Example 1 (1999 National High School Competition Question) If $\left(\log _{2} 3\right)^{x}-\left(\log _{5} 3\right)^{x} \geqslant\left(\log _{2} 3\right)^{-y}-\left(\log _{5} 3\right)^{-y}$, then ().
A. $x-y \geqslant 0$
B. $x+y \geqslant 0$
C. $x-y \leqslant 0$
D. $x+y \leqslant 0$
Solution: Choose B. Reason: Since $0<\log _{5} 3<1<\log _{2} 3$, we know that $y_{1}=\left(\log _{2} 3\right)^{x}$ is an increasing function on $\mathbf{R}$.
Furthermore, since $\left(\log _{5} 3\right)^{x}$ is a decreasing function on $\mathbf{R}$, we know that $y_{2}=-\left(\log _{5} 3\right)^{x}$ is an increasing function on $\mathbf{R}$, thus $y=$ $y_{1}+y_{2}=\left(\log _{2} 3\right)^{x}-\left(\log _{5} 3\right)^{x}$ is an increasing function on $\mathbf{R}$.
Given $\left(\log _{2} 3\right)^{x}-\left(\log _{5} 3\right)^{x} \geqslant\left(\log _{2} 3\right)^{-y}-\left(\log _{5} 3\right)^{-y}$, we get $x \geqslant -y$, which means $x+y \geqslant 0$.