What is absolute value in math?
The absolute value of a number is its distance from zero on the number line, so it is never negative. It is written with two vertical bars: |−6| = 6 and |6| = 6. A positive number and 0 keep their value; a negative number loses its minus sign.
Absolute Value in a picture
Formal definition and notation
For any real number x, the absolute value is |x| = x when x ≥ 0 and |x| = −x when x < 0. On the number line it is the distance between x and 0, and the distance between any two numbers a and b is |a − b|. Because a distance is never negative, |x| ≥ 0 for every x, and |x| = 0 only when x = 0.
| Symbol | Read it as | Meaning |
|---|---|---|
| |x| | the absolute value of x | the distance from x to 0, for example |−9| = 9 |
| |a − b| | the absolute value of a minus b | the distance between a and b on the number line |
| −|x| | the opposite of the absolute value of x | the minus sign is outside the bars, so −|−4| = −4 |
Key facts
- Absolute value is never negative: |x| ≥ 0. It equals 0 only for x = 0.
- A number and its opposite have the same absolute value: |−12| = |12| = 12.
- The distance between two numbers is the absolute value of their difference: from −2 to 5 is |5 − (−2)| = 7.
- Bigger absolute value does not mean bigger number: |−10| > |−3|, but −10 < −3.
- The equation |x| = 6 has two solutions, 6 and −6. The equation |x| = −6 has none.
Absolute value rule
- x
- any number: positive, negative or zero
- −x
- the opposite of x; when x is negative, −x is positive, so |−7| = −(−7) = 7
Work out what is inside the bars first, then take the distance from 0: |2 − 10| = |−8| = 8.
Try it: measure the distance
Drag the two points. The readout gives each point’s absolute value and the distance between them, which is the absolute value of their difference. Try putting one point on 0.
Absolute Value examples, step by step
Example 1: Opposites
Find |−15| and |15|.
- −15 is 15 units to the left of 0, so |−15| = 15.
- 15 is 15 units to the right of 0, so |15| = 15.
- Both numbers are the same distance from 0, in opposite directions.
Answer: Both absolute values are 15.
Example 2: Work inside the bars first
Evaluate |4 − 9| − |−3|.
- Inside the first bars: 4 − 9 = −5, so |4 − 9| = |−5| = 5.
- The second absolute value: |−3| = 3.
- Subtract: 5 − 3 = 2.
Answer: The value is 2.
Example 3: Distance in a real situation
At 6 a.m. the temperature in Fargo was −11 degrees Fahrenheit. By 3 p.m. it was 14 degrees Fahrenheit. By how many degrees did the temperature rise?
- The change is the distance between −11 and 14 on the number line.
- Distance = |14 − (−11)|.
- Subtracting a negative means adding: 14 + 11 = 25, and |25| = 25.
- Check with the number line: 11 degrees up to 0, then 14 more, for 25 in all.
Answer: The temperature rose 25 degrees Fahrenheit.
Common mistakes
Do not keep the negative sign inside the bars. Absolute value is a distance, and a distance is never negative.
The bars act like grouping symbols. Finish the work inside them, then take the absolute value of the result.
Only the minus sign inside the bars is removed. The minus sign outside the bars still takes the opposite of the result.
Two numbers sit at any positive distance from 0, one on each side.
Not to be confused with
The opposite flips the sign; absolute value removes it. The opposite of −4 is 4 and the opposite of 4 is −4, but |−4| and |4| are both 4.
The additive inverse of 7 is −7, the number that adds to 7 to make 0. The absolute value of 7 is 7, its distance from 0.
Quick check: 3 questions
What is |−11| + |3|?
Answer: B. |−11| = 11 and |3| = 3, so the sum is 14. Choice A adds −11 + 3 first and drops the sign, and the negative choices keep a minus sign that absolute value removes.Show answer and explanation
Which number has the greatest absolute value: 7, −3, −12, 10?
Answer: C. −12 is 12 units from 0, farther than any other choice. 10 is the greatest number, but greatest value and greatest absolute value are different questions.Show answer and explanation
Which values of x make |x| = 6 true?
Answer: A. Both 6 and −6 are 6 units from 0. Choices B and C each miss one side of 0. There would be no solution only if the right side were negative, as in |x| = −6.Show answer and explanation
Absolute Value: frequently asked questions
Can absolute value be negative?
No. Absolute value is a distance from 0, and a distance is never negative, so |x| ≥ 0 for every number. That is why |x| = −3 has no solution. An expression such as −|5| is negative, but there the minus sign sits outside the bars: it takes the opposite of the absolute value, −|5| = −5.
What is the absolute value of 0?
The absolute value of 0 is 0, because 0 is zero units away from itself. It is the only number whose absolute value is 0. Every other number, positive or negative, has a positive absolute value, such as |0.5| = 0.5 and |−0.5| = 0.5.
Why do we use absolute value in real life?
Use it whenever only the size of a change or a difference matters, not its direction. A debt of 30 dollars is written −30, and its size is |−30| = 30 dollars. A temperature change from −4 to 9 degrees is |9 − (−4)| = 13 degrees. Tolerances such as ‘within 2 millimeters’ are also written with absolute value.
Practice and tools
Sources
- Common Core State Standards, Grade 6 The Number System (6.NS.C.7.c, 6.NS.C.7.d)
- Common Core State Standards, Grade 7 The Number System (7.NS.A.1.c)
- OpenStax, Prealgebra 2e, 3.1 Introduction to Integers
Updated November 2026·Reviewed by the Effortless Math team·Standards: CCSS 6.NS.C.7.c, CCSS 6.NS.C.7.d, CCSS 7.NS.A.1.c
Cite this entry
Effortless Math Team. “Absolute Value.” Effortless Math Dictionary. Updated November 2026. https://www.effortlessmath.com/math-dictionary/absolute-value/
Flashcard
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