Simple inequality question with division.

Question

In this inequality, why do the typical rules for inequalities not hold up? $$(x/y) > 1 \quad x > y$$

However, it leaves out : $$-x < -y$$

When there is division involved in inequalities, do I need to be extra careful?

Answer 1

Yes, you need to be careful when multiplying OR dividing an inequality by a variable, in this case $y$, since, when $y \lt 0$, it changes the direction of the inequality. When $y \gt 0$, the direction remains as is. And of course, we want to "rule out" or consider the case $y = 0$.

$$\dfrac xy > 1, \;y\neq 0 \implies \begin{cases} x > y, & y > 0 \\ \\ x < y, & y < 0\end{cases}$$

So your initial work leaves out the case when $y \lt 0$.

Note that $-x \lt -y \iff x \gt y$, so be careful with your use of the negative sign to indicate when a variable is less than 0.