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Thread: absolute value

  1. #1
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    absolute value

    How can I show that $\displaystyle x+x|y|-y-|x|y =0 \implies x=y$?

    $\displaystyle |x| = \{ x$ if $\displaystyle x \geq 0$ and $\displaystyle -x$ if $\displaystyle x < 0\}$, right?

    So I break this into 4 cases,
    if $\displaystyle x,y \geq 0 $, then $\displaystyle x+xy-y-xy = 0 \implies x=y$
    if $\displaystyle x,y < 0 $, then $\displaystyle (-x)+(-x)(-y)-(-y)-(-x)(-y) = -x+xy+y-xy = 0 \implies x=y$
    I have trouble with cases $\displaystyle x \geq 0$ and $\displaystyle y <0$, and $\displaystyle x<0$ and $\displaystyle y \geq 0$, some help please.
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  2. #2
    MHF Contributor red_dog's Avatar
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    If $\displaystyle x\geq 0, \ y\leq 0$, then we have $\displaystyle x-2xy-y=0$

    But: $\displaystyle x\geq 0, \ -2xy\geq 0, \ -y\geq 0$

    A sum of positive numbers is 0 if all the numbers are 0. Then $\displaystyle x=y=0$

    In the same way you can proove if $\displaystyle x\leq 0, \ y\geq 0$
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