9. 25 Let $a, b, c$ be real numbers. Then, for any real number $x$, the inequality
$$
a \sin x + b \cos x + x > 0
$$

always holds if and only if
(A) $a, b$ are both 0, and $c > 0$.
(B) $\sqrt{a^{2}+b^{2}} = c$.
(C) $\sqrt{a^{2}+b^{2}} < c$.
(China High School Mathematics League, 1994)
［Solution］When $a, b$ are not both zero, we have
$$
\begin{array}{l}
a \sin x + b \cos x + c > 0 \\
\Leftrightarrow \sqrt{a^{2} + b^{2}} \sin (x + \varphi) + c > 0 \\
\Leftrightarrow \sin (x + \varphi) > -\frac{c}{\sqrt{a^{2} + b^{2}}}.
\end{array}
$$
The necessary and sufficient condition for (2) to hold is
$-\frac{c}{\sqrt{a^{2} + b^{2}}} < 1$, which is included in (C).
Therefore, the answer is $(C)$.