Answer
$5.1$ hours
Work Step by Step
Let's note:
$x$=the time (in hours) the outlet pipe needs to empty the pool when the inlet pipe does not work
$y$=the time (in hours) the inlet pipe needs to fill the pool when the outlet pipe does not work
In one hour the outlet pipe empties $1/x$ from the pool, while the inlet pipe fills $1/y$ from the pool, so when they work together in one hour they fill $(1/y)-(1/x)$ which is $1/8$ from the pool.
Write the two equations:
$$\begin{cases}
\dfrac{1}{y}-\dfrac{1}{x}&=\dfrac{1}{8}\\
x&=y+2.
\end{cases}$$
Multiply the first equation by $8xy$:
$$\begin{cases}
8x-8y&=xy\\
x&=y+2.
\end{cases}$$
Substitute $x=y+2$ in the first equation and solve for $y$ using the Quadratic Formula:
$$\begin{align*}
8x-8y&=(y+2)y\\
8(y+2)-8y&=y^2+2y\\
16&=y^2+2y\\
y^2+2y-16&=0\\
y&=\dfrac{-2\pm\sqrt{(2^2-4(1)(-16)}}{2(1)}\\
&\approx \dfrac{-2\pm 8.246}{2}\\
y_1&\approx -5.1\\
y_2&\approx 3.1.
\end{align*}$$
Because $y>0$, the only solution that fits is $y=3.1$.
Determine the time (in hours) the outlet pipe needs to empty the pool alone:
$$x=y+2=3.1+2=5.1.$$
