If $z$ is a complex number then
the minimum value of $| z | + | z - 1 | + | 2 z - 3 |$ is

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Solution

$| z | + | z - 1 | + | 2 z - 3 |$
$= z | + | z - 1 | + | 3 - 2 z | \geq | z + z - 1 + 3 - 2 z |$
$\Rightarrow | z | + | z - 1 | + | 3 - 2 z | \geq 2$

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