Find all values of $x$ satisfying

$$
\frac{x+\sqrt{x+1}}{x-\sqrt{x+1}}=\frac{11}{5}
$$
Solution

The equation can be rewritten as

$$
\begin{aligned}
\frac{x+\sqrt{x+1}}{x-\sqrt{x+1}} & =\frac{11}{5} \\
5 x+5 \sqrt{x+1} & =11 x-11 \sqrt{x+1} \\
16 \sqrt{x+1} & =6 x \\
8 \sqrt{x+1} & =3 x
\end{aligned}
$$

Squaring both sides of the last equation, we get

$$
\begin{aligned}
9 x^{2} & =64(x+1) \\
9 x^{2}-64 x-64 & =0
\end{aligned}
$$

The roots of this equation are $x=8$ and $x=-\frac{8}{9}$. Of these solutions, only $x=8$ satisfies the given equation.

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