Question

Solve: ${\left(\frac{1}{3}\right)}^2{3}^{x^2}-\frac{1}{{243}^x}<0$.

Answer 1

${\left(\frac{1}{3}\right)}^2{3}^{x^2}-\frac{1}{{243}^x}<0$

$\frac{3^{x^2}}{3^2}-\frac{1}{(3^5{)}^x}<0$

$3^{x^2-2}-3^{-5x}<0$

$3^{x^2-2}<3^{-5x}$

$x^2-2<-5x$

$x^2+5x-2<0$

$x<\frac{-5\pm \sqrt{33}}{2}$

$\therefore x<\frac{-5-\sqrt{33}}{2}$.