Answer
$51$ miles per hour or $59$ miles per hour
Work Step by Step
William wants to get $22$ mile per gallon during his trip. We therefore substitute $22$ for $m$ and solve for $s$. $$\begin{aligned}
m(s) &= -\frac{1}{2} \lvert s-55\rvert+24\\
22 &= -\frac{1}{2} \lvert s-55\rvert+24\\
(-2)\cdot 22 &=(-2)\cdot -\frac{1}{2} \lvert s-55\rvert+(-2)\cdot 24\\
-44 &= \lvert s-55\rvert-48\\
48-44&= \lvert s-55\rvert\\
4&= \lvert s-55\rvert.
\end{aligned}$$ Rewrite into two equations and solve.
$$\begin{aligned}
s-55&= 4\\
& = 4+55\\
& = 59
\end{aligned}$$ or $$\begin{aligned}
s-55&= -4\\
& = -4+55\\
& = 51.
\end{aligned}$$ Check the solution. $$\begin{aligned}
m(59) &= -\frac{1}{2} \lvert 59-55\rvert+24\\
& = -\frac{1}{2} \lvert 4\rvert+24\\
& = -\frac{1}{2}\cdot 4+25\\
& = -2+24\\
& = 22
\end{aligned}$$ and $$\begin{aligned}
m(51) &= -\frac{1}{2} \lvert 51-55\rvert+24\\
& = -\frac{1}{2} \lvert -4\rvert+24\\
& = -\frac{1}{2}\cdot 4+24\\
& = -4+24\\
& = 22.
\end{aligned}$$ Hence, William must drive at $51$ miles per hour or at $59$ miles per hour to achieve a $20$ miles per gallons during his journey.
