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Math Help - absolute value proof

  1. #1
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    absolute value proof

    Prove that if c is a positive real number and x is any real number, then -c </= x </= c if and only if |x|</= C.

    </= greater than or equal to

    Thank you!
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  2. #2
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    That's the absolute value definition.
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  3. #3
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    yeah, but i cant figure out how to formalize it into a proof

    i get x </= |c| , so

    -|c| </= x </= |c|

    and then my head hurts and i want to give up on life
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  4. #4
    is up to his old tricks again! Jhevon's Avatar
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    Quote Originally Posted by annalif View Post
    yeah, but i cant figure out how to formalize it into a proof

    i get x </= |c| , so

    -|c| </= x </= |c|

    and then my head hurts and i want to give up on life
    yeah. we often take this fact for granted around here. it's hard to come up with a rigorous proof. but here is a possible one for the reverse implication.

    Recall that |x| = \left \{ \begin {array}{cc} x & \mbox { if } x \ge 0 \\ -x & \mbox { if } x < 0 \end {array} \right.

    Thus, by the definition of |x|:

    |x| \le C

    \implies x \le C or -x \le C

    \implies x \le C or x \ge -C

    combining these two inequalities, we obtain:

    -C \le x \le C


    now try to go the other way. i would probably try to reverse this exact proof. so split the inequality into two, and work on each case by case (if x > 0 or x < 0)
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