How can I locate this complex number without actually plotting it on a graph.

Question

The question is $$\log_{\frac{1}{√3}}\frac{|z|^2–|z|+1}{2+|z|}>–2.$$ I tried to solve this and simplified it to $$|z|^2-4|z|-5<0.$$

But I wasn't able to locate this number, please tell me the procedure as well as the answer how to do it without actually plotting it on the graph as use of graph is not allowed. Assume $z=x+iy$.

Answer 1

Hint: $|z|\ge 0$, and $$|z|^2-4|z|-5 = (|z|-2)^2-9<0 \implies -3<|z|-2<3 \implies ?$$

Answer 2

$|z|$ is the distance from $0$ to the point $z,$ denote it $r.$
Then $|z|^2-4|z|-5<0$ rewrites $r^2-4r-5<0$ and is equivalent to $$(r-5)(r+1)<0.$$ Since $(r+1)$ is clearly positive (it is a distance + $1$), we solve $$r-5<0$$ or, in terms of $z,$ $$|z|<5$$

Geometrically: the solutions form the circle centered in $0$, with radius $5,$ without the boundary circumference.