Question

Solve the following inequality:
$\sqrt{9^x+3^x-2}⩾9-3^x$

Answer 1

$\sqrt{9^x+3^x-2}\ge 9-3^x$

$let{3}^x=t$

$\sqrt{t^2+t-2}\ge 9-t$

$t^2+t-2\ge {(9-t)}^2$

$t^2+t-2\ge t^2+81-18t$

$19t\ge 83$

$t\ge \frac{83}{19}$

$3^x\ge \frac{83}{19}$

$x\log 3\ge \log \left(\frac{83}{19}\right)$

$x\ge \frac{\log 83-\log 19}{\log 3}$