## Task B-4.1.

Solve the inequality $3^{3 x}+3^{4} \leqslant 3^{2 x+2}+3^{x+2}$.
## Solution.

Let's introduce the substitution $t=3^{x}$. Then the given inequality can be written as

$$
t^{3}-9 t^{2}-9 t+81 \leqslant 0
$$

If we factor out $t$ from the first and third terms, and 9 from the second and fourth terms, respectively, we get:

$$
\begin{aligned}
& t\left(t^{2}-9\right)-9\left(t^{2}-9\right) \leqslant 0 \\
& (t-9)\left(t^{2}-9\right) \leqslant 0 \\
& (t-9)(t-3)(t+3) \leqslant 0
\end{aligned}
$$

Since $t+3$ is always positive, it follows that $(t-9)(t-3) \leqslant 0$.

The solution to this inequality is $t \in[3,9]$,

so the solution to the given inequality is $x \in[1,2]$.