#math/calculus An interval records whether each endpoint is included. | Inequality | Interval | Endpoints | |---|---|---| | $a<x<b$ | $(a,b)$ | neither included | | $a\leq x\leq b$ | $[a,b]$ | both included | | $a\leq x<b$ | $[a,b)$ | only $a$ included | | $a<x\leq b$ | $(a,b]$ | only $b$ included | An infinite endpoint always uses a parenthesis, as in $(-\infty,a]$ or $(b,\infty)$. The **absolute value** $|x|$ is the distance from $x$ to $0$: $ |x|=\begin{cases}x,&x\geq 0,\\-x,&x<0.\end{cases} $ More generally, $|x-a|$ is the distance from $x$ to $a$. For $r>0$, $ |x-a|<r\iff a-r<x<a+r, \qquad |x-a|>r\iff x<a-r\text{ or }x>a+r. $ Useful identities include $|xy|=|x||y|$, $|x/y|=|x|/|y|$ for $y\neq 0$, and the triangle inequality $|x+y|\leq |x|+|y|$.