# Sketching a set in the complex plane?

Given the set $2 < |z-1| < 5$, how could you sketch this set in the complex plane? I understand how to do it if it were just $|z-1| < 5$, but I'm stuck on how to do it with two inequalities. Thanks for any help!

• Think of the points in the complex plane whose distance to $1$ is greater than $2$ but less than $5$. – dxiv Feb 26 '18 at 0:10

$$2 < |z-1| < 5 \iff 2 < |z-1|, \text {and } |z-1| < 5$$
As you know $|z-1| < 5$ is the set of points inside the circle of radius $5$ and centered at $z=1.$
On the other hand, the set $2 < |z-1|$ is the set of points outside the circle of radius $2$ and centered at $z=1.$
Thus $$2 < |z-1| < 5$$ is the region between the two circles.