((1)/(9))^x-10((1)/(3))^x+3>0

To solve the inequality \( \left(\frac{1}{9}\right)^{x}-10\left(\frac{1}{3}\right)^{x}+3>0 \), we can start by making a substitution to simplify the expression. Let \( y = \left(\frac{1}{3}\right)^x \). Then, the inequality becomes .Next, we can solve the quadratic equation to find the critical points. The solutions to this equation are and .Now, we can test the intervals between these critical points to determine where the inequality is satisfied. We find that the inequality is satisfied when .Finally, we can substitute back \( y = \left(\frac{1}{3}\right)^x \) to find the solution in terms of . The solution to the inequality is \( x \log_{\frac{1}{3}}(5 + 2\sqrt{22}) \).