Question

solve for the value of x :both 18 and 20​

Answer 1

We have

$log_{ \frac{1}{2} } \bigg \{ log_{3} \bigg( \frac{x + 1}{x - 1} \bigg) \bigg \} \geqslant 0$

$\implies log_{3} \bigg( \frac{x + 1}{x - 1} \bigg) \leqslant 1$

$\implies \frac{x + 1}{x - 1} \leqslant 3$

$\implies \frac{x + 1}{x - 1} - 3 \leqslant 0$

$\implies \frac{x + 1 - 3x + 3}{x - 1} \leqslant 0$

$\implies \frac{ - 2x + 4}{x - 1} \leqslant 0$

$\implies \frac{ 2x - 4}{x - 1} \geqslant 0$

$\implies \frac{ 2(x - 2)}{x - 1} \geqslant 0$

$\implies \frac{x - 2}{x - 1} \geqslant 0$

$\implies \: x \in( - \infty ,1) \: \cup \: [ 2, \infty )$