What is the fastest method to find which of $\frac {3\sqrt {3}-4}{7-2\sqrt {3}} $ and $\frac {3\sqrt {3}-8}{1-2\sqrt {3}} $ is bigger manually? What is the fastest method to find which number is bigger manually?
$\frac {3\sqrt {3}-4}{7-2\sqrt {3}} $ or $\frac {3\sqrt {3}-8}{1-2\sqrt {3}} $

Claim: 
$$\frac {3\sqrt {3}-4}{7-2\sqrt {3}}<\frac {3\sqrt {3}-8}{1-2\sqrt {3}}$$
Proof:
Let $u=\sqrt 3-1$. Notice $1>u>\frac12$.
If
$$\frac{3u-1}{5-2u}<\frac{3u-5}{-2u-1}$$
$$\Leftarrow-6u^2-u+1>-6u^2+25u-25$$
$$\Leftarrow26>26u$$
$$\Leftarrow1>u=\sqrt3-1$$
which is true.
