Lesson 2: Absolute Value

Property

The absolute value of a number is its distance from $0$ on the number line. The absolute value of a number $n$ is written as $|n|$.

Property of Absolute Value
$|n| \geq 0$ for all numbers. Absolute values are always greater than or equal to zero!

Examples

• The absolute value of $-18$ is its distance from 0, so $|-18| = 18$.
• To simplify $30 - |12 - 4(5)|$, we calculate inside the bars first: $30 - |12 - 20| = 30 - |-8| = 30 - 8 = 22$.
• Compare $|-11|$ and $-|-11|$. We get $11$ and $-11$. So, $|-11| > -|-11|$.

Property

The distance between two points $x$ and $a$ is given by $|x - a|$.

Examples

• The statement "$x$ is five units from the origin" can be written using absolute value notation as $|x| = 5$. The solutions are $x=5$ and $x=-5$.
• The statement "$p$ is two units from 7" can be written as $|p - 7| = 2$. The solutions are $p=5$ and $p=9$.
• The statement "$a$ is within four units of $-3$" can be written as $|a - (-3)| < 4$, which simplifies to $|a + 3| < 4$. This means $a$ is between $-7$ and $1$.

Explanation

Absolute value is a tool for measuring distance on a number line without worrying about direction. The expression $|x - a|$ asks the question, "How far apart are $x$ and $a$?" The result is always a positive number.