Cases environment with angle brackets

The following code updates cases to take an optional argument. The optional argument is used as the left delimiter instead of the traditional (default) \lbrace:

\documentclass{article}

\usepackage{amsmath,etoolbox}

\makeatletter
\renewenvironment{cases}[1][\lbrace]{%
  \def\@ldelim{#1}
  \matrix@check\cases\env@cases
}{%
  \endarray\right.%
}
\patchcmd{\env@cases}{\lbrace}{\@ldelim}{}{}
\makeatother

\begin{document}

\[
  f(x) = \begin{cases}
    a & \text{if $x < 0$} \\
    b & \text{otherwise}
  \end{cases}
\]

\[
  f(x) = \begin{cases}[\langle]
    a & \text{if $x < 0$} \\
    b & \text{otherwise}
  \end{cases}
\]

\end{document}

It should function as expected, even if nested:

\[
  f(x) = \begin{cases}
    a & \text{if $x < 0$} \\
    b & \begin{cases}[\langle]
      {} + c & \text{if $x < 7$} \\
      {} + d & \text{otherwise}
    \end{cases}
  \end{cases}
\]

However, you may be better off using a regular array. It depends on how you nest the content.

If I understand well, what you want, you can, with some limitations, with the empheq package (don't load amsmath: it loads mathtools, which loads it). The limitation is due to the \langle not being extensible as much as you please. Actually, it works with two-line cases, possiblt three lines playing with the vertical spacing.

Here's an example:

\documentclass{article}
\usepackage{empheq}

\begin{document}

\begin{empheq}[left={f(x) = \empheqlangle}]{alignat*= 2}
 & \smash{x^2} + 1&\quad & \text{if }x > 0 \\
 & 1-x && \text{if }x < 0
\end{empheq}

\end{document}