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For example, I want to write

\begin{equation}
 \begin{split}
    \left\{ &ax + b =\\
            & cx + d \right\}
  \end{split}
\end{equation}  

Latex tells me I annot split the brackets like this, for there is a \left without a \right, and vice versa.

How to treat this issue?

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Instead of \left\{ you should use \bigl\{ (or \Bigl\{, \biggl\{ ...) and \bigl\{ (or \Bigr\{, \biggr\{ ...) instead of \right\} :

\documentclass[twocolumn]{article}
\usepackage{amsmath}

\begin{document}
\begin{equation}
 \begin{split}
    \bigl\{ &ax + b =\\
            & cx + d \bigr\}
  \end{split}
\end{equation}
\end{document}

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