Package ulem
addresses this problem. It also works for plain TeX (\bf
):
\input ulem.sty
The following {\bf\uline{Approximations}} and
{\bf\uline{Definitions}} is applied in further work.
\bye

If you are using LaTeX, then know, that \bf
is deprecated since decades.
\documentclass{article}
\usepackage{ulem}
\begin{document}
The following \textbf{\uline{Approximations}} and
\textbf{\uline{Definitions}} is applied in further work.
\end{document}
Package soul
also allows hyphenation:
\documentclass{article}
\usepackage[a6paper]{geometry}
\usepackage{soul}
\begin{document}
The following \textbf{\ul{Approximations}} and
\textbf{\ul{Definitions}} is applied in further work.
\end{document}

Typography: Underlining comes from the time of mechanical typewriters. Font switches were tricky at best. But LaTeX allows for different font shapes, and
bold is more than enough. The usual emphasis command uses italics. Also several ways for emphasis (bold + underline, bold + italics, ...) should not be used.
\documentclass{article}
\begin{document}
The following \textbf{Approximations} and
\textbf{Definitions} is applied in further work.
The following \emph{Approximations} and
\emph{Definitions} is applied in further work.
\end{document}

\underline
is supposed to work. Don't use underlining in any case whatsoever.\vphantom{p}
to even things out. but i agree that underlining is to ba avoided unless there's no other possibility (as to leave a space for a signature, for example).