\documentclass[preview,border=1pt]{standalone}
\usepackage{amsmath,amssymb,amsthm}
\usepackage{mathtools}

\begin{document}

\DeclarePairedDelimiter\abs{|}{|}
\DeclarePairedDelimiterX\ip[2]{\langle}{\rangle}{#1,#2}
\[ \abs{-\frac12} = \abs*{-\frac12} = \abs[\Big]{-\frac12} \]
\[ \ip{\frac{x}{2}}{y} = \ip*{\frac{x}{2}}{y} = \ip[\Big]{\frac{x}{2}}{y}\]
\end{document}