$3^{|x|}-8-\frac{3^{|x|}+9}{9^{|x|}-4\cdot 3^{|x|}+3}\le \frac{5}{3^{|x|}-1}.$

$\text{Пусть } 3^{|x|}=t. \text{ Тогда неравенство примет вид:}$

$t-8-\frac{t+9}{t^2-4t+3}-\frac{5}{t-1}\le 0 \iff t-8-\left(\frac{t+9}{(t-1)(t-3)}+\frac{5}{t-1}\right)\le 0 \iff$ $\iff t-8-\frac{t+9+5t-15}{(t-1)(t-3)}\le 0 \iff t-8-\frac{6(t-1)}{(t-1)(t-3)}\le 0 \iff$ $\iff \left[ \begin{aligned} &\frac{t^2-3t-8t+24-6}{t-3}\le 0,\\ &t-1\ne 0 \end{aligned} \right. \iff \left[ \begin{aligned} &\frac{t^2-11t+18}{t-3}\le 0,\\ &t\ne 1 \end{aligned} \right. \iff \left[ \begin{aligned} &\frac{(t-2)(t-9)}{t-3}\le 0,\\ &t\ne 1. \end{aligned} \right.$

$\text{Решим полученную систему:}$

$\left[ \begin{aligned} &\left\{ \begin{aligned} &t\ne 1,\\ &t\le 2 \end{aligned} \right.\\ &3<t\le 9 \end{aligned} \right. \iff \left[ \begin{aligned} &\left\{ \begin{aligned} &3^{|x|}\ne 3^0,\\ &3^{|x|}\le 3^{\log_3 2} \end{aligned} \right.\\ &3^1<3^{|x|}\le 3^2 \end{aligned} \right. \iff$ $\iff \left[ \begin{aligned} &\left\{ \begin{aligned} &|x|\ne 0,\\ &|x|\le \log_3 2 \end{aligned} \right.\\ &1<|x|\le 2 \end{aligned} \right. \iff \left[ \begin{aligned} &\left\{ \begin{aligned} &x\ne 0,\\ &-\log_3 2\le x\le \log_3 2 \end{aligned} \right.\\ &-2\le x<-1,\ \ 1<x\le 2. \end{aligned} \right.$

$\text{Таким образом, решением исходного неравенства является:}$

$x\in[-2;-1)\cup[-\log_3 2;0)\cup(0;\log_3 2]\cup(1;2].$

$\textbf{Ответ: } [-2;-1)\cup[-\log_3 2;0)\cup(0;\log_3 2]\cup(1;2].$

Задание №30109 ЕГЭ по Математике