Properties of absolute values

We use absolute values to get distance between real numbers. The distance between $a$ and $b$ is $\vert a-b\vert$. The distance from $a$ to $b$ is the same as the distance from $b$ to $a$ so we should get $\vert-x\vert=\vert x\vert$. Distance needs to satisfy the triangle inequality: the distance from $a$ to $c$ can't get any smaller if we insist on going through $b$, so $\vert a-c\vert=\vert a-b+b-c\vert\leq \vert a-b\vert+\vert b-c\vert$, with equality happening if $b$ was between $a$ and $c$. This law is usually stated as

$$\vert a+b\vert\leq \vert a\vert+\vert b\vert$$

Multiplying should stretch distances, so

$$\vert kx\vert=\vert k\vert\vert x\vert$$

We can interpret inequalities like $\vert x-a\vert<d$ as saying that the distance from $x$ to $a$ is less than $d$, so $a$ is in the center and we allow wiggle room of size $d$.

$$\vert x-a\vert<d \mbox{ means } x\in (a-d,a+d)$$

Saying that $\vert x-a\vert>0$ only rules out $x=a$ so

$$\vert x-a\vert>0 \mbox{ means } x\neq a$$