Logarithmic inequality $5\times \frac{3^{x-2}}{3^{x}-2^{x}}\geq1+(\frac{2}{3})^x$

Question

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

Answer 1

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 $x$.