Absolute Value

Definition of Absolute Value

The absolute value of x is the distance from zero to the number x on the real number line. Since distances are always non-negative, this means that the absolute value of x is never negative. Here are some examples:

$\mid-9\mid=9$

$\mid0\mid=0$

$\mid23\mid=23$

Here is another way of writing the definition of absolute value is:

$\mid x\mid=\lbrace a\:if\:a\ge0,\:and\:-a\:if\:a\le0$

Note: Some of you may have learned, or taught yourself, that to get the absolute value you just "drop the sign". If this is the way you think of absolute value, then please take the time now to learn the definition above, as the "drop the sign" way of thinking about absolute value will not work once we get into solving absolute value equations and inequalities later in this course.

Properties of Absolute Value

1. $\mid a\mid\ge0$                          
2. $\mid-a\mid=\mid a\mid$
3. $\mid a\mid\cdot\mid b\mid=\mid ab\mid$
4. $\frac{\mid a\mid}{\mid b\mid}=\mid\frac{a}{b}\mid\:\:\left(b\ne0\right)$
5. $\mid a+b\mid\le\mid a\mid+\mid b\mid\:\:\left(triangle\:inequality\right)$

Examples:

1. $\mid-7\mid=7$ and $7\ge0$
2. $\mid-9\mid=9$ and $\mid9\mid=9$, so $\mid-9\mid=\mid9\mid$
3. $\mid-3\mid\cdot\mid5\mid=3\cdot5=15$ and$\mid-3\cdot5\mid=\mid-15\mid=15$ so $\mid-3\mid\cdot\mid5\mid=\mid-3\cdot5\mid$
4. $\frac{\mid5\mid}{\mid7\mid}=\frac{5}{7}$ and $\mid\frac{5}{7}\mid=\frac{5}{7}$ so $\frac{\mid5\mid}{\mid7\mid}=\mid\frac{5}{7}\mid$
5. $\mid a+b\mid=\mid-8+13\mid=\mid5\mid=5$ and $\mid a\mid+\mid b\mid=\mid-8\mid+\mid13\mid=8+13=21$, so $\mid a+b\mid\le\mid a\mid+\mid b\mid$