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.

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

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

\end{document}


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

\documentclass[leqno]{article}

\usepackage{amsmath}

\makeatletter
\newcommand{\ltxlabel}{$email protected]} \makeatother \begin{document} See \eqref{eqn:sum_x_n_1}. \[ \refstepcounter{equation}\ltxlabel{eqn:sum_x_n_1} \makebox[\linewidth]{% \rlap{(\theequation)}\hfill \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}.
$$\sum_{n = 0}^\infty x^n = \frac{1}{1 - x} \label{eqn:sum_x_n_2}$$

\end{document}


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

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