Help with inequality with one unknown

Question

Please could you help how to solve the inequality $(\sqrt{x-9})(2^{x-8}+3^{x-9}-9)\geq 0$

Answer 1

Let $x = t+9$, then the radical needs $t \geqslant 0$. $t=0$ is a solution obviously, for others, we are left to solve $2\cdot 2^t+3^t\geqslant 9$. Note LHS is increasing, and starting from $3<9$, so there is a unique $a>0$ s.t. $t \in [a, \infty)$ are all solutions, i.e. the full solution set is $x\in \{9\}\cup[a+9, \infty)$, where $2\cdot 2^a+3^a=9$.

To find $a$, you will need to use numerical methods, it’s about $1.288...$.