Absolute Value Equation Solver

Solve equations of the form |ax + b| = c with step-by-step solutions and verification.

Enter Equation Coefficients

|2x + (-3)| = 7
Coefficient of x
Constant inside |...|
Equals value

Solution

Number of Solutions
2
Solution(s) x = 5 or x = -2

Step-by-Step Solution

Solving Absolute Value Equations

An absolute value equation contains an expression inside absolute value bars. The most common form is |ax + b| = c.

The Three Cases

Case 1: c > 0 (Two Solutions)

Split into two equations:

ax + b = c ⇒ x = (c - b)/a
ax + b = -c ⇒ x = (-c - b)/a

Case 2: c = 0 (One Solution)

Only one equation:

ax + b = 0 ⇒ x = -b/a

Case 3: c < 0 (No Solution)

Since |...| ≥ 0 always:

No real solution exists

Graphical Interpretation

The graph y = |ax + b| is a V-shape with vertex at (-b/a, 0). The equation |ax + b| = c finds where this V intersects the horizontal line y = c.

  • c > 0: Line intersects V at two points (two solutions)
  • c = 0: Line touches the vertex (one solution)
  • c < 0: Line is below the V (no intersection, no solution)

Worked Example

Solve |2x - 3| = 7:

Case 1: 2x - 3 = 7 ⇒ 2x = 10 ⇒ x = 5
Case 2: 2x - 3 = -7 ⇒ 2x = -4 ⇒ x = -2

Verify: |2(5) - 3| = |7| = 7 ✓
Verify: |2(-2) - 3| = |-7| = 7 ✓

Common Mistakes

  • Forgetting to check if c < 0 before solving
  • Only finding one solution when c > 0
  • Not verifying solutions by substitution
  • Sign errors when handling the negative case