What Is Absolute Value?
The absolute value of a number is how far that number sits from zero, regardless of which side it is on. Direction is dropped; only the distance remains.
Because distance cannot be negative, the absolute value of any number is zero or positive.
$|7| = 7$ and $|-7| = 7$. Both are 7 units from zero.
The formal definition is a two-case (piecewise) rule:
$$|x| = \begin{cases} x & \text{if } x \geq 0 \ -x & \text{if } x < 0 \end{cases}$$
The second case looks strange at first. When $x$ is negative, $-x$ is positive. For example, if $x = -7$, then $-x = -(-7) = 7$. So the rule simply strips the sign and returns the positive size.
The Absolute Value Symbol
Absolute value is written with two vertical bars around the number or expression: $|x|$, read aloud as "the absolute value of $x$" or "mod $x$."
Table 1: Reading the absolute value symbol.
Expression | Read as | Value |
|---|---|---|
$\lvert 4 \rvert$ | absolute value of 4 | 4 |
$\lvert -4 \rvert$ | absolute value of negative 4 | 4 |
$\lvert 0 \rvert$ | absolute value of 0 | 0 |
$\lvert x \rvert$ | absolute value of $x$ | $\geq 0$ always |
The bars are not parentheses. Everything inside is evaluated first, then the sign is stripped.
Properties of Absolute Value
These rules follow from "distance from zero" and are the ones used most when simplifying.
Table 2: Key properties of absolute value.
Property | Statement | Quick check |
|---|---|---|
Non-negativity | $\lvert a \rvert \geq 0$ | $\lvert -7 \rvert = 7 \geq 0$ |
Zero only at zero | $\lvert a \rvert = 0$ exactly when $a = 0$ | $\lvert 0 \rvert = 0$ |
Symmetry | $\lvert -a \rvert = \lvert a \rvert$ | $\lvert -3 \rvert = \lvert 3 \rvert = 3$ |
Product | $\lvert ab \rvert = \lvert a \rvert \lvert b \rvert$ | $\lvert (-2)(3) \rvert = 2 \cdot 3 = 6$ |
Quotient | $\left\lvert \dfrac{a}{b} \right\rvert = \dfrac{\lvert a \rvert}{\lvert b \rvert}$, $b \neq 0$ | $\left\lvert \dfrac{-6}{2} \right\rvert = \dfrac{6}{2} = 3$ |
Triangle inequality | $\lvert a + b \rvert \leq \lvert a \rvert + \lvert b \rvert$ | $\lvert 3 + (-5) \rvert = 2 \leq 8$ |
The triangle inequality is the one students meet last: the size of a sum is never more than the sum of the sizes, because cancellation can only shrink the total.
The Absolute Value Function And Its Graph
The absolute value function is $f(x) = |x|$. Its graph is a V shape: two straight rays meeting at a single corner point called the vertex.
The general transformed form is:
$$f(x) = a,|x - h| + k$$
Here $(h, k)$ is the vertex. If $a > 0$ the V opens upward and the vertex is the lowest point; if $a < 0$ it opens downward and the vertex is the highest point. The value of $|a|$ controls how steep the rays are.
Examples of absolute value
Example 1: Absolute value of a positive number
Find $|12|$.
12 is already 12 units from zero, on the positive side.
Final answer: $|12| = 12$.
Example 2: Absolute value of a negative number
Find $|-9|$.
$-9$ sits 9 units from zero. Strip the sign:
$|-9| = -(-9) = 9$
Final answer: $|-9| = 9$.
Example 3: Absolute value of an expression
Evaluate $|3 - 8|$.
Work inside the bars first:
$3 - 8 = -5$
Then take the absolute value:
$|-5| = 5$
Final answer: $|3 - 8| = 5$.
Example 4: Solving a simple absolute value equation
Solve $|x| = 6$.
A number is 6 units from zero in two places, to the right and to the left:
$x = 6 \quad \text{or} \quad x = -6$
Final answer: $x = 6$ or $x = -6$.
Example 5: A wrong path first, then the fix
Solve $|x - 2| = 4x + 1$.
The intuitive-but-wrong move is to drop the bars and solve one equation:
$x - 2 = 4x + 1$, which gives $-3 = 3x$, so $x = -1$.
Check it. Substitute $x = -1$ into the original right side: $4(-1) + 1 = -3$. But the left side $|x - 2|$ can never be negative, so a solution that makes the right side negative is impossible. That makes $x = -1$ extraneous, slipping in only when the bars were dropped.
The correct way solves two cases and checks both:
Case 1: $x - 2 = 4x + 1 \Rightarrow x = -1$ (rejected, as shown above).
Case 2: $x - 2 = -(4x + 1) \Rightarrow x - 2 = -4x - 1 \Rightarrow 5x = 1 \Rightarrow x = \tfrac{1}{5}$.
Check $x = \tfrac{1}{5}$: right side $= 4(\tfrac{1}{5}) + 1 = \tfrac{9}{5}$, which is positive, and the left side matches.
Final answer: $x = \dfrac{1}{5}$. The other case produced an extraneous solution.
Example 6: An equation with no solution
Solve $|x| = -3$.
Absolute value is a distance, so it is never negative. No number has a negative distance from zero.
Final answer: no solution.
Why Absolute Value Exists
Absolute value exists because we often care about size without caring about direction. How far is the store, how big is the error, how much did the temperature change? In each case the sign is noise, and what matters is the magnitude.
The idea shows up across fields:
Engineering tolerance. A bolt machined to within $|d - 10| \leq 0.05$ mm is acceptable whether it comes out slightly over or slightly under the target.
Navigation and GPS. Position error is reported as a distance, never as a signed direction. Being 3 metres off is 3 metres off, north or south.
Finance. The size of a price swing matters for risk whether the price rose or fell.
Physics. Speed is the absolute value of velocity; it keeps the magnitude and discards the direction.
One symbol that answers "how far, how big, how much" without the distraction of a sign. That is why absolute value reaches well past a single algebra chapter.
The Mathematicians Behind Absolute Value
Karl Weierstrass (1815–1897, Germany) introduced the now-universal $|x|$ vertical-bar notation around 1841 while building calculus on rigorous foundations. He spent years as an unknown schoolteacher before a single paper revealed a master had been teaching teenagers all along.
Jean-Robert Argand (1768–1822, France) extended the idea of "size" to complex numbers, where the absolute value becomes the distance from the origin in a two-dimensional plane.
Common Mistakes With Absolute Value
Mistake 1: Dropping the sign too early
Where it slips in:
A student writes $|3 - 8| = |3| - |8|$ and gets $-5$, taking the bars apart before working inside them.
Don't do this:
Do not split the absolute value across a subtraction or addition.
The correct way:
Evaluate everything inside the bars first, then strip the sign: $|3 - 8| = |-5| = 5$.
Mistake 2: Believing absolute value can be negative
Where it slips in:
Writing $|x| = -3$ and then solving for a value of $x$.
Don't do this:
Do not look for a number whose absolute value is negative.
The correct way:
A distance is never negative, so $|x| = -3$ has no solution. Stop as soon as the right side is negative.
Mistake 3: Solving $|x| = k$ as one case instead of two
Where it slips in:
Solving $|x| = 6$ and writing only $x = 6$, forgetting the negative side.
Don't do this:
Do not report a single answer for an absolute value equation that has two.
The correct way:
Set up both cases: $x = 6$ or $x = -6$. Two points are the same distance from zero.
Mistake 4: Accepting extraneous solutions
Where it slips in:
Splitting an equation like $|x - 2| = 4x + 1$ into cases, solving each, and reporting every answer without checking.
Don't do this:
Do not skip the substitution check at the end.
The correct way:
Substitute each candidate back into the original equation. Reject any that make the right side negative, as in Example 5.
Practice Questions on Absolute Value
Try these, then check your answers below.
Evaluate $\lvert -15 \rvert$.
Evaluate $\lvert 2 - 9 \rvert$.
Solve $\lvert x \rvert = 10$.
Solve $\lvert x \rvert = -4$.
Solve $\lvert 2x - 1 \rvert = 7$.
Answers
$\lvert -15 \rvert = 15$.
$2 - 9 = -7$, so $\lvert -7 \rvert = 7$.
$x = 10$ or $x = -10$.
No solution — absolute value can never equal a negative number.
$2x - 1 = 7 \Rightarrow x = 4$, or $2x - 1 = -7 \Rightarrow x = -3$. So $x = 4$ or $x = -3$.
What To Explore Next
Absolute value opens onto three connected ideas:
Absolute value inequalities. $|x| < 3$ describes everything within 3 units of zero, the natural step after solving equations.
Graphing transformed V-shapes. Once $f(x) = |x|$ is clear, shifting and stretching it with $a,|x - h| + k$ is the next move.
Distance on the number line and beyond. The same idea defines distance between any two points and, later, the size of a complex number.
Want your child to see why the graph folds into a V, drawn live point by point? A Bhanzu trainer builds the picture before the rule, so the symbol never feels arbitrary.
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