How to think about $|a| \leq b$

Question

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 $|a| \leq b$?

Answer 1

Sorry, but I think you messed up with the equations.

$$a\le|b|$$ is what is says, i.e.

$$a\le b$$ when $b$ is positive and $a\le-b$ otherwise.

Then

$$|a|\le b$$ is equivalent to

$$-b\le a\le b,$$ which is void for negative $b$.

Answer 2

You’ve got things backwards.

• $|a|\leq b$ is equivalent $-b\leq a\leq b$.

  You in fact have that: you say “$a\leq b$ and $-a\leq b$”. Multiplying the second inequality by $-1$ you would get “$a\leq b$ and $a\geq -b$”. Putting the two together you get $-b\leq a\leq b$.

  In partiular, if $b\lt 0$, then you can never get the two inequalities satisfied at the same time, since $-b$ is not less than or equal to $b$.

  When $b\geq 0$, you should imagine $b$ and $-b$ marking the outer edges of the region where $a$ must lie for the inequality to hold.

• On the other hand, $|a|\geq b$ is equivalent to $a\geq b$ or $-a\geq b$ (equivalently, $a\geq b$ or $a\leq -b$).

  When $b\lt 0$, the inequality always holds, because one of the clauses will necessarily hold.

  When $b\geq 0$, you should imagine $b$ and $-b$ again marking the edges of a region; but in this case, $a$ must be somewhere outside the region, meaning it must either be past $b$, or before $-b$.

Answer 3

Always, $|a| \leq b$ is equivalent to $-b \leq a \leq b.$