Equation with complex numbers and absolute value

Question

I'm trying to solve this equation algebraically.

$|z + i| < |z − i|$

The result should be:

$y<0$

My result is:

$y>ix$

Answer 1

let $z=x+iy$ then we get $$\sqrt{x^2+(y+1)^2}<\sqrt{x^2+(y-1)^2}$$ can you solve this?

Answer 2

You shouldn't have any $i$'s when finding the length. Remember that $|x+iy|=\sqrt{x^2+y^2}$. $|z+i|=|x+i(y+1)|$ and $|z-i|=|x+i(y-1)|$.

Answer 3

$|z+i|^2<|z-i|^2 \Longrightarrow |x+iy+i|^2<|x+iy-i|^2 \Longrightarrow x^2+(y+1)^2 < x^2 + (y-1)^2$

solving the last equation you will get $y<0$