\documentclass[12pt]{article}
\usepackage{amsmath,mathtools}
\usepackage[top=1in,bottom=1in,left=1.25in,right=1.25in]{geometry}

\pagestyle{empty}

\DeclareMathOperator{\speed}{speed}
\newcommand{\deriv}{\frac{d}{d\,t}}
\DeclarePairedDelimiter\abs{\lvert}{\rvert}

\begin{document}

\begin{center}
\textbf{Speeding up and slowing down}
\end{center}

\medskip

Consider an object moving on a line, with velocity function $v(t)$ and acceleration function $a(t)$.  To say the object is \textbf{speeding up} means that its speed is increasing; to say it is \textbf{slowing down} means that its speed is decreasing.

Now the speed at time $t$ is given by

\begin{equation*}
\speed(t) = \abs{v(t)} =
\begin{cases*}
\phantom{-\,}v(t) & if $v(t) > 0$,
\\
-\,v(t) & if $v(t) < 0$.
\end{cases*}
\end{equation*}
Hence its rate of change is
\begin{align*}
\deriv \speed(t) &=
\begin{cases*}
\phantom{-\,}v\,'(t) & if $v(t) > 0$,
\\
-\,v\,'(t) & if $v(t) < 0$,
\end{cases*}
\\
&=
\begin{cases*}
\phantom{-\,}a(t) & if $v(t) > 0$,
\\
-\,a(t) & if $v(t) < 0$.
\end{cases*}
\end{align*}
Then
\begin{align*}
\deriv \speed(t) > 0 &\quad\text{if } v(t) > 0 \text{ and } a(t) > 0
\\
                  &\quad\text{or if } v(t) < 0 \text{ and } -\,a(t) > 0,
\shortintertext{that is,}
\deriv \speed(t) > 0 &\quad\text{if } v(t) > 0 \text{ and } a(t) > 0
\\
                  &\quad\text{or if } v(t) < 0 \text{ and } a(t) < 0.
\shortintertext{Similarly,}
\deriv \speed(t) > 0 &\quad\text{if } v(t) > 0 \text{ and } a(t) < 0
\\
                  &\quad\text{or if } v(t) < 0 \text{ and } -\,a(t) < 0,
\shortintertext{that is,}
\deriv \speed(t) < 0 &\quad\text{if } v(t) > 0 \text{ and } a(t) < 0
\\
                  &\quad\text{or if } v(t) < 0 \text{ and } a(t) > 0.
\end{align*}

\noindent
Thus
\begin{align*}
\deriv \speed(t) > 0 &\quad \text{ if } v(t) \text{ and } a(t) \text{ have the same sign},
\\
\deriv \speed(t) < 0 &\quad \text{ if } v(t) \text{ and } a(t) \text{ have opposite signs}.
\end{align*}
So this means:
\begin{equation*}
\boxed{
\begin{aligned}
\text{object is speeding up} &\quad \text{ if } v(t) \text{ and } a(t) \text{ have the same sign},
\\
\text{object is slowing down} &\quad \text{ if } v(t) \text{ and } a(t) \text{ have opposite signs}.
\end{aligned}
}
\end{equation*}

\end{document}