Question

If x is a real number, prove that -X<=|x|​

Answer 1

Step-by-step explantion:

Definition of absolute value of x:

|x|=x assuming x has taken a positive value.

|x|=-x assuming x has taken a negative value.

|x|=0 assuming x is 0.

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`-x \le |x|`

Add
`x` on both sides:

`0 \le |x|+x`

`|x|+x \ge 0`

Let
`h(x)=|x|+x`.

Assume x is positive, then
`h(x)=x+x=2x` where
`2x` is positive since
`x` is positive. Also
`h(x)=2x>0` for positive x.

Assume
`x` is negative, then
`h(x)=-x+x=0`.

Assume
`x=0`, then
`h(0)=|0|+0=0+0=0`.

The result for h is positive when x is positive but 0 elsewhere for x.

-------------------------------------End of Proof--------------------------------------