Results 1 to 2 of 2

Math Help - [SOLVED] real analysis - absolute value

  1. #1
    Newbie
    Joined
    Apr 2009
    Posts
    7

    [SOLVED] real analysis - absolute value

    If x, y, and are elements of R and x <=z, show that x <= y <= z if and only if |x - y| + |y - z| = |x - z|.
    Follow Math Help Forum on Facebook and Google+

  2. #2
    Newbie
    Joined
    Aug 2011
    Posts
    10

    Re: [SOLVED] real analysis - absolute value

    \textbf{Proof.}

    (\Rightarrow) \text{We have that } x \leq y \leq z \text{.}

    \text{i) Supose } \left | x - y \right | + \left | y - z \right | < \left | x - z \right | \text{.}

    \Rightarrow y - x + z - y < z - x

    \Rightarrow z - x < z - x

    \Rightarrow \text{Contradiction!}

    \text{ii) Supose } \left | x - y \right | + \left | y - z \right | > \left | x - z \right | \text{.}

    \Rightarrow y - x + z - y > z - x

    \Rightarrow z - x > z - x

    \Rightarrow \text{Contradiction!}

    \text{Hence, } \left | x - y \right | + \left | y - z \right | = \left | x - z \right | \text{.}


    (\Leftarrow) \text{We have that } \left | x - y \right | + \left | y - z \right | = \left | x - z \right | \text{.}

    \text{i) Supose } y < x \leq z \text{.}

    \Rightarrow x - y + z - y = z - x

    \Rightarrow 2x = 2y

    \Rightarrow x = y

    \Rightarrow \text{Contradiction!}

    \text{ii) Supose } x \leq z < y \text{.}

    \Rightarrow y - x + y - z = z - x

    \Rightarrow 2y = 2z

    \Rightarrow y = z

    \Rightarrow \text{Contradiction!}

    \text{Hence, } x \leq y \leq z \text{.}
    \text{Q.E.D.}
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Finding real solutions with absolute value
    Posted in the Algebra Forum
    Replies: 3
    Last Post: August 7th 2011, 07:50 PM
  2. Replies: 8
    Last Post: April 7th 2009, 01:15 PM
  3. [SOLVED] real analysis test question
    Posted in the Number Theory Forum
    Replies: 4
    Last Post: March 6th 2009, 08:30 PM
  4. Replies: 4
    Last Post: October 13th 2008, 08:07 AM
  5. [SOLVED] Real Analysis
    Posted in the Advanced Math Topics Forum
    Replies: 1
    Last Post: February 9th 2008, 06:56 PM

Search Tags


/mathhelpforum @mathhelpforum