\documentclass[12pt,reqno]{amsart}
\begin{document}
\section{Assignment: Absolute Values}
\par\vspace{0.2 cm}\noindent
For $a\in\mathbb{R}$ the \textit{positive part} is defined by
\[
a^+:=\frac{|a|+a}{2}
\vspace{0.2 cm}
\]
and the \textit{negative part} by
\[
a^-:=\frac{|a|-a}{2}
\vspace{0.2 cm}
\]
\par\vspace{1 cm}\noindent
a) Prove that $a=a^+-a^-$ and $|a|=a^++a^-$.
\par\vspace{1 cm}\noindent
b) Prove that
\[
a^+=\left\{
\begin{array}{rl}
a & a\geq 0\\
0 & a\leq 0\\
\end{array}
\right.
\quad\mbox{and}\quad
a^-=\left\{
\begin{array}{rl}
0 & a\geq 0\\
-a & a\leq 0\\
\end{array}
\right.
\]

\end{document}
