Question

Solve the following inequality:
${\log }_2\frac{3x-1}{2-x}<1$

Answer 1

${\log }_2\left(\frac{3x-1}{2-x}\right)<1$

$\frac{(3x-1)}{(2-x)}<2$ also, $\frac{(3x-1)}{(2-x)}>0$

$(3x-1)(2-x)<2{(2-x)}^2(3x-1)(2-x)>0$

$((3x-1)-2(2-x))(2-x)<0(x-2)(3x-1)<0$

$(3x-1-4+2x)(2-x)<0xϵ(\frac{1}{3},2)$

$(5x-5)(2-x)<0$

$5(x-2)(x-1)>0$

$xϵ(-\infty ,1)\bigcup (2,\infty )$

Taking intersection we get $xϵ(\frac{1}{3},1)$