Absolute Value — Definition and Properties

Absolute value means distance from $0$ on the number line. For real numbers, that makes $|x|$ always nonnegative.

That is why $|5| = 5$ and $|-5| = 5$. The numbers are on opposite sides of $0$, but they are the same distance from it.

Absolute Value Definition

For a real number $x$,

$$|x| = \begin{cases} x, & x \ge 0 \\ -x, & x < 0 \end{cases}$$

This does not mean absolute value "makes everything positive." It leaves nonnegative numbers alone and changes the sign of negative numbers.

Think Of It As Distance

The best mental model is distance. If you read $|x|$, think "the distance from $x$ to $0$."

The same idea explains expressions like $|a-b|$. That is the distance between $a$ and $b$ on the number line.

For example,

$$|2 - 7| = |-5| = 5$$

so the distance between $2$ and $7$ is $5$.

Key Absolute Value Properties

These are the properties you will use most often:

1. $|x| \ge 0$ for every real number $x$.
2. $|x| = 0$ only when $x = 0$.
3. $|-x| = |x|$.
4. $|ab| = |a||b|$ for real numbers $a$ and $b$.
5. If $b \ne 0$, then $\left|\frac{a}{b}\right| = \frac{|a|}{|b|}$.

The condition $b \ne 0$ matters in the last property because division by $0$ is undefined.

Worked Example: Solve $|x - 3| = 5$

This equation asks for numbers whose distance from $3$ is $5$.

If a number is $5$ units to the right of $3$, then

$$x - 3 = 5$$

so

$$x = 8$$

If a number is $5$ units to the left of $3$, then

$$x - 3 = -5$$

so

$$x = -2$$

So the solutions are

$$x = 8 \quad \text{or} \quad x = -2$$

This two-case idea is the main pattern to remember. If $|u| = k$ and $k > 0$, solve both $u = k$ and $u = -k$.

Common Absolute Value Mistakes

One common mistake is thinking $|x|$ can be negative. It cannot. For real numbers, absolute value is always at least $0$.

Another common mistake is solving only one case. In the example above, stopping at $x = 8$ misses the second point that is also $5$ units from $3$.

Students also mix up $|-x|$ and $-|x|$. They are not the same. In fact, $|-x| = |x|$, but $-|x|$ is zero or negative.

When Absolute Value Is Used

Absolute value shows up whenever size matters more than direction.

You see it in distance on a number line, error or deviation from a target, equations and inequalities, and formulas where only magnitude should remain. In later math, it also appears in coordinate geometry, calculus, and complex numbers, though the exact meaning depends on the setting.

Quick Check For Equations Like $|u| = k$

If you see an equation like $|u| = k$, check the right side first.

If $k < 0$, there is no real solution because an absolute value cannot equal a negative number. If $k = 0$, then the inside must be $0$. If $k > 0$, expect two cases unless both cases lead to the same value.

Try A Similar Absolute Value Problem

Try solving $|x + 4| = 9$. Read it as "distance from $-4$ is $9$," then write the two matching cases. If you want to check yourself after solving, plug both answers back into the original equation.