Is { $ x \in \mathbb R \mid |x - a| > b$} equal to { $ x \in \mathbb R \mid |x - a| < b$}?

Question

This is where $a \in \mathbb R$ and $b > 0$.

I found that for both of them, $a \le x \le a$ which means that $x = a$. Am I wrong? Thanks.

Answer 1

No, they are not equal. The first set is the set of all real numbers $x$ such that $x$ is a distance more than $b$ away from $a$. The second set is the opposite; it is the set of all $x$ such that $x$ is a distance less than $b$ away from $a$.

Answer 2

The first set does not contain $a$, whereas the second does. So they cannot be equal.

Answer 3

No, they are actually disjoint which means they do not have anything in common.

Note that $$|x-a|<b\implies a-b<x<a+b $$

While $$|x-a|>b\implies x<a-b \text { or } x>a+b $$

The first set is bounded and the second on is unbounded.