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Math Help - solution set of |x-1|+|x+1| < 6

  1. #1
    Junior Member
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    solution set of |x-1|+|x+1| < 6

    Hi, I am having some problems with an exercise.

    I have solved the first one, but I don't know how to go about the 2nd one .

    The first one is:
    Find the solution set of |x-1|+|x+1|<1

    And I got this:
    |x|=
    -2x, x<=-1
    2x, x>=1
    2, -1<x<1
    (I got this from drawing the graph for |x-1| and |x+1| and adding the parallel lines)

    For the second exercise I have:
    Find the solution set of
    |x-1|+|x+1|<6

    I am pretty sure the result cannot be the same from looking at the graph, but I have no clue how to work it out using calculus :-/.

    Any help would be very much appreciated, thanks!
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  2. #2
    Senior Member yeKciM's Avatar
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     |x-1|+|x+1| < 6

     x-1 =0 \Rightarrow x = 1

     x+1 =0 \Rightarrow x = -1

    meaning you need to look at the regions (-\infty , -1 ) \cup  (-1, 1) \cup  (1 , +\infty) of the region  (-\infty , +\infty)

    meaning you say now
    1st case is  x\in (-\infty , -1 ) so you will have  (x-1)<0 and (x+1) <=0 so equation will be

     -(x-1) - (x+1) < 6

    solve than you see if solution is in that interval ? if not than that is not the solution (or solutions) than next interval and so on ... later solutions are all that satisfy conditions
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  3. #3
    Math Engineering Student
    Krizalid's Avatar
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    no calculus need to solve this.

    we just focus on where the absolute values get zero, that happens when x=1 and x=-1, note that both points satisfy the inequality, hence, in order to remove the absolute value, we study the cases when (-\infty,-1],\,[-1,1],[1,\infty).
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