If $\left|z+1\right|=3$, determine the greatest and least possible values for $|z|$

Question

If $\left|z+1\right|=3$, determine the greatest and least possible values for $|z|$

According to the answer, the greatest value of $|z|$ is $2$ and the least is $-4$.

But why is that?

My attempt:

if $\left|z+1\right|=3$, then $-(z+1) = 3$ or $(z+1) = 3$

therefore $z = -4$ and $|z| = 4$ and $z = 2$ and $|z| = 2$

which corresponds to the value $4$ to be the greatest and $2$ to be the least. I want to know why this thinking is wrong?

So I googled to find help. I watched this video by Eddie Woo --> https://www.youtube.com/watch?v=e3YNDyeiSiY&ab_channel=EddieWoo. The solution is really long and some commenters says that he is wrong.

Answer 1

|z| can never be a negative value. Thus |z| = -4 can never be a possible solution. Thus your answer seems to be correct if the values for |z| are asked ...

Answer 2

By the triangle inequality: $$3=|z+1|\geq|z|-1,$$ which gives $$|z|\leq4.$$ The equality occurs for $z=-4$, which says that we got a maximal value.

Also, by the triangle inequality again: $$3=|z+1|\leq|z|+1,$$ which gives $$|z|\geq2.$$ The equality occurs for $z=2$, which says that we got a minimal value.

Answer 3

I would argue that the problem is mistaken. If it asks for $|z|$, then because the solutions are $-4$ and $2$, the greates solution would be $|-4|=4$