Which is bigger $\frac{1}{{\sqrt{1001}}+ {\sqrt{1000}}}$ or $\frac {1}{10}$. What is the best way to find the inequality? I know the answer but my solution is messy.
Using the hint in the comments. Since
\begin{align}
\sqrt{1001} + \sqrt{1000} &amp;&gt;\sqrt{1000}+\sqrt{1000}\\
&amp;=10\sqrt{10}+10\sqrt{10}\\
&amp;=20\sqrt{10}\\
&amp;&gt;20
\end{align}
We have that
$$\frac{1}{\sqrt{1001}+\sqrt{1000}}&lt;\frac{1}{20}&lt;\frac{1}{10}.$$
For the time being, leave the relation $@$ undefined.
$\frac1{10}@\frac1{\sqrt{1001}+\sqrt{1000}}$
$\sqrt{1001}+\sqrt{1000}@10$
Compute $\sqrt{1001}$ and $\sqrt{1000}$.
Call their sum $x$.
$x@10$
Should $@$ be $\lt$, $=$ or $\gt$?