Problem 386

Solution. Since

$$25^{x}=\left(5^{2}\right)^{x}=5^{2 x}, \quad 125^{3 x-2}=\left(5^{3}\right)^{3 x-2}=5^{9 x-6}$$

the given inequality is equivalent to the inequality

$$5^{2 x}>5^{9 x-6} \Leftrightarrow 2 x>9 x-6 \Leftrightarrow x<6 / 7$$

Therefore, the interval ( $-\infty ; 6 / 7$ ) is the set of all solutions to the original inequality.

The solution of any non-strict exponential inequality differs from the solution of the corresponding strict inequality only by including in the set of all its solutions the roots of the corresponding equation.