Question

Resolva a inequação
$log_{ \frac{1}{3} }(x - 1) \geqslant log_{3}(4)$

Answer 1

$log_{\frac{1}{3}}(x-1) \geqslant log_34\\\\ \frac{log_3(x-1)}{log_3 \frac{1}{3}} \geqslant log_34\\\\ \frac{log_3(x-1)}{-1} \geqslant log_34\\\\ -log_3(x-1)} \geqslant log_34\\\\ log_3(x-1) \leqslant -log_34\\\\ log_3(x-1) \leqslant log_34^{-1}\\\\ log_3(x-1) \leqslant log_3 \frac{1}{4}\\\\ (x-1) \leqslant \frac{1}{4}\\\\ x \leqslant \frac{1}{4}+1\\\\ x \leqslant \frac{5}{4}$

Pela condição de existência:

$(x-1)\ \textgreater \ 0\\\\ x\ \textgreater \ 1\\\\ Assim:\\\\ x\ \textgreater \ 1 \ e \ x \leqslant \frac{5}{4}\\\\ ou \\\\ 1\ \textless \ x \leqslant \frac{5}{4}$