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  1. #1
    Junior Member Freaky-Person's Avatar
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    help

    ||x - 4|- 3| + 4 < |2x + 5|
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  2. #2
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    Quote Originally Posted by Freaky-Person View Post
    ||x - 4|- 3| + 4 < |2x + 5|
    ||x-4|-3|+4<|2x+5|
    Again, you need to consider to cases for absolute for one non-negative and one negative. That is the standard way of doing these nasty ones.

    2x+5\geq 0
    x\geq -2.5 for non-negative value.
    x<-2.5 for negative value.

    |x-4|-3\geq 0
    |x-4|\geq 3
    x-4\geq 3 \mbox { or }x-4\leq -3
    x\geq 7 \mbox{ or }x\leq 1 for non-negative value.

    |x-4|-3<0
    |x-4|<3
    -3<x-4<3
    1<x<7
    For negative value.

    Now we divide the problem into cases,
    x<-2.5 thus, 2x+5<0, |x-4|-3>0
    -2.5\leq x\leq 1 thus, 2x+5,|x-4|-3>0
    1<x<7 thus, 2x+5>0, |x-4|-3<0
    7\leq x thus. 2x+5>0, |x-4|-3>0.

    And you do what I did before.
    In each of these cases the absolute sign disappearsn and you deal with a different inequality. If the set of x contradicts the condition from the intersection.
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