Question

If $\left\{x\in N:2+{\log }_2\sqrt{x+1}>1-{\log }_{1/2}\sqrt{4-x^2}\right\}$, then

Answer 1

$2+{\log }_2\sqrt{x+1}>1-{\log }_{1/2}\sqrt{4-x^2}$
$1+{\log }_2\sqrt{x+1}>{\log }_2\sqrt{4-x^2}$
${\log }_{2}2+{\log }_2\sqrt{x+1}>{\log }_2\sqrt{4-x^2}$

${\log }_2\left(\sqrt{4(x+1)}\right)>{\log }_2\sqrt{4-x^2}$
$4x+4>4-x^2$
$x^2+4x>0$
$⟹x>0$ and $x+4>0$