How do I identify the set of points satisfying $|z-1|+|z+1|\leq 2$? How do I identify the set of points satisfying $|z-1|+|z+1|\leq 2$?
My idea is:     $|z+1|^{2}=|z|^2+1+2x$
$|z-1|^{2}=|z|^2+1-2x$
$|z-1|+|z+1|\leq 2$ $\implies$ $|z+1|\leq 2 -|z-1|$
$(|z+1|)^2\leq (2 -|z-1|)^2$ (* is this step correct?)(because $-3&lt;2$ but $9 \nleq 4$)
I am confused at this step of my solution. Can anyone suggest me how do I fix this problem?
Think of the triangle inequality.
Since the distance from $z=-1$ to $z=1$ is $2$,
the only points satisfying the (in)equality are on the line segment from $z=-1$ to $z=1$.
$|z+1|+|z-1|=2$ denotes a line segment. But $|z+1|+|z-1|&lt;2$ is a null set and it represents nothing in the Argand plane.
And when $|z+1|+|z-1|&gt;2$, represents region (point) out side  an ellipse.