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Math Help - Simple Inequality

  1. #1
    Super Member craig's Avatar
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    Simple Inequality

    Just a quick question on a pretty simple inequality question:

    |1-\frac{1}{x}|<1

    -1 < 1-\frac{1}{x}<1

    -2 < \frac{-1}{x}<0

    0 < \frac{1}{x}<2

    Up to here is simple enough. Now if I take the reciprocal of both sides, I can't with the zero on the left had side, so would I just consider:

    \frac{1}{x}<2

    So x > \frac{1}{2}?

    Thanks in advance
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  2. #2
    Senior Member nikhil's Avatar
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    if 1/x>0 then
    x should be >0 (because reciprocal of a positive number is always positve)
    now if 1/x<2 then
    x>1/2
    see
    1/x<2
    (1/x)-2<0
    (1-2x)/x<0
    since x can't be 0 multiplying both side by x
    1-2x<0
    2x>1
    x>1/2
    Last edited by nikhil; May 31st 2010 at 08:15 PM.
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  3. #3
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    Although \frac{1}{x} is NOT "infinity", it is true that \lim_{x\to 0}\frac{1}{x}= \infty so you can write this as \frac{1}{2}< x< \infty which is the same as just saying x> \frac{1}{2}.
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  4. #4
    Super Member craig's Avatar
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    Ahh I get you, cheers for the replies.
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