Absolute value
Definition of absolute value
The absolute value of a number tells how far that number is from zero. It is a simple idea, but it returns in definitions, equations, inequalities, and parameter problems.
Check your knowledge
A short quiz will help you quickly spot which concepts are worth reviewing.
What is absolute value?
The absolute value of a number x is written as |x| and read as the absolute value of x.
Absolute value removes information about direction on the number line, but keeps information about distance from zero. That is why the result of an absolute value is never negative.
Piecewise notation
The number is not negative, so it stays the same.
The number is negative, so we take its opposite.
Intuition: absolute value is distance
In short: |x| is the distance from x to zero on the number line. The numbers 5 and -5 lie on opposite sides of zero, but both are 5 units away from it, so |5| = 5 and |-5| = 5.
Absolute value does not ask whether we move left or right. It asks only how far. That is why the absolute value of a negative number is positive, while a positive number stays unchanged.
This intuition is very useful in expressions such as |x - 3|. The expression |x - 3| means the distance from x to 3, not just two bars around a formula.
Basic properties
The properties below are worth knowing as ideas, not only as formulas. Each one follows from the fact that absolute value measures distance or turns a number into its nonnegative version.
| Property | Name | Plain explanation |
|---|---|---|
| |a| >= 0 | Nonnegativity | Absolute value never gives a negative number because distance cannot be negative. |
| |a| = |-a| | Opposite numbers | The numbers a and -a are the same distance from zero. |
| |ab| = |a| · |b| | Product | You can multiply first and then take absolute value, or take absolute values first and multiply them. |
| |a / b| = |a| / |b|, b != 0 | Quotient | The absolute value of a fraction is the quotient of the absolute values of the numerator and denominator. The denominator cannot be zero. |
| |a| = 0 if and only if a = 0 | Zero | Only the number 0 is at distance 0 from zero. |
| |a + b| <= |a| + |b| | Triangle inequality | The direct route is not longer than a route through an intermediate point. |
| ||a| - |b|| <= |a - b| | Difference of distances | The difference between distances from zero cannot be greater than the distance between the numbers. |
| |a|^2 = a^2 | Square of absolute value | After squaring, the sign disappears anyway, so absolute value does not change the result. |
| |a| < r means -r < a < r, when r > 0 | Less-than inequality | The number a is closer to zero than r units. |
| |a| > r means a < -r or a > r, when r > 0 | Greater-than inequality | The number a is farther than r units from zero, so it lies either on the left or on the right. |
Watch the minus before absolute value
The expression -|a| is always less than or equal to zero. For example, -|-4| = -4, because we evaluate the absolute value first and then keep the minus before it.
Absolute value of a whole expression
In |2x - 6| we check the sign of the whole expression 2x - 6, not only the sign of x. The boundary between cases is x = 3.
Distance between numbers
The distance between a and b on the number line can be written as |a - b| or |b - a|. Both forms give the same result.
Absolute value equations
The equation |A| = c makes sense only for c >= 0. When c > 0, we usually split it into two cases: A = c or A = -c.
When can you remove absolute value bars?
You do not remove absolute value bars simply because they are inconvenient. First you need to know the sign of the expression inside. Only then can you replace the bars with the correct formula.
The inside is nonnegative
|A| = A
If A >= 0, absolute value changes nothing. Example: when x >= 2, |x - 2| = x - 2.
The inside is negative
|A| = -A
If A < 0, we take the opposite expression. Example: when x < 2, |x - 2| = -(x - 2) = -x + 2.
The sign is unknown
|A| = ?
You need to consider cases or use the geometric interpretation. You must not simply delete the bars.
Step-by-step examples
The examples below are simple, but they show the key habit: first evaluate or determine the expression inside the absolute value, then decide whether it stays unchanged or changes sign.
Positive number
|7| = 7
7 is to the right of zero, so its distance from zero is 7.
Negative number
|-4| = 4
-4 lies 4 units from zero, so the result is positive.
Zero
|0| = 0
Zero is 0 units away from zero.
First calculate inside
|2 - 8| = |-6| = 6
First compute 2 - 8 = -6, and only then take the absolute value.
Sum of absolute values
|-3| + |5| = 3 + 5 = 8
Evaluate each absolute value separately, then add the results.
Minus before absolute value
-|-9| = -9
The absolute value of -9 is 9, and the minus before the absolute value remains.
Product
|-2 · 6| = |-12| = 12
You can also use the property: |-2| · |6| = 2 · 6 = 12.
Quotient
|-15 / 3| = |-5| = 5
First divide, then take absolute value. Equivalently: |-15| / |3| = 15 / 3 = 5.
Linear expression
for x = 1: |x - 4| = |1 - 4| = 3
Substitute x and evaluate the inside of the absolute value.
Removing bars on an interval
when x >= 3: |x - 3| = x - 3
On this interval, the inside x - 3 is nonnegative.
The other interval
when x < 3: |x - 3| = -x + 3
On this interval, the inside x - 3 is negative, so we change the sign of the whole expression.
Simple equation
|x| = 5 gives x = -5 or x = 5
We look for numbers that are 5 units away from zero.
Shifted equation
|x - 2| = 4 gives x = -2 or x = 6
We look for numbers that are 4 units away from 2.
Inequality closer to zero
|x| < 3 gives -3 < x < 3
x must lie at a distance less than 3 from zero.
Inequality farther from zero
|x| >= 2 gives x <= -2 or x >= 2
x must be at least 2 units away from zero.
Common student mistakes
Most mistakes with absolute value come from rushing: the bars are removed before the sign of the expression inside is checked.
Deleting bars without a condition
|x - 3| = x - 3 always
|x - 3| = x - 3 only for x >= 3
For x < 3, the expression x - 3 is negative, so absolute value changes its sign.
Losing the minus before absolute value
-|-5| = 5
-|-5| = -5
First evaluate the absolute value: |-5| = 5. The minus before it still remains.
One solution instead of two
|x| = 7, so x = 7
|x| = 7, so x = -7 or x = 7
Two numbers are 7 units away from zero: one on the left and one on the right.
Solving an equation with a negative value
|x - 1| = -3 has a solution
|x - 1| = -3 has no solution
An absolute value cannot be equal to a negative number.
Wrong parentheses when changing sign
|2x - 6| = -2x - 6 for x < 3
|2x - 6| = -(2x - 6) = -2x + 6
When we change the sign of an expression, we change the sign of every term.
Remember
Absolute value is distance
If you are unsure what to do, think about the number line. |x - a| means the distance from x to a.
The result is nonnegative
|A| is never less than zero. Equations such as |A| = -5 immediately have no solutions.
Remove bars only after checking the sign
For A >= 0 we have |A| = A, and for A < 0 we have |A| = -A.
Two cases are normal
In equations and inequalities, absolute value often splits the problem into a few simpler parts.
Most common exam tasks
In exams, absolute value rarely appears only as a definition. Most often, you combine the definition with the number line, an equation, an inequality, or a graph.
Evaluating an expression
Substitute the number, evaluate the inside of the absolute value, and only then take the absolute value.
- |2 - 9|
- |-3| + |4 - 10|
- -|5 - 8|
Equations with one absolute value
Most often, use the fact that |A| = c gives A = c or A = -c when c > 0.
- |x - 4| = 6
- |2x + 1| = 5
- |x + 7| = 0
Inequalities with absolute value
For the < sign, the solution is usually an interval between two numbers; for the > sign, it is a union of two intervals.
- |x| < 4
- |x - 2| >= 5
- |3x + 6| <= 12
Geometric interpretation
The task asks for points at a given distance from a number, or for an interval of such points.
- |x - 3| = 7
- |x + 1| < 2
- |x - a| = r
Expressions with several absolute values
Mark the zeros of the expressions inside the absolute values and split the number line into intervals.
- |x - 1| + |x + 2|
- |x| + |x - 3| = 5
- |x + 4| - |x - 2|
Graphs of functions with absolute value
Often you reflect the part of the graph below the x-axis or analyze shifts such as |x - a|.
- y = |x|
- y = |x - 2|
- y = |f(x)|
Next step
Once the definition is clear, move to the geometric interpretation. There, absolute value becomes distance on the number line, which makes equations and inequalities much easier.
Go to the number line