If $A > 0, |x| < \sqrt{A}$ prove that $\frac{1}{2}\left|x + A\frac{1}{x}\right| > \sqrt{A}$

Question

If $A > 0, |x| < \sqrt{A}$ prove that $\frac{1}{2}\left|x + A\frac{1}{x}\right| > \sqrt{A}$

My work:

\begin{align*} |x| <& \sqrt{A} \\ \frac{1}{|x|} >& \frac{1}{\sqrt{A}} \\ A\frac{1}{|x|} >& \sqrt{A} \\ \end{align*}

\begin{align*} \frac{1}{2}\left|x + A\frac{1}{x}\right| &\le \frac{1}{2} \left(|x| + A \frac{1}{|x|}\right)\\ \end{align*}

Answer 1

squaring the given inequality we get $$\frac{1}{4}\left(x^2+\frac{A^2}{x^2}+2A\right)\geq 4A$$and this is equivalent to $$\left(x-\frac{A}{x}\right)^2\geq 0$$ since $$|x|<\sqrt{A}$$ it must be $$A>0$$