A question related to this does exists before. Main point: When triangle inequality in complex numbers is applicable? Is there a condition? Or this is applicable for all complex numbers? If so, why we have the following ambiguity?
Here's the question,
If a complex number $z$ lies in the interior or on the boundary of a circle of radius $3$ units and centre $(– 4, 0)$, find the greatest and least values of $|z +1|$ .
So, if we visualize a diagram, we get the biggest complex arrow $z$ = $-7$
Therefore, the maximum value is $|-7+1| = 6$
But, if we go with the triangle inequality,
$$|z+1|\leq |z| + 1\leq 7+1 = 8$$
So, why the maximum value of $|z+1|$ is $8$ now.
Where I went wrong with the triangle inequality?
There is no further condition to apply triangle inequality with complex number. In your case, for $z=-7$, it holds, because $6\leq 8$. Nothing went wrong!
Note that if you take $z=7$ (which is outside your disc) then $$8=|7+1|=|z+1|\leq |z|+1 =7+1=8.$$ In your disc, equality holds for example when $z=-7$ and $w=-1$, then $$8=|(-7)+(-1)|=|z+w|\leq |z|+|w|=|-z|+|-w|=8.$$ Please see also Equality of triangle inequality in complex numbers