Does the inequality still hold if I invert both sides with cyclic sums?

Question

For example if I invert both sides of

$$ \sum_{\sigma} \frac{x(x+2)}{2x^2 + 1} \geq 0$$

So that it becomes

$$\sum_{\sigma}\frac{2x^2+1}{x(x+2)} \leq 0$$

Will the inequality hold?

Answer 1

Hint:

Does $x\geq 0$ imply $\dfrac{1}{x}\leq 0$?