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Inequality question

  1. Aug 28, 2007 #1
    Does anyone can solve the following equation?

    |x − 1| = 1 − x

    Thanks

    Tom
     
  2. jcsd
  3. Aug 28, 2007 #2
    Yes I can. Thanks for inquiring!
     
  4. Aug 28, 2007 #3
    Sorry, that's unfair - it's your first post. Can you show us the work you have so far? Do you understand what absolute value means?
     
  5. Aug 28, 2007 #4
    Square both sides. And see what you get.
     
  6. Aug 28, 2007 #5

    symbolipoint

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    A more introductory way is to consider (x -1) as separately positive, and negative.

    If x - 1 is positive, then x - 1 = 1 - x;
    If x - 1 is negative, then x - 1 = -(1 - x )
     
  7. Aug 28, 2007 #6

    symbolipoint

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    Actually, that is not an inequality question; but an absolute value equation.
     
  8. Aug 28, 2007 #7
    If you have |a| = -a, what is the only way this can be true?
     
  9. Aug 29, 2007 #8
    This is the solutions which is done by myself:

    |x − 1| = 1 − x
    x − 1 = ±(1 − x)

    First
    x − 1 = +(1 − x)
    = 1 − x

    Second
    x − 1 = −(1 − x)
    = −1 + x
    = x − 1

    By |x| = ±x properties.

    That's absolutely correct. I mean ABSOLUTE VALUE QUESTION. Thanks for remind me about that.

    Why should I square-ing both sides of them? This is the most question I want to know in the ABSOLUTE VALUE. As I can read on Calculus book, there is no rules about SQUARE on absolute value except the |x| = √x^2 one. Can you explain me further about this?
     
  10. Aug 29, 2007 #9

    symbolipoint

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    tomcenjerrym - Your first condition yields x=0 as a solution; and your second condition allows ALL real numbers as solutions. All real numbers will satisfy the equation.
     
  11. Aug 30, 2007 #10
    No they won't. Take x=5 as an example |5-1|=|4|=4, but 1-5=-4, so in this case |x-1| does not equal 1-x, hence it is obviously not true for all real numbers. There is, however, a subset of the real numbers (with more than a single element) that satisfies the above equation.
     
  12. Aug 30, 2007 #11

    symbolipoint

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    You're correct. I was not careful enough when I solved the problem. We must watch around the critical point. The first part indicates x=1. When we check a value less than 1, we find equality; when we check a point greater than 1, we do not find equality.

    The solution seems to be x<=1
     
  13. Aug 30, 2007 #12
    I like nicktacik way to solve the problem. It’s simple and easy to understand. Thank you very much nicktacik and thank you to everyone too.
     
  14. Aug 30, 2007 #13
    The title "Inequality Question" is thus not entirely wrong, since the solution is an inequality.
     
  15. Aug 31, 2007 #14

    cks

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    nicktacik,

    ya, his answer was kind of making me suddenly awake!
     
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