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  4. The inequality |z-4|<|z-2| represents the region given by

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Q. The inequality $|z-4|<|z-2|$ represents the region given by

Complex Numbers and Quadratic Equations

Solution:

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If $z$ satisfies $|z-4|=|z-2|$, then $z$ lies on the perpendicular bisector of the segment joining $z=2$ and $z=4$.
i.e., $|z-4|=|z-2| \Rightarrow \operatorname{Re}(z)=3$.
As $z=0$ does not satisfy $|z-4|<|z-2|$, we get $|z-4|<|z-2|$ represents the region $\operatorname{Re}(z)>3$.