Last updated: 2026-05-19

The absolute value equation

An absolute value equation in the form $|ax + b| = c$ asks for every value of $x$ whose distance from the origin — after applying the linear transformation $ax + b$ — equals exactly $c$. This calculator handles all three outcome types: two solutions, one solution, and no real solution.

How the formula works

The key insight is that $|expression| = c$ means the expression equals either $c$ or $-c$, because both are the same distance from zero. This splits one absolute value equation into two ordinary linear equations:

Solving each for $x$ (assuming $a \neq 0$):

The three cases

Case 1: c > 0 — two solutions

When $c$ is positive, the two equations above give distinct values of $x$. Both satisfy the original equation.

Worked example: Solve $|2x - 3| = 7$.

Identify: $a = 2$, $b = -3$, $c = 7$.

Check: $|2(5) - 3| = |7| = 7$ ✓ and $|2(-2) - 3| = |-7| = 7$ ✓

Case 2: c = 0 — one solution

When $c = 0$, the two equations collapse to a single equation $ax + b = 0$. The two formulas give the same result:

Example: $|2x - 4| = 0 \Rightarrow x = 4 / 2 = 2$.

Case 3: c < 0 — no real solution

Absolute value is always non-negative: $|ax + b| \geq 0$ for every real $x$. A negative $c$ asks when a non-negative quantity equals a negative number — which never happens. The equation has no real solution.

Example: $|2x - 3| = -5$ has no solution.

The distance interpretation

On the number line, $|u|$ is the distance from the point $u$ to zero. Setting $u = ax + b$:

asks: "For which $x$ does the output of the linear function $ax + b$ land exactly $c$ units away from zero?" Since distance is symmetric, there are two such landing spots ($+c$ and $-c$), giving at most two values of $x$.

This geometric picture also explains why $c < 0$ yields no solution: distance is never negative, so there are no landing spots.

The degenerate case a = 0

If $a = 0$, the equation becomes $|b| = c$, which no longer involves $x$ at all. The calculator treats $a = 0$ as an invalid input and shows an error, because the equation is not an absolute value equation in $x$ — it is just an arithmetic statement.