TITLE: Definition of abs() function
QUESTION [5 upvotes]: Let $\text{abs}(a)$ denote the absolute value of $a$. Is it true that $\text{abs}(a)\geq{-a}$? I suppose that $\text{abs}(a)>{-a}$, but my math book says the other way. Please help me to understand is it a misprint in my book, or my misunderstanding. Thank you in advance.

REPLY [7 votes]: We have
$$\operatorname{abs}(a)=\max(a,-a)=\left\{\begin{array}{cl}a\;&\text{if}\; a\ge0\\-a\;&\text{otherwise}\end{array}\right.$$

REPLY [4 votes]: The $abs$ function is defined by:
$\forall{x}\in\mathbb{R},\,abs(x)=|x|=\left\{
     \begin{array}{lr}
       x & : x\ge0\\
       -x & : x <0
     \end{array}
   \right.$
So $\forall x\in\mathbb{R},\,|x|\ge0$
Let $a\in\mathbb{R}$.
If $a\ge0$ then $|a|=a$ and so $a\le|a|$
If $a<0$ then $|a|=-a>0>a$ and so $a\le|a|$
Now, if $a\ge0$ then $-a\le0\le|a|$
If $a<0$ then $-a=|a|$ and so $-a\le|a|$.