Lesson 4: Absolute Value

Property

The absolute value of a number is its distance from zero on the number line.
The absolute value of a number $n$ is written as $|n|$ and $|n| \geq 0$ for all numbers.
Absolute values are always greater than or equal to zero.

Examples

• The absolute value of $-9$ is $9$, because $-9$ is $9$ units away from $0$. This is written as $|-9| = 9$.
• The absolute value of $25$ is $25$, because $25$ is $25$ units away from $0$. This is written as $|25| = 25$.
• The equation $|x| = -4$ has no solution. Absolute value represents distance, which cannot be a negative number.

Explanation

Think of absolute value as a 'distance-meter' from zero. Since distance can't be negative, the absolute value of any number, positive or negative, will always be a positive result or zero. It simply tells you how far away you are.

Property

For any number $x$, it is always true that $x \leq |x|$ because absolute value represents the distance from zero on the number line.
When comparing absolute values of different numbers, we are comparing their distances from zero, which may not preserve the original ordering of the numbers.

Examples

Property

To find the distance, $D$, between two numbers, $a$ and $b$, on a number line, calculate the absolute value of their difference:

Examples