Example 6. Solve the inequality

$$
\sqrt{6-x}\left(5^{x^{2}-7.2 x+3.9}-25 \sqrt{5}\right) \geqslant 0
$$
Solution. The domain of definition of inequality (1) is determined by the condition $6-x \geqslant 0$, i.e., $x \leqslant 6$. Inequality (1) is equivalent to a combination consisting of an equation and a system of two inequalities:

$$
\sqrt{6-x}=0, \quad\left\{\begin{array}{l}
5^{x^{2}-7.2 x+3.9}-25 \sqrt{5} \geqslant 0 \\
x<6
\end{array}\right.
$$

From the equation $\sqrt{6-x}=0$ we find $x=6$.

Since $25 \sqrt{5}=5^{2} \cdot 5^{1 / 2}=5^{2.5}$, the first inequality of the system is equivalent to the inequality

$$
5^{x^{2}-7.2 x+3.9} \geqslant 5^{2.5}
$$

which is equivalent to the inequality

$x^{2}-7.2 x+3.9 \geqslant 2.5 \Leftrightarrow x^{2}-7.2 x+1.4 \geqslant 0 \Leftrightarrow(x-1 / 5)(x-7) \geqslant 0$.

Therefore, the first inequality of the system is equivalent to the combination of simple inequalities

$$
x \leqslant 1 / 5, \quad x \geqslant 7
$$

Considering the condition $x<6$, we obtain the solution of the system, which is the interval $-\infty<x \leqslant 1 / 5$.

Therefore, the solutions of inequality (1) are the number $x=6$ and all numbers in the interval $-\infty<x \leqslant 1 / 5$.