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Math Help - Inequalities: how do I know when to take the union of x values or not?

  1. #1
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    Inequalities: how do I know when to take the union of x values or not?

    "Find the set of values of x for which |x+2| > 1/(x -2). Hence, solve |(2x-1)/x| > -x/(1+2x)."

    The answer to the first part is x < 2 or x > 2.24. The answer to the second part is x < -1/2 or x > - 0.447, x not equals zero.

    I need help with the second bit only.

    I know that you need to replace x with -1/x, so: -1/x > -2 and 1/x < - 2.24. Then I solved using the graphical method, so I got x > 0, x < -1/2 for 1/x > -2 and - 0.447 < x < 0 for 1/x < - 2.24. When I take the union of these values I get the answer to the second part.

    Can someone explain briefly why I apparently need to take the union of the values on the number line as opposed to leaving them as they are? Seeing as the answer to the first part is x <2 or x > 2.24 anyway.

    (Apologies for such a nitty gritty question it's been eating away at me)
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  2. #2
    MHF Contributor Unknown008's Avatar
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    The answer to the second part clearly says that x is not equal to 0, which is that your '3 parts' solution is saying:

    I'm putting it this way:

    -0.5 > x; -0.447 < x < 0; x > 0

    So, by the second and third ranges, x is greater than -0.447, but less than 0, and greater than 0, which is what the answer tells you; it's greater than -0.477 but not 0.
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  3. #3
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    thanks (:
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