How can I prevent LaTeX from breaking inline formulas globally?

Increase the penalty for line breaks at relation symbols and binary operators:

\relpenalty=9999
\binoppenalty=9999

If you simply include these commands in your preamble (between \documentclass{...} and \begin{document}), it will prevent line breaks in most cases, but in extreme situations, they can still be broken. If you set

\relpenalty=10000
\binoppenalty=10000

equations will never be broken - this may, however, destroy your layout!

\mbox{$equation$} is not an answer, unfortunately: the spacing in the equation doesn't grow or shrink together with the other spaces in the line. The "correct" answer is

$LHS=\nobreak RHS$

TeX will choose another break point, if there is one, or warn about an Overfull \hbox

Diabonas's answer shows how to avoid such breaks globally, and correctly warns about the dangers of doing so.