Write out proofs of the following properties of the absolute

Write out proofs of the following properties of the absolute value, using whatever cases seem most natural to you. a) |x| = |x| for all real numbers x.

Solution

As we know we have three cases for value of x i.e. x>0, or x=0, or x<0, so

If x>0, then |x|=x, and since -x<0, |-x|=-(-x)=x, so |x|=|-x|.

If x=0, then |x|=0 and -x=0, so |-x|=0, so |x|=|-x|.

If x<0, then |x|=-x, and since -x>0, |-x|=-x, so |x|=|-x|.