Question

Solve the following inequality:
${\log }_{1/2}\left({\log }_2\frac{x}{1+x}\right)>0$

Answer 1

${\log }_{1/2}\left({\log }_2\frac{x}{(1+x)}\right)>0$

$-{\log }_2{\log }_2\left(\frac{x}{1+x}\right)>0$

$\frac{x}{1+x}>0$

${\log }_2{\log }_2\left(\frac{x}{1+x}\right)<0x(1+x)>0$

$xϵ(-\infty ,-1)\bigcup (0,\infty )$

${\log }_2\frac{x}{(1+x)}<1$

$\frac{x}{(1+x)}<2$

$x(1+x)<{(1+x)}^2\left(2\right)$

$x^2+x<(1+2x+x^2)2$

$x^2+x<(2x^2+4x+2)$

$(x^2+3x+2)>0$

$(x+1)(x+2)>0$

$xϵ(-\infty ,-2)\bigcup (-1,\infty )$

Taking interaction we get $xϵ(-\infty ,-2)\bigcup (0,\infty )$