What would be the graph of the following set:
$$\{z \in \mathbb{C} \mid |z-1|\lt |z|\}$$
I tried out writing $z$ as $x+iy$, squaring both sides and ultimately came up with the condition on $x$ as $x \lt \frac{1}{2}$.
Help me judge where I am wrong and correct me. Thanks
$(x-1)^2+y^2 < x^2+y^2\\\Rightarrow -2x+1<0\\\Rightarrow 2x>1 \\\Rightarrow x > \frac{1}{2}$