How can I solve an equation like this:

$\displaystyle

|z+1| + |z-1| = 3, z \in \mathbb C

$

I've tried expanding z as

$\displaystyle

z = x + iy; x,y \in \mathbb R

$

and computing the equation

$\displaystyle

\sqrt{(x+1)^2 + y^2} + \sqrt{(x-1)^2 + y^2} = 3

$

but this way it quickly gets terribly complicated. Plus I have two variables in one expression.

Any other ideas?