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.
why can we not cancel out m by dividing it?
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