Example 7. Solve the inequality

$$
\frac{1}{5} \cdot 5^{2 x} 7^{3 x+2} \leq \frac{25}{7} \cdot 7^{2 x} 5^{3 x}
$$
Solution. Dividing both sides of the inequality (2) by the expression $\frac{25}{7} \cdot 7^{2 x} 5^{3 x}$, which is positive for any real $x$, we obtain the inequality

$$
\frac{5^{2 x-1} 7^{3 x+2}}{5^{2} 7^{2 x-1} 5^{3 x}} \leqslant 1
$$

equivalent to (2), i.e., the inequality

$$
7^{x+3} 5-x-3<1
$$

The last inequality is equivalent to the inequality

$$
\left(\frac{7}{5}\right)^{x+3} \leqslant\left(\frac{7}{5}\right)^{0} \Leftrightarrow x+3 \leqslant 0 \Leftrightarrow x \leqslant-3
$$

Thus, the set of all solutions to the inequality (2) is the interval $(-\infty ;-3]$.