Solution of linear algebraic equation

Question

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

Answer 1

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!)