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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- September 20th 2007, 06:50 PMannalifabsolute 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! - September 20th 2007, 06:51 PMKrizalid
That's the absolute value definition.

- September 20th 2007, 06:58 PMannalif
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 - September 20th 2007, 07:04 PMJhevon
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

Thus, by the definition of :

or

or

combining these two inequalities, we obtain:

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)