CAT 2023 Slot 2 QA Question 3: Any non-zero real numbers $x$, $y$ such that $y
eq 3$ and $\dfrac{x}{y}

Question

Any non-zero real numbers $x$, $y$ such that $y 
eq 3$ and $\dfrac{x}{y} < \dfrac{x + 3}{y - 3}$, will satisfy the condition

Accepted Answer

Bring the two fractions to one side and simplify.

$\dfrac{x + 3}{y - 3} - \dfrac{x}{y} > 0$
$\dfrac{y(x + 3) - x(y - 3)}{y(y - 3)} > 0$
$\dfrac{3(x + y)}{y(y - 3)} > 0$
$\dfrac{x + y}{y(y - 3)} > 0$

Now consider the sign of $y$.

If $y > 3$, then $y(y - 3)$ is positive. Hence, $x + y > 0$, so $-x

Any non-zero real numbers xxx, yyy such that y≠3y \neq 3y=3 and xy<x+3y−3\dfrac{x}{y} < \dfrac{x + 3}{y - 3}yx​<y−3x+3​, will satisfy the condition