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Describe the set whose points satisfy the following region?

|z − 2| > |z − 3|.

My atempt

The open half-plane: Re z > 5/2; a region, My guess is that if this region takes all the imaginary axis

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You should draw a picture. Which points verify the relation $|z-2|=|z-3|$? Once this is settled you can wonder, of the space that remains, which half consists of those points with $|z-2|>|z-3|$. – Olivier Bégassat May 10 '12 at 19:51
up vote 2 down vote accepted

$|z-2|$ is the distance from $z$ to $2$. $|z-3|$ is the distance from $z$ to $3$. So the region you want consists of all points that are closer to $3$ than they are to $2$.

The set of points whose distance to $2$ and to $3$ is the same is a straight line, perpendicular to the line segment joining $2$ and $3$ (i.e., vertical), and going through its midpoint $2.5$. What happens to a point on the right of that line? On the left? How do you describe the points on the line? How do you describe the points on the right? How do you describe the points on the left?

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