Any non-zero real numbers x,y such that $$y\neq3$$ and $$\frac{x}{y}<\frac{x+3}{y-3}$$, Will satisfy the condition.
It is given thatĀ $$\frac{x}{y}<\ \frac{\ x+3}{y-3}$$, which can be written asĀ $$\frac{x}{y}-\frac{\ x+3}{y-3}<0$$
=>Ā $$\ \frac{\ x\left(y-3\right)-y\left(x+3\right)}{y\left(y-3\right)}<0$$
=>Ā $$\ \frac{\ xy-3x-xy-3y}{y\left(y-3\right)}<0$$
=>Ā $$\ \frac{\ -3\left(x+y\right)}{y\left(y-3\right)}<0$$
=>Ā $$\ \frac{\ 3\left(x+y\right)}{y\left(y-3\right)}>0$$
From this inequality, we can say that, when $$y<0=>y(y-3)>0$$. Now to satisfy the given equation $$\ \frac{\ 3\left(x+y\right)}{y\left(y-3\right)}>0$$,Ā
$$(x+y)$$ must be greater than zero Hence, $$x>0$$ andĀ $$\left|x\right|\ >\left|y\right|$$
Therefore, the magnitude of $$x$$ is greater than the magnitude of $$y$$.Ā
Hence, $$x>y$$, andĀ $$\left|x\right|\ >\ \left|y\right|$$ =>$$-x\ <\ y$$Ā (Since the magnitude of $$x$$ is greater than the magnitude of $$y$$.)
The correct option is B.
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