What is an absolute value equation?

An absolute value equation contains an expression inside absolute value bars, such as |ax + b| equals c. The absolute value of a number is its distance from zero, always non-negative. Solving requires splitting the equation into two cases because the expression inside can be either positive or negative.

How do you solve an absolute value equation?

First isolate the absolute value expression by moving constants and dividing out coefficients. If the right side is positive, split into two cases: the expression inside equals the right side, or equals its negative. Solve each case as a linear equation. The solutions of both cases are the final answers.

When does an absolute value equation have no solution?

When the isolated absolute value equals a negative number. Since an absolute value is always greater than or equal to zero, no real value of the variable can make |expression| equal a negative number. The solver detects this case immediately and reports no solution.

What happens when the absolute value equals zero?

When |expression| equals zero, the expression inside must equal exactly zero. This gives a single solution rather than two, because there is only one value (zero) whose absolute value is zero. The two cases collapse into one.

How do I solve |A| = |B|?

When both sides are absolute values, two cases arise: A equals B, or A equals negative B. Solve each as a linear equation. Equating A with positive B gives one candidate; equating A with negative B gives another. Both are valid solutions of the original equation.

Absolute Value Equation Solver - Step-by-Step

Getting Started

Demo|x − 3| = 5, Solve, Clear, bars equal to zero

Step 0 of 4

The solver shows an equation display at the top with a blinking yellow caret marking the cursor position. Click anywhere in the display to place the cursor, or use arrow keys, Home, and End for keyboard navigation. Type directly or use the button panel below.

Three quick experiments:

• Type

|x - 3| = 5

and press Enter — the right panel splits the equation into

x - 3 = 5

and

x - 3 = -5

, then solves both branches to give

x = 8

or

x = -2

.
• Click the "No Solution" example template — an equation with a negative right side loads. The solver detects this immediately and reports no solution.
• Enter

|2x + 1| = 0

to see the zero case: only one solution exists because zero has only one absolute value preimage.

The Solve button is disabled until you enter something. Pressing Enter on the display is equivalent to clicking Solve. The special row offers the absolute value bars button (rendered as

|

|

). An expression without an equals sign is refused with "Input must be an equation (use = sign)", one without bars with "No absolute value found", and a non-linear expression inside the bars with its own message.

|2x + 1| = 0

gives the single solution

x = -0.5

. What the bars mean is explained in what is absolute value.

Building Equations with Buttons and Keyboard

DemoA coefficient and a constant, undo, delete, Solve

Step 0 of 4

The button panel groups inputs by type. All inputs can be made either by clicking or typing on the keyboard.

• Variable row —

x

,

y

,

n

. The solver picks up whichever variable appears in the equation.
• Number row — digits 0 through 9 and the decimal point.
• Operator row — multiplication, division, plus, minus, equals.
• Special row — absolute value bars button, and parentheses.

The absolute value bars button inserts a single

|

character; click it twice to surround an expression. Keyboard users can type the vertical bar character directly. For multi-term expressions inside the bars, type them between the two bars:

|2x + 3|

.

Keyboard shortcuts: type letters and digits directly, use star or slash for multiplication and division, and press Enter to solve. Ctrl+Z undoes up to fifty edits back; Backspace deletes the character before the cursor; Delete removes the one after it. The Clear button empties the display entirely; the curved-arrow button steps through the undo stack one edit at a time; the ⌫ button deletes the character before the caret. Why the bars make two cases out of one equation is explained in what is absolute value.

Try an Example — Eight Form Templates

DemoOpen the templates, load two forms

Step 0 of 4

Click the "Try an Example" header to expand the template panel. Each card generates a random equation of that form. Clicking again produces a new random version.

• Simple —

|x + a| = b

. Coefficient of 1 on the variable; the base case.
• With Coefficient —

|ax + b| = c

. Requires solving each branch as a two-step linear equation.
• Equals Zero —

|ax + b| = 0

. Single solution; the two cases collapse.
• No Solution —

|expression| =

negative. The solver detects and rejects immediately.
• Isolate First —

|x + a| + b = c

. Requires subtracting

b

before splitting into cases.
• Coefficient Outside —

a \cdot |x + b| = c

. Requires dividing by

a

before splitting.
• Abs Both Sides —

|ax + b| = |cx + d|

. Uses the

A = B

or

A = -B

case split.
• Fraction Inside —

|x/a + b| = c

. Tests fractional-coefficient handling inside the bars.

Roughly 80 percent of generated equations produce clean integer solutions; the rest exercise the special cases and edge scenarios. Loading a template clears the previous solution; click Solve to work the new equation. The split applied to every template is justified in the two-case split.

Reading the Step-by-Step Solution

DemoBars on both sides, then no solution

Step 0 of 3

The solution panel shows each algebraic move as a labeled step. The step sequence depends on the structural case.

• Rearrange — if the absolute value is on the right side, the solver swaps sides for clarity.
• Isolate Absolute Value — subtracts an additive constant or divides by a coefficient outside the bars to leave

|\text{expression}| = c

alone on one side.
• No Solution (when applicable) — if the isolated right side is negative, the solver flags impossibility and stops.
• Zero Case (when applicable) — if the isolated right side equals zero, the inner expression must equal zero. Single solution returned.
• Split into Two Cases — for

|A| = c > 0

, splits into

A = c

and

A = -c

.
• Case 1: Positive — solves the linear equation

A = c

for the variable.
• Case 2: Negative — solves the linear equation

A = -c

for the variable.
• Verify Both Solutions — substitutes each candidate back into the original to confirm the absolute value evaluates to the right side.

For

|A| = |B|

equations, the solver uses Case 1: A = B and Case 2: A = -B instead of the positive/negative split.

|x - 1| = |2x + 4|

gives

x = -5

or

x = -1

;

|x| + 3 = 1

isolates to

|x| = -2

and ends in No solution. The box under the solver, How the solver works, summarises the method in a few lines and links to the theory below. The three stages are set out in the solving process explained.

The Two-Case Split — Why It Works

The absolute value function strips off the sign of its input.

|5| = 5

and

|-5| = 5

. Two different inputs can produce the same output.

The principle. For any real number

A

,

|A| = \begin{cases} A & \text{if } A \geq 0 \\ -A & \text{if } A < 0 \end{cases}

So

|A| = c

does not determine the sign of

A

. Both

A = c

and

A = -c

satisfy the equation. The solver must consider both possibilities and produce both solutions (or whichever turns out to be valid).

Example.

|x - 4| = 7

. The expression inside is

x - 4

. Two cases:

• Case 1:

x - 4 = 7

, so

x = 11

.
• Case 2:

x - 4 = -7

, so

x = -3

.

Check:

|11 - 4| = |7| = 7

✓ and

|{-3} - 4| = |{-7}| = 7

✓. Both solutions are valid.

Edge cases the solver detects automatically.

• $|A| = c$ with $c < 0$ : no solution. An absolute value cannot be negative.
• $|A| = 0$ : one solution. The inner expression must be exactly zero, so the two cases collapse.
• $|A| = |B|$ : two cases on inputs rather than outputs. Either

A = B

or

A = -B

. The principle is the same: both sign assignments of the right-hand expression must be checked.

For deeper coverage see absolute value and piecewise definition of absolute value.

The split is worked in the basic equation and solving equations of the form |f(x)| = k; when a case can fail is absolute value and extraneous solutions.

What is Absolute Value?

The absolute value of a real number

x

, written

|x|

, is its distance from zero on the number line:

|x| = \begin{cases} x & \text{if } x \geq 0 \\ -x & \text{if } x < 0 \end{cases}

Equivalently,

|x| = \sqrt{x^2}

. Absolute value is always non-negative:

|x| \geq 0

for every real

x

, with equality only when

x = 0

.

Key properties.

• Non-negativity:

|x| \geq 0

, with equality iff

x = 0

.
• Symmetry:

|-x| = |x|

. The function is even.
• Multiplicativity:

|xy| = |x| \cdot |y|

.
• Triangle inequality:

|x + y| \leq |x| + |y|

, with equality when

x

and

y

have the same sign.

Graph. The graph of

y = |x|

is a V-shape with vertex at the origin and slopes of

\pm 1

on the two branches. The graph of

y = |ax + b|

is a V with vertex at

x = -b/a

.

Geometric interpretation.

|x - a|

is the distance between

x

and

a

on the number line. The equation

|x - a| = c

asks: which points are exactly

c

units from

a

? Answer:

a + c

and

a - c

.

For deeper coverage see absolute value function, distance on the number line, and piecewise functions.

The definition is given in definition of absolute value, the distance reading in geometric interpretation, and the place of these equations among the equation types in absolute value equations.

The Solving Process Explained

Three stages solve any absolute value equation. The first is mandatory; the second branches on the structural case.

• Stage 1: Isolate the absolute value. Move every term outside the bars to the other side. Subtract additive constants, then divide by any coefficient multiplying the absolute value expression. The goal is to reach the form

|A| = c

where

A

is the expression inside the bars and

c

is a single number on the other side.

• Stage 2: Examine the right side.

• If $c < 0$ : no solution. Stop.
• If $c = 0$ : the inner expression equals zero. Solve

A = 0

as a linear equation. Single solution.
• If $c > 0$ : split into two cases.

• Stage 3: Solve the two cases (when $c > 0$ ).

• Case 1:

A = c

. Solve as a linear equation in

x

.
• Case 2:

A = -c

. Solve as a linear equation in

x

.

Both solutions are valid by construction. No extraneous-check step needed for the standard form because the squaring trap of radical equations does not arise here.

Variant: $|A| = |B|$ . Both sides absolute values. The analogous split: Case 1 is

A = B

, Case 2 is

A = -B

. Each yields a linear equation. Both solutions are valid.

Variant with coefficient outside:

a \cdot |A| + d = c

. Subtract

d

, divide by

a

, then proceed with the standard split.

For comprehensive treatment see solving absolute value equations and case analysis.

The stages are worked in solving equations of the form |f(x)| = k and equations with absolute value on both sides; several bars at once are treated in equations with multiple absolute value terms, and the forms are summarised in summary of absolute value equation forms.

Related Concepts

Absolute Value Function —

f(x) = |x|

. V-shaped graph with vertex at the origin.

Absolute Value Inequalities — equations with

<

,

>

,

\leq

, or

\geq

instead of equals. Two subtypes:

|A| < c

becomes a compound inequality

-c < A < c

;

|A| > c

becomes a union of two inequalities

A < -c

or

A > c

.

Piecewise Functions — functions defined by different rules on different intervals. The absolute value is the canonical example:

|x|

equals

x

on

[0, \infty)

and

-x

on

(-\infty, 0)

.

Distance Formula — the geometric interpretation of absolute value.

|x - a|

is the distance between points

x

and

a

on the real line.

Triangle Inequality —

|x + y| \leq |x| + |y|

. The fundamental inequality relating absolute values; appears throughout analysis.

Modulus — the absolute value generalized to complex numbers:

|a + bi| = \sqrt{a^2 + b^2}

. The geometric interpretation as distance from origin extends to the complex plane.

Norm — the further generalization to vectors and other spaces. The absolute value is the one-dimensional norm.

Linear Equations — the equation type that results after splitting cases. The solver routes each case through a linear-equation step.

Compound Inequalities — used to express the solution set of an absolute value inequality. The form

|x - a| \leq c

corresponds to the compound inequality

a - c \leq x \leq a + c

.

The related tools are the absolute value inequality solver, the linear equation solver and the complex numbers calculator for the modulus; the lessons are geometric interpretation and absolute value equations.