That is because we don’t know whether $m$ is positive or not. If we knew that $m$ was positive, then we could divide by $m$ and conclude that $x \gt y$. In general, one cannot divide an inequality by a variable without knowing if it is positive or negative.

Another way to think about this is by looking at an example with numbers. Suppose we have the inequality $6 \gt 3$, if we divide both sides by $-3$ we would get $-2 \gt -1$, which is incorrect, and to correct the outcome we have to flip the sign, or $-2 \lt -1$. So in general, if you choose to divide an inequality by a quantity it is important to know if it is positive or negative so one can change the greater than or less than sign appropriately.

Suzanne Shrestha says

why can we not cancel out m by dividing it?

GMAT Quantum says

Hi Suzanne,

That is because we don’t know whether $m$ is positive or not. If we knew that $m$ was positive, then we could divide by $m$ and conclude that $x \gt y$. In general, one cannot divide an inequality by a variable without knowing if it is positive or negative.

Another way to think about this is by looking at an example with numbers. Suppose we have the inequality $6 \gt 3$, if we divide both sides by $-3$ we would get $-2 \gt -1$, which is incorrect, and to correct the outcome we have to flip the sign, or $-2 \lt -1$. So in general, if you choose to divide an inequality by a quantity it is important to know if it is positive or negative so one can change the greater than or less than sign appropriately.

I hope this makes sense.

Dabral