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anonymous

  • 5 years ago

im trying to prove the inverse triangle inequality\[||x|-|y||\leq|x-y|\]like this:\[x=x-y+y,\]\[|x|=|x-y+y|,\]\[|x|\leq|x-y|+|y|,\]\[|x|-|y|\leq|x-y|.\]i get stuck here; dont know if its legit or not to take the absolute value of the LHS and preserve the inequality. help?

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  1. watchmath
    • 5 years ago
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    your statement is equivalent to \(-|x-y|\leq |x|-|y|\leq |x-y|\) now use triangle inequality

  2. anonymous
    • 5 years ago
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    im a little confused... how can i do that?

  3. watchmath
    • 5 years ago
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    Do you know that the inequality \(|x|\leq a\) is equivalent to \(-a\leq x\leq a\) ?

  4. anonymous
    • 5 years ago
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    i agree, but how can i apply the triangle inequality to the above system?

  5. anonymous
    • 5 years ago
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    i thought i could only apply it to an expression.

  6. watchmath
    • 5 years ago
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    ok, let's look at the first inequality \(-|x-y|\leq |x|-|y|\). Can you rewrite the inequality so that there is no minus (but all plus) by moving around the expressions?

  7. watchmath
    • 5 years ago
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    Can you see that it is equivalent to \(|y|\leq |x|+|x-y|\)?

  8. anonymous
    • 5 years ago
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    \[-|x-y|\leq|x|-|y|\implies|y|\leq|x|+|x-y|?\]

  9. watchmath
    • 5 years ago
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    great!

  10. watchmath
    • 5 years ago
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    Now if we apply the triangle inequality to \(|y|=|(y-x)+x|\) what do we get?

  11. anonymous
    • 5 years ago
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    \[|y|\leq|y-x|+|x|\]

  12. watchmath
    • 5 years ago
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    yes, but that is the same as \(|y|\leq |x|+|x-y|\) right ? since |y-x|=|x-y|

  13. anonymous
    • 5 years ago
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    i agree.

  14. watchmath
    • 5 years ago
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    so if you want a more linear argument we can like this for the first inequality: \(|y|=|(y-x)+x|\leq |y-x|+|x|=|x|+|x-y|\) It follows that \(-|x-y|\leq |x|-|y|\)

  15. watchmath
    • 5 years ago
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    you can do something similar to the 2nd one

  16. anonymous
    • 5 years ago
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    that makes sense. and with this, we can conclude that\[||x|-|y||\leq|x-y|?\]

  17. watchmath
    • 5 years ago
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    yes, if you prove two inequalities: \(-|x-y|\leq |x|-|y|\) and \(|x|-|y|\leq |x-y|\) you conclude the original inequality

  18. anonymous
    • 5 years ago
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    beautiful. i see it now. thank you very much for your time and help. i appreciate it a lot!

  19. watchmath
    • 5 years ago
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    you are welcome :)

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