Question

If $|z+4|\le 3$, then find the greatest and least values of |z + 1|.

Answer 1

Given that, $|z+4|\le 3$

For the greatest value of |z + 1|

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

So, greatest value of |z + 1| = 6

For, now, least value of |z + 1|, we know that the least value of the modulus of a complex number is zero. So, the least value of |z + 1| is zero.