Question

Qual é o conjunto solução da inequação
$log_{ \frac{1}{2} }(x - 1) - log_{ \frac{1}{2} }(x + 1) < log_{ \frac{1}{2} }(x - 2)+1$

Answer 1

$\log _{\frac{1}{2}}\left(x-1\right)-\log _{\frac{1}{2}}\left(x+1\right)\ \textless \ \log _{\frac{1}{2}}\left(x-2\right)+1$

⇒$\log _{\frac{1}{2}}\left(\frac{x-1}{x+1}\right)\:\ \textless \ \log _{\frac{1}{2}}\left(x-2\right)+1$

⇒$\log _{\frac{1}{2}}\left(\frac{x-1}{x+1}\right)-\log \:_{\frac{1}{2}}\left(x-2\right)\:\ \textless \ 1$

⇒$\log \:_{\frac{1}{2}}\left(\frac{\frac{x-1}{x+1}}{x-2}\right)\ \textless \ 1$

⇒$\log \:_{\frac{1}{2}}\left(\frac{x-1}{\left(x+1\right)\left(x-2\right)}\right)\ \textless \ 1$

⇒$\frac{x-1}{\left(x+1\right)\left(x-2\right)}\ \textless \ \left(\frac{1}{2}\right)^1$

⇒$\frac{x-1}{\left(x+1\right)\left(x-2\right)}-\frac{1}{2}\ \textless \ 0$

⇒$\frac{-x^2+3x}{2\left(x+1\right)\left(x-2\right)}\ \textless \ 0$

Ache os valores de x que satisfaçam a inequação:

⇒$-x\left(x-3\right)\ \textless \ 0$ ⇒ $x\ \textless \ 0 ou x\ \textgreater \ 3$

⇒$x-2\ \textless \ \:0$⇒ $x\ \textless \ 2$

⇒$x+1$⇒$x\ \textless \ -1$

Fazendo a intersecção na reta real, temos que S={X e Rl $2\ \textgreater \ x\ \textgreater \ 3$}