I'm trying to sketch the following sets in the complex plane:

$\displaystyle a)~Re\left(\frac{1}{z}\right) < 1$

$\displaystyle b)~ |z+i|\ge|z-i|$

For $\displaystyle a)$ I found the real part of $\displaystyle \frac{1}{z}$ by setting $\displaystyle z = x + iy$ and got:

$\displaystyle \frac{x}{x^2+y^2}$

I then set this = 1 to try and graph something from which I can determine the values that are < 1:

$\displaystyle \frac{x}{x^2+y^2}=1$

$\displaystyle x = x^2+y^2$

$\displaystyle y = \sqrt{x-x^2}$

So I would have the top half of a circle starting at (0,0) and ending at (0,1), so my set would be drawn to represent [0,1) right?

The answer I have has a circle though with the equation $\displaystyle |y| \ge \sqrt{x-x^2}$ and I'm not sure how this point was achieved?

b) I'm not sure how to proceed since $\displaystyle z$ is abitrary but $\displaystyle i$ is defined and I can plot it at (0,1), so I don't see how I can compare them.

Thanks