Absolute Value Calculator
Find the absolute value of any number, solve absolute value equations and inequalities, or calculate the distance between two points on a number line. Pick a mode, enter your values, and see every step of the work.
Formula
Worked example
Basic: |(-7.5)| = 7.5 because -7.5 is negative, so we negate it. Distance: |(-3) - 5| = |-8| = 8. Equation: |2x - 3| = 7 splits into 2x - 3 = 7 (x = 5) and 2x - 3 = -7 (x = -2). Inequality: |x - 2| < 5 gives -5 < x - 2 < 5, so -3 < x < 7.
What absolute value means
The absolute value of a number is its distance from zero on the number line, written with vertical bars as |x|. Distance is never negative, so the result is always zero or a positive number regardless of the input sign. For a positive number the absolute value is the number unchanged. For a negative number it is the number with its minus sign removed. Because |x| only measures magnitude, both a number and its opposite, such as 6 and -6, share the same absolute value of 6. The concept generalises naturally: |a - b| gives the distance between any two points a and b on the number line, without caring which is larger.
Solving absolute value equations
An absolute value equation such as |ax + b| = c has two cases unless c is zero. In the first case the expression inside the bars equals c, giving ax + b = c. In the second case it equals -c, giving ax + b = -c. Solve each linear equation separately to get two solutions. If c is negative there are no real solutions, because an absolute value can never be negative. If c is zero both cases collapse to the same equation and there is exactly one solution. Always substitute each answer back into the original equation to verify it.
Solving absolute value inequalities
Absolute value inequalities split based on direction. When you have |ax + b| < c (a "less than" inequality) the solution is a bounded interval: -c < ax + b < c, which you solve as a compound inequality to get a range of x values. When you have |ax + b| > c (a "greater than" inequality) the solution is an unbounded union: ax + b < -c or ax + b > c, which you solve as two separate inequalities. In both cases you divide by a at the end, remembering to flip the inequality sign if a is negative. The calculator shows the critical boundary values and the full solution set.
Where absolute value is used
Absolute value appears wherever size matters but direction does not. In statistics it underlies the mean absolute deviation, a measure of spread that averages how far data points fall from the mean. In physics it converts a signed displacement into a distance, or a velocity into a speed. Engineers use it to express tolerances: |measured - expected| <= tolerance means the reading is acceptable whether it falls above or below the target. The distance formula on the number line, |a - b|, generalises in higher dimensions to the Euclidean distance between two points.
Absolute value quick reference
| Problem type | Form | Number of solutions | Notes |
|---|---|---|---|
| Single number | |x| | One result (>= 0) | Always non-negative |
| Distance | |a - b| | One result (>= 0) | Same as |b - a| |
| Equation (c > 0) | |ax + b| = c | Two solutions | x = (c-b)/a and x = (-c-b)/a |
| Equation (c = 0) | |ax + b| = 0 | One solution | ax + b = 0 |
| Equation (c < 0) | |ax + b| = c | No real solution | Absolute value is never negative |
| Inequality (less than) | |ax + b| < c | Bounded interval | -c < ax + b < c |
| Inequality (greater than) | |ax + b| > c | Two unbounded rays | ax + b < -c or ax + b > c |
Common patterns and how the calculator handles each case.
Frequently asked questions
Can an absolute value ever be negative?
No. Absolute value measures distance from zero, and distance is never negative. The result of |x| is always zero or a positive number, even when the input is negative.
What is the absolute value of zero?
It is zero. Zero is already at the origin of the number line, so its distance from zero is nothing. Zero is the only number whose absolute value equals the number itself.
Why do |5| and |-5| give the same result?
The numbers 5 and -5 lie the same distance from zero in opposite directions. Absolute value ignores direction and keeps only the magnitude, so both return 5.
How many solutions does an absolute value equation have?
When the right-hand side c is positive there are typically two solutions, because the expression inside can equal either c or -c. When c is zero there is exactly one solution. When c is negative there are no real solutions, because an absolute value can never be negative.
What is the difference between a "less than" and "greater than" absolute value inequality?
A "less than" inequality, such as |x| < 5, gives a bounded interval of solutions (here -5 < x < 5) because x must be close to zero. A "greater than" inequality, such as |x| > 5, gives an unbounded union of two rays (here x < -5 or x > 5) because x must be far from zero.
How does the distance mode differ from just subtracting two numbers?
Subtracting gives a signed result: 3 - 8 = -5, meaning 3 is below 8. The distance |3 - 8| = 5 gives the magnitude without the sign, which is the same regardless of which point you subtract from which. The calculator also shows the midpoint between the two values.