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Given the set $2 < |z-1| < 5$, how could you sketch this set in the complex plane? I understand how to do it if it were just $|z-1| < 5$, but I'm stuck on how to do it with two inequalities. Thanks for any help!

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    $\begingroup$ Think of the points in the complex plane whose distance to $1$ is greater than $2$ but less than $5$. $\endgroup$ – dxiv Feb 26 '18 at 0:10
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$$ 2 < |z-1| < 5 \iff 2 < |z-1|, \text {and } |z-1| < 5$$

As you know $ |z-1| < 5$ is the set of points inside the circle of radius $5$ and centered at $z=1.$

On the other hand, the set $ 2 < |z-1| $ is the set of points outside the circle of radius $2$ and centered at $z=1.$

Thus $$ 2 < |z-1| < 5 $$ is the region between the two circles.

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  • $\begingroup$ Oh okay that makes so much sense, thank you! $\endgroup$ – lbh Feb 26 '18 at 0:30

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