TITLE: Verify and Improve: $|x+\frac{1}{x}|\geq 6$
QUESTION [5 upvotes]: Exercise:

Determine the intervals in which the following inequality is satisfied:
  $$|x+\frac{1}{x}| \geq 6$$


Attempt:
I apologize for not MathJax-ifying this; the following easy-to-read formatting is beyond my expertise.


Request:
Can I improve upon the solution to make it more elegant and concise?

REPLY [3 votes]: My answer tries to explain more precisely the symmetry involved.
Let's consider:
$$
f(x) \equiv x+\frac{1}{x}=-f(-x)
$$
Hence, the LHS of the inequality is an odd function of x. So, under an absolute value:
$$
|f(x)|=|f(-x)|\implies |f(x)|=|f(|x|)| \,\,\,\forall x
$$
Hence, you can replace $x\to|x|$ and solve accordingly as other user have suggested.