Why some times we use "or", and sometimes "and" in absolute value inequalities?

Question

I was watching a video about how to solve absolute value inequalities, but I didn't understand one step. He said that if the sign in the inequality is $<$, we should use AND between the two cases, but if the sign is $>$ we should use OR.

like this: $|1-\frac{x}{2}| < 1$

It will be: $1-\frac{x}{2} < 1$ and $1-\frac{x}{2} > -1$

I don't know where did this rule come from.

Answer 1

That rule comes from the theorem

"Let $a \in \mathbb{R}$ and $M \geq 0$. Then $|a| \leq M$ if and only if $-M \leq a \leq M$."

In your case,

$$\left|1-\frac{x}{2}\right| < 1 \iff -1 < 1-\frac{x}{2} < 1.$$

We can rewrite that compound inequality as

$$-1 < 1-\frac{x}{2} \text{ and } 1-\frac{x}{2} < 1.$$

In general, the statement "$a < b < c$" is always understood as the statement "$a<b \text{ and } b < c$."

Answer 2

Consider the definition of absolute value:

$|a| = \begin{cases} a & \mbox{if } a \geq 0 \\ -a & \mbox{if } a < 0 \end{cases}$

Then look at what happens with $|a| < M$ for some $M > 0$. We'll look for value solutions under two cases - when $a \geq 0$ and when $a < 0$:

• If $a \geq 0$, then $|a| = a$, so $a < M$. In other words, $0 \leq a < M$ is one range of solutions.

• If $a < 0$, then $|a| = -a$, so $-a < M$, which becomes $a > -M$ since you swap the inequality when you multiply by $-1$. In other words, $-M < a < 0$ is the other range of solutions.

So overall, the full range of values which satisfies $|a| < M$ is the union of $0 \leq a < M$ and $-M < a < 0$, which you can simplify to $-M < a < M$.

By comparison, if you look at $|a| > M$, then you get:

• If $a \geq 0$, then $|a| = a$ and so $a > M$. Notice that while this range is technically defined as $a \geq 0$ AND $a > M$, since $M > 0$ this reduces to just saying $a > M$.

• If $a < 0$, then $|a| = -a$ and so $-a < M$ or equivalently $a < -M$.

So overall the full range of solutions to $|a| > M$ is the union of $a > M$ and $a < -M$, but there's no neat way to write that so we just say that either $a > M$ OR $a < -M$.

You can also apply the intuition that $|a|$ represents the "distance" of $a$ from zero (and similarly $|a - b|$ is the "distance" of $a$ from $b$). Then saying $|a| < M$ means the distance between $a$ and zero is less than $M$, meaning it's trapped between the points $-M$ and $M$; whereas $|a| > M$ means $a$ is a distance greater than $M$ from zero, so it is locked out of the region between $-M$ and $M$.