Example 4. Solve the inequality

$$
2^{x+2}-2^{x+3}-2^{x+4}>5^{x+1}-5^{x+2}
$$
Solution. Since

$$
\begin{gathered}
2^{x+2}=4 \cdot 2^{x}, \quad 2^{x+3}=8 \cdot 2^{x}, \quad 2^{x+4}=16 \cdot 2^{x} \\
5^{x+1}=5 \cdot 5^{x}, \quad 5^{x+2}=25 \cdot 5^{x}
\end{gathered}
$$

then the given inequality is equivalent to the inequality

$$
2^{x}(4-8-16)>5^{x}(5-25) \Leftrightarrow 2^{x}(-20)>5^{x}(-20) \Leftrightarrow 2^{x}<5^{x}
$$

Thus, the interval $(0 ;+\infty)$ is the set of all solutions to the given inequality.