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Thread: |2x-1|-|x|< 4

  1. #1
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    |2x-1|-|x|< 4

    How do I solve this |2x-1|-|x|< 4 ? Do i simply treat it like an equation, meaning that it has three different solutions? or do other rules apply?
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  2. #2
    Super Member wingless's Avatar
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    Use cases.
    $\displaystyle |2x-1|-|x|<4$
    There are three cases,
    i. $\displaystyle x\ge\frac{1}{2}$
    ii. $\displaystyle 0 < x < \frac{1}{2}$
    iii. $\displaystyle x \le 0$

    -------------
    i. $\displaystyle x\ge\frac{1}{2}$
    The inequality becomes,
    $\displaystyle 2x - 1 - x < 4$
    $\displaystyle x < 5$
    The interval here is, $\displaystyle \frac{1}{2}\le x < 5$

    -------------
    ii. $\displaystyle 0 < x < \frac{1}{2}$
    $\displaystyle -(2x-1) - x < 4$
    $\displaystyle x > -1$
    Interval: $\displaystyle 0 < x < \frac{1}{2}$

    -------------
    iii. $\displaystyle x \le 0$
    $\displaystyle -(2x -1)-(-x)<4$
    Interval: $\displaystyle -3 < x \le 0$

    -------------
    -------------

    When we put the intervals together,
    $\displaystyle \frac{1}{2}\le x < 5$
    $\displaystyle 0 < x < \frac{1}{2}$
    $\displaystyle -3 < x \le 0$
    It makes,
    $\displaystyle -3<x<5$
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  3. #3
    Senior Member Peritus's Avatar
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    the first element is positive if x > 0.5 and negative otherwise, the second element is positive if x > 0 and negative otherwise.

    1. x => 0.5, in this range both elements are non negative so:

    2x-1 - x < 4
    x < 5

    so the solution is: 0.5 <= x < 5

    2. 0 <= x <= 0.5 in this range the first element is negative and the second is non negative so:

    -(2x-1) - x < 4
    x > -1

    thus the answer is 0 <= x <= 0.5

    3. x <= 0 in this range the first element is negative and the second is non positive so:

    -(2x-1) + x < 4
    x > -3

    thus the solution is -3 < x <= 0

    The final solution is the intersection of these 3 solutions thus:

    -3 < x < 5
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  4. #4
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    Quote Originally Posted by weasley74 View Post
    How do I solve this |2x-1|-|x|< 4 ? Do i simply treat it like an equation, meaning that it has three different solutions? or do other rules apply?
    If you like - or allowed to - you can solve it graphically.
    Attached Thumbnails Attached Thumbnails |2x-1|-|x|&lt; 4-betrag_kleinerzahl.gif  
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