The inequality  $|z-4|<|z-2|$ represents the region most appropriately given by 

$\begin{array}{l}|\mathrm{z}-4|<|\mathrm{z}-2|\Rightarrow |\mathrm{z}-4{|}^2<|\mathrm{z}-2{|}^2\Rightarrow (\mathrm{z}-4)(\overline{\mathrm{z}}-4)<(\mathrm{z}-2)(\overline{\mathrm{z}}-2)\\ \Rightarrow \mathrm{z}\overline{\mathrm{z}}-4(\mathrm{z}+\overline{\mathrm{z}})+16<\mathrm{z}\overline{\mathrm{z}}-2(\mathrm{z}+\overline{\mathrm{z}})+4\\ \Rightarrow \underset{\_}{\underset{\_}{2(\mathrm{z}}}+\overline{\mathrm{z}})>12\\ \Rightarrow 4\mathrm{Re}\left(\mathrm{z}\right)>12\Rightarrow \mathrm{Re}\left(\mathrm{z}\right)>3\end{array}$