Why these conclusions from the definition? Absolute value

Question

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!

Answer 1

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, $|x| \geq x$