Absolute Value Calculator
Find the absolute value of a number or expression, the distance between two values, and solutions to |x| = a.
Calculator
Calculate absolute value
Absolute value removes the sign and gives distance from zero.
Supports integers, decimals, fractions, negatives, grouped numbers like 1,000, and simple expressions like 3-8.
Results
Your result
Absolute value of a number
Absolute value
7
|-7| = 7
Original input
-7
Sign of input
Negative
Fraction form
7
Formula used
|x| = x when x >= 0, and |x| = -x when x < 0
-7 is 7 units from zero, so |-7| = 7.
Interpretation
What the result means
In plain words
-7 is 7 units from zero, so |-7| = 7.
Distance is non-negative
Absolute value is never negative because it represents distance on the number line.
Signs and magnitude
Negative inputs become positive magnitudes; positive inputs stay the same.
Distance formula
|a - b| gives the distance between a and b, no matter which value is larger.
Formula
Absolute value formulas
Absolute Value Definition
|x| = x if x >= 0
Negative Input Rule
|x| = -x if x < 0
Distance From Zero
Distance from zero = |x|
Distance Between Two Numbers
Distance = |a - b|
Basic Absolute Value Equation
|x| = a means x = a or x = -a, when a >= 0
Notation
Symbols and notation
x
The input number or expression inside the absolute value bars.
|x|
The absolute value of x, or its distance from zero.
a and b
Two values being compared on a number line.
|a - b|
The distance between a and b.
Bars are not parentheses
Parentheses group expressions; absolute value bars measure distance or magnitude.
Non-negative output
Absolute value output is always zero or positive.
Number line
Number line explanation
-5 and 5 are on opposite sides of zero, but both are 5 units from zero. That is why |-5| = 5 and |5| = 5.
Examples
Worked examples
Find |-9|
Rule: Negative numbers become positive magnitude.
Substitution: |-9| = 9
Answer: 9
-9 is 9 units from zero.
Find |12|
Rule: Positive numbers keep the same value.
Substitution: |12| = 12
Answer: 12
12 is already 12 units from zero.
Find |0|
Rule: Zero is zero units from zero.
Substitution: |0| = 0
Answer: 0
Zero is the only number whose absolute value is zero.
Find |-3.75|
Rule: Remove the negative sign.
Substitution: |-3.75| = 3.75
Answer: 3.75
Distance does not depend on direction.
Find |-2/5|
Rule: A negative fraction becomes positive.
Substitution: |-2/5| = 2/5
Answer: 2/5
The magnitude of the fraction is 2/5.
Distance between -4 and 7
Rule: Distance = |a - b|
Substitution: |-4 - 7| = |-11|
Answer: 11
-4 and 7 are 11 units apart.
Solve |x| = 6
Rule: |x| = a gives two values when a is positive.
Substitution: x = 6 or x = -6
Answer: x = 6 or -6
Both values are 6 units from zero.
Explain why |x| = -3 has no solution
Rule: Absolute value cannot be negative.
Substitution: No distance can equal -3
Answer: No real solution
Distances are zero or positive.
Equations
Absolute value equations and inequalities
|x| = a
Two solutions when a is positive: x = a and x = -a.
|x| = 0
One solution: x = 0.
|x| = negative
No real solution because absolute value cannot be negative.
|x| < a
x is within a units of zero when a is positive.
|x| > a
x is more than a units from zero.
Screening idea
These are beginner rules, not a full algebra solver.
Mistakes
Common mistakes
Thinking absolute value always changes a number
Positive numbers and zero stay the same.
Writing |-5| = -5
The correct value is 5 because -5 is 5 units from zero.
Confusing bars with parentheses
Absolute value bars change the result to a distance or magnitude.
Forgetting distance meaning
Distance does not have a negative direction.
Solving |x| = -4
There is no real solution because absolute value cannot be negative.
Mishandling |3 - 8|
First simplify inside the bars: 3 - 8 = -5, then |-5| = 5.
FAQ
Frequently asked questions
Related tools
More tools to try
Related reading
Learn the formula, assumptions, examples, or context behind this tool.