# Why these conclusions from the definition? Absolute value [closed]

I know the definition is \begin{align} \lvert x \rvert = \begin{cases} x, \quad &x\geq 0 \\ -x, \quad &x<0 \end{cases} \end{align}

But why can we conclude that $x \leq \lvert x\rvert$?

And if $x \leq \lvert x\rvert$ is true, why can we also write $-\lvert x \rvert \leq x \leq \lvert x \rvert$?

I don't understand these conclusions just from the definition.

Thanks!

## closed as unclear what you're asking by Did, Leucippus, Shailesh, Namaste, RamiroAug 10 '17 at 1:37

Please clarify your specific problem or add additional details to highlight exactly what you need. As it's currently written, it’s hard to tell exactly what you're asking. See the How to Ask page for help clarifying this question. If this question can be reworded to fit the rules in the help center, please edit the question.

• Excuse my snarkiness, but It seems you haven't thought about the definition. – Bobson Dugnutt Mar 12 '17 at 15:47

## 1 Answer

There are two possible cases:

Case $1$:

$x \geq 0 \Rightarrow |x| = x$

Case $2$:

$x < 0 \Rightarrow -x > 0 \Rightarrow |x|=-x>x$

Therefore, by 'exhaustion' of all cases, $|x| \geq x$

• And, similarly, $x\ge-\lvert x\rvert$ – egreg Mar 12 '17 at 16:09