Question

Solve the following inequality:
$(0.5{)}^{{\log }_3{\log }_{1/5}(x^2-\frac{4}{5})}<1$

Answer 1

$2^{-{\log }_3{\log }_{1/5}(x^2-\frac{4}{5})}<1$

Taking ${\log }_2$ both sides

$-{\log }_3{\log }_{1/5}(x^2-\frac{4}{5})<0{\log }_3{\log }_{1/5}(x^2-\frac{4}{5})>0{\log }_{1/5}(x^2-\frac{4}{5})>1-\frac{\log (x^2-\frac{4}{5})}{\log 5}>1-\log (x^2-\frac{4}{5})>\log 5\log (x^2-\frac{4}{5})<\log \frac{1}{5}(x^2-\frac{4}{5})<\frac{1}{5}{x}^2-1<0x\in (-1,1)$

Also, $(x^2-\frac{4}{5})>0$

$x\in (-\infty ,\frac{-2}{\sqrt{5}})\cup (\frac{2}{\sqrt{5}},\infty )$

Taking intersection , we get $x\in (-1,\frac{-2}{\sqrt{5}})\cup (\frac{2}{\sqrt{5}},1)$