Describing and sketching the set of complex numbers satisfying certain conditions

Question

I need to describe and sketch the set of complex numbers satisfying the conditions (|z|-1)(|z|-2) ≤ 0 and Im(z) > 0.

This is what I have done:

Solved the inequality for (|z|-1)(|z|-2) ≤ 0 which turns out to be 1 ≤ |z| ≤ 2. I then let |z| = √(x^2 + y^2) and squared the inequality giving 1 ≤ x^2 + y^2 ≤ 4 and let Im(z) = y giving y > 0.

Then sketching it we have,

Do the description and sketch look correct for this question?

Answer 1

You have that

1. The inequality $|z|\leq 2$ represent the full circle with radius $2$ and center in $0$ (interior and boundary).

2. The inequality $|z| \geq 1$ represent the complement of the open circle with radius $1$ and center in $0$.

3. The inequality $Im(z)>0$ represent the open upper half-plane.

Thus your pic is not correct even though the idea is right.

Answer 2

You have the right idea. However, with $1 \le x^2 + y^2 \le 4$, the inner circle has a radius of $1$ and the outer circle has a radius of $2$, but your diagram shows boundary circles with radii of $2$ and $4$ instead. Note you can also determine what the correct circle radii will be from $1 \le |z| \le 2$ since it means it's distance from the origin for the points of $z$ are between $1$ and $2$.