Proof of an expression involving absolute value

AI Thread Summary
The discussion centers on proving the inequality -|x| ≤ x ≤ |x| for any real number x using only the definition of absolute value. Participants suggest breaking the proof into three cases: x greater than zero, x less than zero, and x equal to zero. This approach, referred to as proof by exhaustion, allows for a clear demonstration of the inequality in each scenario. The initial steps involve proving the left side of the inequality first, followed by the right side. Overall, the methodical breakdown of cases is emphasized as an effective strategy for this proof.
mindauggas
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Homework Statement



Prove: for any real number x, -|x|\leqx\leq|x|.

Homework Equations



Only the definition of absolute value is allowed.

The Attempt at a Solution



Don't even know where to start
 
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First prove -lxl≤x then prove x≤lxl
 
Also consider breaking the problem into two cases, where x ≥ 0 and x < 0.
 
Last edited:
pretty much what mtayab and LCKurtz said
first take x greater than zero, take x less than zero and take x equal to zero

this method is called a proof by exhaustion
 

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