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.
Best Answer
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 "$a<b \text{ and } b < c$."