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How would I do this?

I know it's a circle of radius half around 3 and a larger circle of radius 1 around 3. But why?

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    It's not. It's a half-open annulus. https://en.wikipedia.org/wiki/Annulus_%28mathematics%292012-11-03

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With complex numbers, $|a-b|$ will give you the distance between $a$ and $b$. Hence $|2z-6|=2|z-3|$ is twice the distance from $z$ to $3$. Divide your inequalities by $2$, and you get ${1\over 2}<|z-3|\leq 1$ In other words, the distance between $z$ and $3$ should lie between $1\over 2$ and $1$, which is the same as the answer you gave.

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    It's worth noting that the $z$ in the set **may** lie on the larger circle, but **cannot** lie on the smaller one.2012-11-03