Solve |a+b|/2geqsqrt{|ab|}

$$\frac { | a + b | } { 2 } \geq \sqrt { | a b | }$$

Evaluate (complex solution)

$$\frac{|a+b|}{2}\geq \sqrt{|a||b|}$$

Solve for b

$$\left\{\begin{matrix}b\in [(2\sqrt{2}-3)a,\infty)\cup (-\infty,-(2\sqrt{2}+3)a]\text{, }&a>0\\b\neq -a\text{, }&a=0\\b\geq -(2\sqrt{2}+3)a\text{, }&a\leq 0\\b\leq (2\sqrt{2}-3)a\text{, }&a<0\end{matrix}\right.$$

Solve for a

$$\left\{\begin{matrix}a\in [-(2\sqrt{2}+3)b,\infty)\cup (-\infty,(2\sqrt{2}-3)b]\text{, }&b<0\\a\in (-\infty,-b)\cup \mathrm{R}\setminus -b\text{, }&b=0\\a\in [(2\sqrt{2}-3)b,\infty)\cup (-\infty,-(2\sqrt{2}+3)b]\text{, }&b>0\\a\in [(2\sqrt{2}-3)b,0]\text{, }&b\geq 0\end{matrix}\right.$$