-8x+16+y^2<x^2-4x+4+y^2$
$\Rightarrow -4x<-12$ or $x>3$ i.e. $Re\left(z\right)>3.$