((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 \( y^2 - 10y + 3 > 0 \).<br /><br />Next, we can solve the quadratic equation \( y^2 - 10y + 3 = 0 \) to find the critical points. The solutions to this equation are \( y = 5 - 2\sqrt{22} \) and \( y = 5 + 2\sqrt{22} \).<br /><br />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 \( y < 5 - 2\sqrt{22} \) or \( y > 5 + 2\sqrt{22} \).<br /><br />Finally, we can substitute back \( y = \left(\frac{1}{3}\right)^x \) to find the solution in terms of \( x \). The solution to the inequality is \( x < \log_{\frac{1}{3}}(5 - 2\sqrt{22}) \) or \( x > \log_{\frac{1}{3}}(5 + 2\sqrt{22}) \).