Official solution
We know minimum value of
\[\,\,\,\left| {{Z_1} + {Z_2}} \right|\,\,\,\] is \[\,\,\,\left| {\left| {{Z_1}} \right| - \left| {{Z_2}} \right|} \right|\]
Thus minimum value of
\[\,\,\,\left| {Z + {1 \over 2}} \right|\,\,\,\] is \[\,\,\,\left| {\left| Z \right| - {1 \over 2}} \right|\]
\[\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, \le \left| {Z + {1 \over 2}} \right| \le \left| Z \right| + {1 \over 2}\]
Since, \[\,\,\,\left| Z \right| \ge 2\]
\[\therefore\] \[\,\,\,2 - {1 \over 2} < \left| {Z + {1 \over 2}} \right| < 2 + {1 \over 2}\]
\[ \Rightarrow {3 \over 2} < \left| {Z + {1 \over 2}} \right| < {5 \over 2}\]