Absolute Value Equation Solver

The absolute value of a number is its distance from zero, so it is never negative. To solve |ax + b| = c, split it into two equations, ax + b = c and ax + b = −c, and solve each. If c is negative there is no solution. This solver shows both cases and graphs the V-shaped function.

Solves |ax + b| = c.
Try:

How to use the absolute value equation solver

  1. Enter a, b and c from |ax + b| = c.
  2. Press Calculate.
  3. Read the solutions, the two cases and the graph.

Formula

∣u∣=c⇒u=c or u=−c|u| = c \Rightarrow u = c \text{ or } u = -c

Worked example: |2x − 3| = 7

  • a: 2
  • b: -3
  • c: 7

✓ Answer checked

x=−2  or  x=5x = -2 \;\text{or}\; x = 5
−4−2246824681012x = -2x = 5|ax + b|y = 7
The V-shaped graph meets the horizontal line at the solutions.
Step-by-step working (3 steps)

  1. Split into two cases |u| = c ⇒ u = ±c

    If |u| = c, then u = c or u = −c.

    ∣2x−3∣=7\left|2x - 3\right| = 7

  2. Case 1

    2x - 3 = 7

    x=5x = 5

  3. Case 2

    2x - 3 = -7

    x=−2x = -2
How this was checked
  • ✓ Each solution makes the absolute value exactly equal to c.

Frequently asked questions

How do I solve |2x − 3| = 7?

2x − 3 = 7 gives x = 5, and 2x − 3 = −7 gives x = −2.

Why does |3x − 1| = −2 have no solution?

An absolute value can never be negative.

When is there only one solution?

When c = 0, because both cases give the same equation ax + b = 0.

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