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General inequality question

  1. Jan 14, 2009 #1

    Simfish

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    Gold Member

    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).
     
  2. jcsd
  3. Jan 14, 2009 #2

    HallsofIvy

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    Science Advisor

    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.
     
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