Absolute Value Equation Calculator

Absolute value measures distance from zero, which means it's always zero or positive — and that one property is what makes absolute value equations behave differently from ordinary linear equations. Most have two solutions instead of one, some have none at all, and inequalities split into two different shapes depending on the direction of the sign. This calculator covers the five absolute value problems that come up most often in algebra: evaluating |x|, solving |x| = a, solving |ax + b| = c, solving |ax + b| = |cx + d|, and solving absolute value inequalities.

Quick Answer

|x| = a → x = a or x = −a (two solutions, only when a ≥ 0)
Example: |2x − 3| = 7 → 2x − 3 = 7 gives x = 5, and 2x − 3 = −7 gives x = −2
Pick the exact type of problem you have in the tool below — each one is explained with formulas further down the page.

Find the absolute value of any number — e.g. |−7| = 7.

Number (x):

Solve the simplest absolute value equation, |x| = a.

a (the value on the right side):

Solve an absolute value equation of the form |ax + b| = c — e.g. |2x − 3| = 7.

a (coefficient of x):

b (constant inside the bars):

c (value on the right side):

Solve an equation with an absolute value on each side, |ax + b| = |cx + d|.

a (coefficient of x, left side):

b (constant, left side):

c (coefficient of x, right side):

d (constant, right side):

Solve an absolute value inequality of the form |ax + b| < c or |ax + b| > c.

a (coefficient of x):

b (constant inside the bars):

Inequality sign:

c (value on the right side):

Result:

What Absolute Value Means

The absolute value of a number, written |x|, is its distance from zero on the number line — and distance is never negative. So |5| = 5 and |−5| = 5, since both numbers sit exactly 5 units from zero, just in opposite directions. This single fact — that an absolute value expression can equal a positive number in two different ways (from a positive input or a negative one) — is the reason absolute value equations typically split into two separate equations to solve.

|x| is always ≥ 0, no matter what x is
An absolute value equation set equal to a negative number has no solution
An equation like |expression| = k (k > 0) always splits into two cases: expression = k and expression = −k
Absolute value inequalities produce a range of solutions, not a single number

Solving |x| = a

RULE If a > 0: x = a or x = −a | If a = 0: x = 0 | If a < 0: no solution

Example: Solve |x| = 9
x = 9 or x = −9

This is the base case every other absolute value equation reduces to. Since |x| can never be negative, an equation like |x| = −4 is automatically impossible — there is no number whose distance from zero is negative four.

Solving |ax + b| = c

RULE If c < 0: no solution. Otherwise, split into: ax + b = c and ax + b = −c

Example: Solve |2x − 3| = 7
2x − 3 = 7 → x = 5
2x − 3 = −7 → x = −2

First isolate the absolute value expression completely on one side of the equation. Then write the two linear equations — one for the positive case, one for the negative case — and solve each independently. This can produce zero, one, or two solutions depending on the value of c.

Solving |ax + b| = |cx + d|

RULE ax + b = cx + d or ax + b = −(cx + d)

Example: Solve |3x + 1| = |x + 9|
3x + 1 = x + 9 → 2x = 8 → x = 4
3x + 1 = −(x + 9) → 4x = −10 → x = −2.5

Two absolute value expressions are equal exactly when the expressions inside them are equal, or when one is the exact negative of the other — since |k| = |−k| for any number k. Both resulting linear equations should be solved, and it's always worth checking each answer in the original equation.

Absolute Value Inequalities

RULE — "less than" |ax + b| < c → −c < ax + b < c (a single range)
RULE — "greater than" |ax + b| > c → ax + b > c or ax + b < −c (two separate ranges)

Example: Solve |x − 2| < 5
−5 < x − 2 < 5 → −3 < x < 7

A "less than" absolute value inequality describes values that stay close to a center point, producing one continuous interval. A "greater than" inequality describes values that are far from that point in either direction, producing two separate, non-overlapping ranges. As with equations, if c is negative there is no solution for "less than," and every real number is a solution for "greater than."

The Trap: Forgetting the Negative Case

COMMON MISTAKE
Solving |x − 4| = 10 as just x − 4 = 10, giving only x = 14
Correct: also solve x − 4 = −10, giving the second solution x = −6

The single most common error in absolute value equations is solving only the positive case and stopping there. Because |x − 4| = 10 is true whenever x − 4 equals 10 or −10, skipping the negative case throws away a valid solution. Whenever the right side of an isolated absolute value equation is positive, expect two equations to solve, not one.

Real-Life Uses

Error and tolerance ranges — manufacturing and engineering specs often state a measurement plus or minus a tolerance, which is exactly an absolute value inequality (|measured − target| ≤ tolerance)
Temperature and distance thresholds — describing "within 5 degrees of a target temperature" or "within 3 miles of a location" is naturally an absolute value statement
Statistics and quality control — control charts flag any measurement whose deviation from the mean, in absolute value, exceeds a set limit
Programming and error handling — many numerical algorithms stop once the absolute difference between two successive results falls below a small threshold
Standardized test algebra sections — absolute value equations and inequalities are staples of SAT, ACT, and general algebra coursework

Glossary

Absolute Value: The non-negative distance of a number from zero on the number line, written with vertical bars, as in |x|.
Isolate: To rearrange an equation so the absolute value expression stands alone on one side before splitting it into cases.
Extraneous Solution: A value produced while solving an equation that does not actually satisfy the original equation, and must be discarded after checking.
Compound Inequality: An inequality written with two inequality signs at once, such as −c < x < c, describing a single continuous range.
No Solution: The outcome when an absolute value expression is set equal to (or less than) a negative number, which is impossible.

Frequently Asked Questions

Q: How do you solve an absolute value equation?

A: Isolate the absolute value expression, then split it into two equations — one where the inside equals the positive value, one where it equals the negative value — and solve both.

Q: What is |x| = a and when does it have no solution?

A: It asks for numbers a distance of a from zero. Two solutions if a > 0, one solution (x = 0) if a = 0, and no solution if a < 0.

Q: How do you solve |ax + b| = c?

A: If c is negative, no solution. Otherwise solve ax + b = c and ax + b = −c separately. For |2x − 3| = 7, that gives x = 5 and x = −2.

Q: How do you solve an absolute value equation with absolute values on both sides?

A: Solve ax + b = cx + d and ax + b = −(cx + d) as two separate linear equations.

Q: How do you solve an absolute value inequality?

A: For "less than," rewrite as −c < ax + b < c for one range. For "greater than," split into ax + b > c or ax + b < −c for two ranges.

Q: Why do absolute value equations sometimes produce extraneous solutions?

A: Algebraic manipulation can introduce values that don't satisfy the original equation, so every solution should be checked by substitution before it's accepted.