Absolute Value

Property	Statement
Definition	$
Key equivalence	$
Triangle inequality	$
Product rule	$

Definition

The absolute value of a real number $x$ is the nonnegative real number $|x|$ defined as follows:

|x|=\begin{cases} x & \text{if }x\geq 0\\ -x & \text{if } x<0 \end{cases}

Properties

For arbitrary real numbers $a$ and $b$:

1. $|a|\ge 0$
2. $|a|=0$ if and only if $a=0$
3. $\sqrt{a^2}=|a|$
4. $|-a|=|a|$
5. $|ab|=|a||b|$
6. $\left|{\displaystyle \frac{a}{b}}\right|={\displaystyle \frac{|a|}{|b|}}$ (provided $b\ne 0$)

7. If , the following three equivalences hold:

   $$\boxed{|x|\le r\text{is equivalent to}-r\le x\le r}$$$$\boxed{r\le |x|\text{is equivalent to}x\le -r\text{or}r\le x}$$$$\boxed{|x|=r\text{is equivalent to}x=r\text{or}x=-r}$$
8. $|a+b|\le |a|+|b|$     (the triangle inequality)

The three equivalences in property (7) are extremely useful for solving absolute value equations and inequalities. The first equivalence says that $|x|\le r$ describes a closed interval centered at 0; the second says that $r\le |x|$ describes the two rays outside that interval.

Frequently Asked Questions

Why is it called the triangle inequality?

The name comes from geometry: in any triangle, the length of one side is at most the sum of the lengths of the other two sides. In the context of real numbers on a line, $|a+b|$ is the distance from $0$ to $a+b$, while $|a|+|b|$ is the total distance traveled if you go $|a|$ units in one direction and then $|b|$ units in another. The inequality says the straight-line distance is never more than the path distance, which is exactly the triangle inequality from geometry.

How do I solve an inequality like ?

Use the first equivalence from property (7) with $r=5$: The solution is the open interval $(-1,4)$.

How does absolute value relate to the epsilon-delta definition of a limit?

In the epsilon-delta definition, the expression means the distance between $f(x)$ and $L$ is less than $\epsilon$. Similarly, means $x$ is within distance $\delta$ of $a$ but is not equal to $a$ (the deleted neighborhood condition). Understanding absolute value thoroughly is essential for reading and writing limit proofs.