<s>[INST] Solving $5\times9^x-8\times15^x+3\times25^x>0$ Question: Solve the inequality:
$5\times9^x-8\times15^x+3\times25^x>0$
So far I have managed to factorise the inequality:
$(5\times3^x-3\times5^x)(3^x-5^x)>0$
From here on, I am stuck.
I would very much appreciate it if you could answer this exam question and lay the working out as if it were an exam. I have an answer from the solutions book so can tell you if your answer is correct using that. Many thanks.
 [/INST] When both factors are positive, the intersection of $x < 0$ and $x < 1$ is $x < 0$. But when both factors of $(5\times3^x-3\times5^x)(3^x-5^x)$ are negative, their product is still positive, so we must consider this case as well.
$5\times3^x-3\times5^x < 0$ gives $\left( \frac{3}{5}\right)^{x-1} < \left( \frac{3}{5}\right)^{0}$ like before. However, since $3/5 < 1$, the $y$-values are getting smaller as the $x$-values increase. Drawing a sketch of the exponential function and the curve, the inequality sign flips to give $x - 1 > 0, x > 1$.
Similarly, $3^x - 5^x < 0$ leads to $3^x < 5^x$ or $\left( \frac{3}{5}\right)^x < \left( \frac{3}{5}\right)^{0}$, and the inequality sign flips again, thus $x > 0$. Since both factors need to be negative for the product to be positive, we need to take the intersection of $x > 1, x > 0$ which is $x > 1$.
Altogether, we have $x < 0$ and $x > 1$.
</s>