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Triangular inequality

  1. Dec 28, 2016 #1
    1. The problem statement, all variables and given/known data
    Hi guys, I would just like someone to go over my method for this derivation/proof ( not sure of the right word to use here). Anyway I think this is right method, but just feel like I am missing something. Could someone please check my method. Thanks in advance.

    2. Relevant equations
    $$|x|= x\geq 0 , -x < 0 $$

    $$|a-b|\leq|a|-|b|$$

    3. The attempt at a solution
    By using the formal definition of the absolute value I get this:

    1.$$-|a|\leq a\leq |a|$$
    2.$$-|b|\leq b\leq |b|$$

    1-2: $$-(|a|-|b|)\leq a-b \leq |a|-|b| $$

    Therefore I get: $$|a-b|\leq|a|-|b|$$

    Is this correct? Is there any improvements that anyone could share. I do have a couple more variations of the triangle inequality to go through but want to try the first before posting.

     
  2. jcsd
  3. Dec 28, 2016 #2

    Ray Vickson

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    This is incorrect. Look at the case ##a=1, b=2##, or better still, the case ##a=1, b=-2##.
     
  4. Dec 28, 2016 #3

    mfb

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    The first line under "relevant equations" looks odd and the second one is wrong.

    You cannot subtract inequalities like that. As an example, 4<5 and 3<7, but 4-3 < 5-7 is wrong.
     
  5. Dec 28, 2016 #4
    I wish the section for the problem statement were filled out explicitly.
     
  6. Dec 29, 2016 #5
    @mfb and @Ray Vickson have already pointed out your mistakes. I just want to add a bit to mfb's reply.

    Inequalities with same "symbol" can be added and inequalities with "different symbol" can be subtracted from one another and not the other way around.
    Like ## 13 < 42## and ## -42 < -1 ## can be added to produce ##-29 < 41## and ## 42 > 13 ## and ## -42 < -1 ## can be subtracted to produce ##84 > 14##.
    When subtracting the symbol of inequality from which the other is subtracted will be the symbol of the resultant inequality.
     
  7. Dec 29, 2016 #6

    mfb

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    You can make an addition out of the subtraction:

    ##-42 < -1## is equivalent to (edit: fixed) ##42 > 1## (reverse the sign on both sides, reverse the direction of the inequality), and that can be added to ##42>13## to ##84>14##.

    And if you don't like the first step, split it in substeps:
    ##a<b##
    subtract a on both sides
    ##0 < b-a##
    subtract b on both sides
    ##-b < -a##
    Now write it in the other direction:
    ## -a > -b##.
     
    Last edited: Dec 30, 2016
  8. Dec 31, 2016 #7
    First thanks for the response and I see the error I made. I have also notice that I put the inequity the wrong way round sorry. So I have had another attempt at the solution which is below:

    Prove: $$|a|-|b|\leq|a-b|$$

    Attempt:$$a=a-b+b$$
    $$|a|=|a-b+b|$$

    Using triangle inequity

    $$|a|\leq|a-b|+|b|$$
    $$|a|-|b|\leq|a-b|$$
     
  9. Dec 31, 2016 #8

    haruspex

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    Looks good.
     
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