Understanding Inequalities: Division by Negative Numbers

In summary, the inequality ty <= 1-sx can be translated to y <= 1/t - s/tx (for t > 0) and y >= 1/t - s/tx (for t < 0) due to the division by negative signs. This holds true for xy <= 1 and follows the rule that if a <= b and c > 0, then ac <= bc while if c < 0, ac >= bc. Additionally, when t > 0, 1/t is also positive and when t < 0, 1/t is also negative, making the division by t equivalent to multiplying by 1/t.
  • #1
Simfish
Gold Member
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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).
 
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  • #2
Yes, If [itex]a\le b[/itex] and c> 0 then [itex]ac\le bc[/itex] while if c< 0, [itex]ac\ge bc[/itex]. 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.
 

FAQ: Understanding Inequalities: Division by Negative Numbers

What is general inequality?

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