Example 1 If
$$
\left(\log _{2} 3\right)^{x}-\left(\log _{5} 3\right)^{x} \geqslant\left(\log _{2} 3\right)^{-y}-\left(\log _{5} 3\right)^{-y} \text {, }
$$
then ( ).
(A) $x+y \geqslant 0$
(B) $x-y \geqslant 0$
(C) $x+y \leqslant 0$
(D) $x-y \leqslant 0$
(1999, National High School Mathematics Joint Competition)
【Analysis】Comparing the left and right sides of the given equation, we find that the bases are the same, but the exponents are different. Therefore, we construct the function
$$
f(x)=\left(\log _{2} 3\right)^{x}-\left(\log _{5} 3\right)^{x} \text {. }
$$

It is easy to see that the constructed function is monotonically increasing.
Thus, the given inequality can be transformed into
$$
f(x) \geqslant f(-y) \text {. }
$$

Therefore, $x \geqslant -y$, which means $x+y \geqslant 0$. Choose (A).