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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!

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

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    $\begingroup$ Excuse my snarkiness, but It seems you haven't thought about the definition. $\endgroup$ – Bobson Dugnutt Mar 12 '17 at 15:47
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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, $|x| \geq x$

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

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