# Determine the set of all complex number z satisfying following conditions

I’m having some troubles of calculating complex numbers where I need to deal with absolute values and inequalities. Here is an example I’ve been working on but I get stuck

Re(2/z)+Im(4/z)<1

I use z=x+iy And find 1/z = (x-iy)/(x^2+y^2)

So

Re(2/z) = 2x/(x^2+y^2) and Im(4/z) = (-4y)/(x^2+y^2)

If I just forget about the inequality and make it equal to 1. I get:

2x-i4y = x^2+y^2

Which is a circumference with a complex radius. Now I’m not sure what to do or if I’m going in the correct direction.

I hope you can help me.

That factor of $i$ you added is spurious. You should get $2x-4y<x^2+y^2$, which is the inside of a circle of real radius.
Solve $$\frac{2x}{x^2+y^2}-\frac{4y}{x^2+y^2}<1$$
Your equation is slightly wrong: that $i$ should not be there. It should be $$2x - 4y = x^2 + y^2$$ Now its radius is a real number, and you should have no trouble plotting it.