Proof of an expression involving absolute value

[mindauggas]

1. The problem statement, all variables and given/known data

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

2. Relevant equations

Only the definition of absolute value is allowed.

3. The attempt at a solution

Don't even know where to start

[mtayab1994]

First prove -lxl≤x then prove x≤lxl

[LCKurtz]

Also consider breaking the problem into two cases, where x ≥ 0 and x < 0.

[genericusrnme]

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