Logarithmic inequality $5\times \frac{3^{x-2}}{3^{x}-2^{x}}\geq1+(\frac{2}{3})^x$ Can somebody help me with this problem?
Solve inequation
$$5\times \frac{3^{x-2}}{3^{x}-2^{x}}\geq1+(\frac{2}{3})^x$$ 
I am trying to multiply both sides of inequation with
$$\frac{3^{x}}{3^{x}+2^{x}}$$
and I get
$$\frac{5\times3^{2x}}{9\times (3^{2x}-2^{2x})} \geq1$$
but I don t know how to continiue

You can rewrite it as follows:
$$5 \geq 
(1 + \left(\frac{2}{3}\right)^x) \frac{3^x - 2^x}{3^{x-2}}=
(1 + \left(\frac{2}{3}\right)^x) ( 9 - \frac{1}{9} \left(\frac{2}{3}\right)^x) $$
Now if we set $a = \left(\frac{2}{3}\right)^x$, we're reducing the problem to this binomial inequality:
$$(1 + a)(9 - \frac{1}{9}a) \leq 5$$
Knowing the range of $a$, you can obtain the range of $x$.
