Is there a mathematical step I'm unconsciously skipping in this inequality

Question

I have the following expression

$$(xy-1)p<0$$ By assuming that $ p>0$ I'll then have

$$xy-1<0$$

which means that $$y<\frac{1}{x}$$

Now if I know make $x=-1$ and $p=1$ I will have

$$-y-1<0$$

which means that $$y>-1$$

However if I replace $x$ in my final answer for the general case I will have $$y<-1$$

What is this paradox? My guess is that I'm unconsciously skipping a math step but I don't know what. Can someone clarify me please?

Answer 1

You cannot divide both sides of the inequality by x cause it's negative, so the sign of the inequality must change

Answer 2

You gave $x$ a negative value. $$xy-1 < 0 \implies xy < 1$$ $$y < \frac{1}{x} \iff x > 0$$ $$y > \frac{1}{x} \iff x < 0$$ In your work, you forgot to mention these two cases, so it would be true only if $x > 0$ (case $1$). If $x < 0$, you would have to reverse the inequality sign since you’re dividing by a negative value (case $2$).