Absolute value calculator, equations and inequalities too.

The absolute value of a number is its distance from 0 on the number line, so it is never negative: |−7| = 7 and |7| = 7. This absolute value calculator gives the absolute value of any number or sum, the distance between two numbers, the size (modulus) of a complex number, and solves absolute value equations and inequalities such as |2x − 3| < 7, with the solution as an inequality, in interval notation and on a number line.

How to use the absolute value calculator

1. Choose Numbers and sums, Solve |ax + b|, |u| and |v|, Distance between or Complex number.
2. For numbers, type one per line. Sums work too, and so do bars: |2 − 9| + 1 or abs(-4). A table shows each result when you type several.
3. To solve, fill in a, b and c for |ax + b|, add a k if a number sits outside the bars (like |2x − 3| + 4), pick the comparison, and read the answer, its interval and the number line. Steps shows the two cases.
4. For |ax + b| against |cx + d|, use |u| and |v|.
5. Copy result copies the answers, and Copy link keeps your inputs.

The rule

If a number is 0 or more, its absolute value is itself. If it is negative, change its sign. In symbols, |x| = x when x ≥ 0 and |x| = −x when x < 0. Examples: |−7.5| = 7.5, |3 − 8| = |−5| = 5 and |0| = 0. Absolute value also keeps the right order of work: do the sum inside the bars first, then drop the sign, so |2 − 9| + 1 = |−7| + 1 = 8.

Distance between two numbers

The distance between a and b is |a − b|, and it does not matter which comes first. For −4 and 9, |−4 − 9| = |−13| = 13, and |9 − (−4)| = 13 too. This is why absolute value shows up in questions about temperature differences, errors and tolerances.

Solving an absolute value equation

For |2x − 3| = 7, the inside must be 7 or −7. So 2x − 3 = 7 gives x = 5, and 2x − 3 = −7 gives x = −2. The two answers are the same distance from the centre of the expression, which is x = 3/2. If the right side is negative, such as |x| = −2, there is no solution, because an absolute value is never negative. If it is 0, you get a single answer.

Solving an absolute value inequality

Less than means between: |2x − 3| < 7 becomes −7 < 2x − 3 < 7, so −4 < 2x < 10 and −2 < x < 5. In interval notation that is (−2, 5), with open circles on the number line. Greater than means outside: |2x − 3| ≥ 7 gives x ≤ −2 or x ≥ 5, which is (−∞, −2] ∪ [5, ∞), with filled circles. When you divide by a negative number, flip the inequality signs. The calculator does this and says so in the steps.

A number outside the bars

For |2x − 3| + 4 < 9, first get the bars alone by taking 4 from both sides: |2x − 3| < 5. Then −5 < 2x − 3 < 5 gives −1 < x < 4. Type 4 in the k box and 9 in c. If k pushes c below 0, the page tells you why there is no solution or why every number works.

Two absolute values

For |x − 2| = |3x + 1| the insides are equal or opposite. x − 2 = 3x + 1 gives x = −3/2, and x − 2 = −(3x + 1) gives x = 1/4. For an inequality such as |x − 2| < |3x + 1|, square both sides: both are never negative, so the order stays. That gives (u − v)(u + v) < 0, a product of two straight lines. Here that is (−2x − 3)(4x − 1) < 0, which is true for x < −3/2 or x > 1/4. The page tests each part between the zeros and the zeros themselves, then draws the answer.

Absolute value of a complex number

For a + bi the absolute value is the distance from the origin on the complex plane: √(a² + b²). For 3 + 4i it is √(9 + 16) = 5, and for 2 + 6i it is √40 = 2√10 ≈ 6.3246. The calculator keeps the exact root.

Common problems

• Dropping the sign too early. |−3 + 5| is |2| = 2, not 3 + 5. Work inside the bars first.
• Forgetting the second case. An equation like |x − 4| = 3 has two answers, 7 and 1.
• No solution is a valid answer. |x| < −1 has none, and |x| > −1 is true for every x. The page says which.

For powers and roots use the exponent calculator, and for rounding a result or comparing two values see the greater than less than calculator. To draw |x| as a V shape, type abs(x) in the graphing calculator.