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Math Help - absolute value

  1. #1
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    absolute value

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

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

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

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

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

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