Simple Absolute Value Proof

Question

Let $\circ$ be an inequality.

Prove $|x| \circ a \equiv -a \circ x \circ a$.

If $x$ is positive, then $|x| \circ a = x \circ a$.

If $x$ is negative, then $|x| \circ a = x \circ a$ ?

Answer 1

I suppose that you have to show: for $x,a \in \mathbb R$ and $a \ge 0$:

$|x| \le a$ implies $-a \le x \le a$.

Case 1: $x \ge 0$. Then we get from $|x| \le a$ that $0 \le x \le a$ and hence $-a \le x \le a$.

Case 2: $x < 0$. Then we get from $|x| \le a$ that $- x \le a$ and hence $-a \le x <0$, thus $-a \le x \le a$.

Answer 2

Note $\circ$ can be either $<$ or $>$.

If $x>0$, then $|x|\circ a \equiv x\circ a$.

If $x<0$, then $ |x|\circ a \equiv -x\circ a$.