Understanding Inequalities: Division by Negative Numbers

[Simfish]

So, say, we have the inequality ty <= 1-sx

Does this inequality always translate to y <= 1/t - s/tx (for t > 0)
y >= 1/t - s/tx (for t < 0)? (due to division by divided signs?)

I'm sure this is true, as I've tested it for xy <= 1. But I just want to be sure, since this nuance is definitely easy to miss (and requires a step that I'm usually not accustomed to taking so far).

 

[HallsofIvy]

Yes, If $a\le b$ and c> 0 then $ac\le bc$ while if c< 0, $ac\ge bc$. If t> 0 then so is 1/t and if t< 0 so is 1/t: dividing by t is the same as multiplying by 1/t.