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Math Help - When? does the inequality solve

  1. #1
    Junior Member Freaky-Person's Avatar
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    When? does the inequality solve

    In what cases would you have to change the inequality sign?

    I mean, if you have an equation like,

    |x| + |2x-5| < 5

    You would have 3 cases right?

    So what would the sign (< and >) look like for those 3 and why does it change some times?
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  2. #2
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    Quote Originally Posted by Freaky-Person View Post
    In what cases would you have to change the inequality sign?

    I mean, if you have an equation like,

    |x| + |2x-5| < 5

    You would have 3 cases right?

    So what would the sign (< and >) look like for those 3 and why does it change some times?
    The traditional way you are taught in high-schools, that means look at the plus minus signs DOES NOT WORK in general. It only works in the problems they give you.
    This happens to be a more advanced problem.
    We need to use the following definition,
    |n|=\left\{ \begin{array}{c}n \mbox{ for }n\geq 0 \\ -n\mbox{ for }n<0 \end{array} \right\}
    Thus, we the the following cases,
    |x|=x , -x
    And it depends on, x\geq 0, x<0
    Also we have,
    |2x-5|=2x-5,-2x+5
    And it depends on, 2x-5\geq 0,2x-5<0
    Solve the inequality,
    x\geq 2.5 and x<2.5.
    In combination with,
    x\geq 0 and x<0.
    We have the following cases,

    1) x<0\to x<2.5
    |x|=-x and |2x-5|=-2x+5

    2) x\geq 0 \mbox{ and }x<2.5
    |x|=x and |2x-5|=-2x+5

    3) x\geq 2.5 \to x\geq 0
    |x|=x and |2x-5|=2x-5.

    Thus, consider the inequality
    |x|+|2x-5|<5
    If, the first case,
    -x-2x+5<5
    -3x+5<5
    -3x<0
    x>0
    But that is a contradiction because x<0.
    If, the second case,
    x-2x+5<5
    -x+5<5
    -x<0
    x>0
    This will be a contradiction unless 0<x<2.5
    If, the third case,
    x+2x-5<5
    3x-5<5
    3x<10
    x<10/3
    This will be a contradiction unless 2.5\leq x<10/3
    Thus,
    x=(0,2.5)\cup [2.5,10/3)=(0,10/3)
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