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Math Help - Inequantity be/tn absolute value of difference and the difference of absolute values

  1. #1
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    Inequantity be/tn absolute value of difference and the difference of absolute values

    The appendix of my calculus book lists Absolute Value Properties:
    1. |-a| = |a|
    2. |ab| = |a||b|
    3. |a/b| = |a|/|b|
    4. |a+b| <= |a|+|b|

    For 4., it is also mentioned that if 'a' and 'b' differ in sign, then |a+b| is less than |a|+|b|. In "all other cases", |a+b| equals |a|+|b|. I verified that 4. holds true a<0, b<0 (with a few examples).

    I questioned about |a-b| and whether it is <,>,=, <=, or greater than or equal to |a|-|b|. I decided to see whether the signs mattered (whether a<0 or a>0 and b<0 and b>0 mattered) and whether the order of "a" and "b" matterred.

    With a couple of examples, I found that |a-b| is greater than or equal to |a|-|b|. I found that if 'a' and 'b' differ in signs, then |a-b| > |a|-|b|. However, whether the inequality was "equal" or "greater than" varied with the order in which 'a' and 'b' is subsituted:

    When a,b are positive numbers and a<b, the inequality statement is true when |a-b| > |a|-|b|. However, if a and b were reversed in the inequality statement, the statement is true when |b-a| = |b|-|a|.

    When a,b are negative numbers and a<b, the inequality statement is true when |a-b| = |a|-|b|. However, if a and b were reversed in the inequality statement, the statement is true when |b-a| > |b|-|a|.

    So am I right to conclude that:
    (1) |a-b|=>|a|-|b|
    (2)

    When a,b are positive numbers and a<b, the inequality statement is true when |a-b| > |a|-|b|. However, if a and b were reversed in the inequality statement, the statement is true when |b-a| = |b|-|a|.

    When a,b are negative numbers and a<b, the inequality statement is true when |a-b| = |a|-|b|. However, if a and b were reversed in the inequality statement, the statement is true when |b-a| > |b|-|a|.
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  2. #2
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    Re: Inequantity be/tn absolute value of difference and the difference of absolute val

    |a+b|\le|a|+|b| holds for all a,b\in\mathbb R.

    On the other hand, we have |a|=|a-b+b|\le|a-b|+|b| so |a-b|\ge |a|-|b| and you can do the same by writing |b|=|b-a+a|.
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  3. #3
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    Re: Inequantity be/tn absolute value of difference and the difference of absolute val

    What do you mean? I asked about whether |a-b|>= |a|-|b| holds true for all real a,b.
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  4. #4
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    Re: Inequantity be/tn absolute value of difference and the difference of absolute val

    It does, as I showed above.
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