How to locally decrease the distance between an equation and its numerical label

The equation number has a minimum distance from the equation, but there's a slick trick in exercise 19.8 of the TeXbook.

\documentclass[letterpaper, 10pt, conference]{IEEEconf}  % Comment this line out if you need a4paper

\usepackage{graphicx} % for pdf, bitmapped graphics files
\usepackage{mathptmx} % assumes new font selection scheme installed
\usepackage{amsmath} % assumes amsmath package installed
\usepackage{amssymb}  % assumes amsmath package installed
\usepackage{amsthm}
\usepackage{lipsum}

\renewcommand{\qedsymbol}{$\blacksquare$}

\newtheorem{thm}{Theorem}[section]
\newtheorem{lem}[thm]{Lemma}
\newtheorem{prop}[thm]{Proposition}
\newtheorem{cor}{Corollary}
\newtheorem{conj}{Conjecture}[section]
\theoremstyle{definition}
\newtheorem{defn}{Definition}[section]
\newtheorem{exmp}{Example}[section]
\newtheorem{rem}{Remark}

\title{My Title*}

\author{Guy$^{1}$
\thanks{*This work was not supported by any organization}% <-this % stops a space
\thanks{$^{1}$Guy is with Hell, {\tt\small [email protected]}}%
}

\begin{document}

\maketitle

\lipsum[2]\nopagebreak
\noindent
\begin{minipage}{.45\linewidth}
\begin{equation}
  \begin{cases}
    \theta_{i}^{h^{-}} = \dfrac{ 2(i-1)\pi}{ m_{c}^{\hat{x}}}\\
    \theta_{i}^{h^{+}} = \dfrac{ 2i\pi}{ m_{c}^{\hat{x}}}\\
\end{cases}
\hspace{10000pt minus 1fil}
\end{equation}
\end{minipage}\hspace{.1\linewidth plus 4pt}%
\begin{minipage}{.45\linewidth}
\begin{equation}
    \begin{cases}
    \theta_{j}^{v^{-}} = \dfrac{ 2(j-1)\pi}{ m_{c}^{\hat{v}}}\\
    \theta_{j}^{v^{+}} = \dfrac{ 2j\pi}{ m_{c}^{\hat{v}}}\\
\end{cases}
\hspace{10000pt minus 1fil}
\end{equation}
\end{minipage}\nopagebreak\vspace{\belowdisplayskip}%
\lipsum[3]

\end{document}

I have used lipsum to emulate some text before and after the equations. Note how to get the same spacing as for a normal equation environment, with no breaks before and after.

The trick is inserting a large glob of horizontal glue, which can be shrinked as much as possible:

\hspace{10000pt minus 1fil}

I have also reduced the width of the minipages, to avoid the left equation number to be too near to the part on the right.

Note also \dfrac, rather than all those \displaystyle tokens.

Add some negative space

\begin{equation}
  \hspace{-2em}
  \begin{cases}
    ...
  \end{cases}
  \hspace{-2em}
\end{equation}

\documentclass[letterpaper, 10pt, conference]{IEEEconf}

\usepackage{graphicx}
\usepackage{mathptmx}
\usepackage{amsmath}
\usepackage{amssymb}

\title{My Title*}

\author{Guy$^{1}$
\thanks{*This work was not supported by any organization}% <-this % stops a space
\thanks{$^{1}$Guy is with Hell, {\tt\small [email protected]}}%
}

\begin{document}

\maketitle
\thispagestyle{empty}
\pagestyle{empty}

\begin{abstract}

ABSTRACT

\end{abstract}

\section{INTRODUCTION}

\noindent\begin{minipage}{.5\linewidth}
\begin{equation}
  \hspace{-2em}
  \begin{cases}
    \theta_{i}^{h^{-}} = \frac{\displaystyle 2(i-1)\pi}{\displaystyle m_{c}^{\hat{x}}}\\
    \theta_{i}^{h^{+}} = \frac{\displaystyle 2i\pi}{\displaystyle m_{c}^{\hat{x}}}
  \end{cases}
  \hspace{-2em}
\end{equation}
\end{minipage}%
\begin{minipage}{.5\linewidth}
\begin{equation}
  \hspace{-2em}
  \begin{cases}
    \theta_{j}^{v^{-}} = \frac{\displaystyle 2(j-1)\pi}{\displaystyle m_{c}^{\hat{v}}}\\
    \theta_{j}^{v^{+}} = \frac{\displaystyle 2j\pi}{\displaystyle m_{c}^{\hat{v}}}
  \end{cases}
  \hspace{-2em}
\end{equation}
\end{minipage}

\end{document}

Edit: Since the question was misleading, the first answer version was accommodate to one column document. Of course meanwhile, after editing of question, appears other answers ...

Reduce width of minipages and push them apart with \hfill and reduce space between equation and equation number by \hspace{-1em}:

\documentclass{article}
\usepackage{mathtools}

\begin{document}

  \noindent$|$\hrulefill\ column width\ \hrulefill$|$

\noindent\begin{minipage}{.48\linewidth} % <-- reduced minipage width
\begin{equation}
  \begin{cases}
\theta_{i}^{h^{-}} = \frac{\displaystyle 2(i-1)\pi}{\displaystyle m_{c}^{\hat{x}}}\\
\theta_{i}^{h^{+}} = \frac{\displaystyle 2i\pi}{\displaystyle m_{c}^{\hat{x}}}
    \end{cases}\hspace{-1em}% <-- added to reduce distance to eq. numbering apart
\end{equation}
\end{minipage}\hfill % <-- added to push minipages appart
\begin{minipage}{.48\linewidth} % <-- reduced minipage width
\begin{equation}
    \begin{cases}
\theta_{j}^{v^{-}} = \frac{\displaystyle 2(j-1)\pi}{\displaystyle m_{c}^{\hat{v}}}\\
\theta_{j}^{v^{+}} = \frac{\displaystyle 2j\pi}{\displaystyle m_{c}^{\hat{v}}}
\end{cases}\hspace{-1em}% <-- added to reduce distance to eq. numbering
\end{equation}
\end{minipage}
\end{document}