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In addition to $x=0$, are $x=\infty$ and $x=-\infty$ solutions of $x=2x$ because $\infty=2\cdot\infty$ and $-\infty=-2\cdot\infty$?

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    Not in $\mathbb{R}$ since $\pm \infty$ are not real numbers. – 2017-02-17
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    Infinity is not a number – 2017-02-17
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    Not to mention, the problem of finding solutions of $x=2x$ can be rephrased as the finding the zeroes of the function $f(x)=2x-x$ by moving the both to the same side, i.e. the zeroes of the function $f(x)=x$. $\infty$ is not considered a zero of that function when viewing it from a complex analysis point of view. – 2017-02-17

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If you're solving the problem in the extended real numbers, then yes, the set of solutions to $x = 2x$ is indeed $\{ 0, +\infty, -\infty\}$. Problems like this arise from time to time when computing limits in calculus, and sometimes the actual answer you're looking for really is $+\infty$ or $-\infty$, so it's good to pay attention to this detail when the extended real numbers are applicable.

If you're just solving in the real numbers, however, then $0$ is the only solution. ($\pm \infty$ aren't real numbers, they're extended real numbers!)