Absolute Value: Distance from Zero, Equations and Inequalities

Q: Can absolute value be negative?

No. It measures a distance, and distances are zero or positive. An equation like |x| = -2 has no solution.

Q: Why does an absolute value equation have two answers?

There are usually two numbers at a given distance from a point: one on each side. That is why |x - 4| = 3 gives both 1 and 7.

Absolute value as distance, why $|a - b|$ is the gap between two numbers, and how that one idea solves absolute value equations and inequalities, and later the language of limits.

4 min read Supports 3 lessons

How far is $-4$ from zero? Four steps. How far is $4$ from zero? Also four steps. Absolute value measures exactly this: how far a number is from zero, ignoring which side it is on.

$|-4| = 4$ : four steps from zero.

|7| = 7 \qquad |-7| = 7 \qquad |0| = 0

Key idea

Absolute value is never negative: $|x| \ge 0$ for every number $x$ .

Distance between two numbers

The real power of absolute value is measuring gaps. The distance between $a$ and $b$ on the number line is $|a - b|$ , whichever is bigger.

Distance

\text{dist}(a, b) = |a - b|

The distance between $3$ and $10$ is $|3 - 10| = |-7| = 7$ .

So $|x - 4|$ reads as "the distance between $x$ and $4$ ". Learning to read it that way makes everything below easy.

Solving absolute value equations

$|x - 4| = 3$ asks: which numbers are exactly 3 away from 4? There are two, one on each side.

The solutions are $x = 1$ and $x = 7$ .

Equations

|x - a| = d \;\Rightarrow\; x = a - d \;\text{ or }\; x = a + d

If $d$ is negative there is no solution, since a distance cannot be negative.

Worked example

Solve $|2x - 3| = 7$ .

1
The inside is either $7$ or $-7$ .

Absolute value inequalities

$|x - 4| < 3$ asks for every number within 3 of 4. That is a single stretch of the number line, from $1$ to $7$ .

$|x - 4| < 3$ means $1 < x < 7$ .

$|x - 4| > 3$ asks for numbers more than 3 away, which splits into two pieces: $x < 1$ or $x > 7$ .

Tip

Less than: one piece in the middle ("within"). Greater than: two pieces heading outwards ("beyond").

Absolute value as a piecewise function

Algebraically, absolute value leaves positive numbers alone and flips the sign of negative ones:

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

The graph of $y = |x|$ is a V with its corner at the origin.

Watch out

$-|5| = -5$ . The minus sign outside the bars is applied after the absolute value.

Looking ahead

In calculus, $|x - a| < \delta$ is the precise way to say " $x$ is close to $a$ ". It is the language that limits are written in, and it all comes back to distance.

Common questions

Can absolute value be negative?: No. It measures a distance, and distances are zero or positive. An equation like $|x| = -2$ has no solution.
Why does an absolute value equation have two answers?: There are usually two numbers at a given distance from a point: one on each side. That is why $|x - 4| = 3$ gives both $1$ and $7$ .

Practise absolute value

The Absolute Value lesson has worked examples and unlimited practice that adapts to you.

Open the lesson

Common questions

Can absolute value be negative?

No. It measures a distance, and distances are zero or positive. An equation like $|x| = -2$ has no solution.

Why does an absolute value equation have two answers?

There are usually two numbers at a given distance from a point: one on each side. That is why $|x - 4| = 3$ gives both $1$ and $7$ .