How do you label an equation with something on the left and on the right?

This could be an overkill. Yet you only need to say \rtag{stuff} to get the a tag on the right. Note also that amsmath makes some efforts that the tag does not overlap with the equation. This solution does not make such efforts.

\newcommand{\rtag}[1]{\begin{tikzpicture}[baseline=(tmp.base),remember picture]
\node[inner sep=0pt](tmp){\vphantom{1}};
 \path (current page text area.east|-tmp.base)
  node[anchor=base east,inner sep=0pt,outer sep=0pt]{(#1)};
 \sum\limits_{n=0}^{\infty} x^n = \frac{1}{1-x} \rtag{$|x|<1$}

  \rtag{$|y|<1$}2\sum\limits_{n=0}^{\infty} y^n = \frac{2}{1-y}


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You have to do some manual intervention since it's not the usual way of doing things:

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\newcommand{\ltxlabel}{\[email protected]}

See \eqref{eqn:sum_x_n_1}.
    $\displaystyle \sum_{n = 0}^\infty x^n = \frac{1}{1 - x}$\hfill
    \llap{($\lvert x\rvert < 1$)}%
And also see \eqref{eqn:sum_x_n_2}.
  \sum_{n = 0}^\infty x^n = \frac{1}{1 - x} \label{eqn:sum_x_n_2}


A more typical way of managing conditions is via a \quad (or \qquad) separation:

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  \sum_{n = 0}^\infty x^n = \frac{1}{1 - x} \qquad (\lvert x \rvert < 1)