Let $a,b \in \mathbb{R}$.

$a \leq |b|$ is equivalent to the expression $-b \leq a \leq b$. Easy, geometrical, elegant, intuitive.

**But what about**

$|a| \leq b$

Suppose $b \geq 0$, then

$|a| \leq b$ seems to be equivalent to $a \leq b \wedge -a \leq b$, for any $a \in \mathbb{R}$.

Suppose that $b < 0$, then,

$|a| \leq b \Leftrightarrow |a| \leq -|b|$,

which is equivalent to $a \leq -|b| \wedge -a \leq -|b|$ and further equivalent to to $a \leq |b| \wedge a \geq |b|$, but the only condition that satisfies this is when $a = 0, b= 0$.

Hence overall,

$$|a| \leq b = \begin{cases} a \leq b \wedge -a \leq b & \text{whenever } b \geq 0\\ N/A &\text{whenever } b < 0 \end{cases}$$

Is this a good way of expressing this relationship? Is there any easier way to think about $|a| \leq b$?