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Math Help - Proving a certain range such that absolute value exists

  1. #1
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    Proving a certain range such that absolute value exists

    Prove that x<-a or x>a for abs(x)>a where a>0.

    Is my proof acceptable?

    abs(x)>a

    x^2>a^2

    x^2-a^2>0

    (x-a)(x+a)>0

    Therefore, x<-a or x>a
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  2. #2
    Super Member Bacterius's Avatar
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    Re: Proving a certain range such that absolute value exists

    Yes it is correct, make sure you clearly state the identity |x| = \sqrt{x^2}.
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  3. #3
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    Re: Proving a certain range such that absolute value exists

    Hi. Thanks for your reply.

    What if the question is phrased this way :

    Prove that abs(x)>a if and only if x>a or x<-a. (where a is any positive number).

    Will the phrasing of the question in such a way change the required proof?
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  4. #4
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    Re: Proving a certain range such that absolute value exists

    Hi. This is the provide solution. Is this correct? I feel that (i) should be : prove | x | > a instead of | x | > 0 . The same goes for (ii).

    i) Prove | x | > 0 => x > a or x < a for any positive number, a.

    For x greater or equal 0, | x | = x. | x | > a => x > a.

    For x less than 0, | x | = -x. | x | > a => -x > a => x < -a.

    ii) Prove x > a or x < a => | x | > 0 for any positive number, a.

    a > 0 and x > a => | x | = x. So x > a | x | > a.

    For a > 0, -a < 0 and x < -a => x < 0 => | x | = -x. So x < -a => -x > a => | x | > a.
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