Question

The greatest and least values of $|z+1|$ if $|z+4|\le 3$
are...and...
$\left(a_1\right)|z-2+i|\le 2,$ find the greatest and least values of $\left|z\right|$.

Answer 1

$|z+1|=|z+4|\le 3$

$|z+1|=|z+4-3|\le |z+4|+|3|\le 3+3=6$

$z=-7$ satisfies $|z+4|=3$

$\therefore max|z+1|=6$

$\therefore min|z+1|=0$

$|z|=|z-2+i|\le 2$

$|z|=|z-2+i-2|\le |z-2+i|+|2|$

$z=-i$ satisfies $|z-2+i|=2$

$\therefore max|z|=i$

$\therefore min|z|=0$