Absolute Value Equations Solver

Solve absolute value equations step by step |ax + b| = c, |ax + b| = |cx + d|, and inequalities

Equation Solver
| x + | =

Absolute Value Rules

  • |x| = c → x = c or x = -c (if c ≥ 0)
  • |ax + b| = c → ax + b = c or ax + b = -c
  • |A| = |B| → A = B or A = -B
  • |x| < c → -c < x < c
  • |x| > c → x < -c or x > c

Key Properties

  • |x| ≥ 0 for all real x
  • |x| = |-x|
  • |x × y| = |x| × |y|
  • |x / y| = |x| / |y| (y ≠ 0)
  • |x + y| ≤ |x| + |y| (Triangle Inequality)

Graphical Meaning

  • |x - a| = distance from x to a
  • |ax + b| = distance scaled by |a|
  • Solutions are intersection points
  • V-shaped graph with vertex at x = -b/a
Frequently Asked Questions
What is an absolute value equation?

An absolute value equation is an equation that contains an absolute value expression, such as |x| = 5 or |2x - 3| = 7. The absolute value of a number represents its distance from zero on the number line, so |x| = 5 means x is 5 units away from 0, giving solutions x = 5 or x = -5.

How do I solve |ax + b| = c?

To solve |ax + b| = c where c ≥ 0:

  1. Set up two equations: ax + b = c and ax + b = -c
  2. Solve each equation separately
  3. Check if both solutions satisfy the original equation

Example: |2x - 3| = 7 → 2x - 3 = 7 → x = 5, or 2x - 3 = -7 → x = -2. Solutions: x = 5, x = -2.

What if c is negative in |ax + b| = c?

Absolute value is always non-negative (≥ 0). So if c is negative, the equation |ax + b| = c has no solution. For example, |2x + 1| = -3 has no solution because absolute value cannot equal a negative number.

How do I solve |ax + b| = |cx + d|?

When two absolute values are equal, set up two cases:

  1. ax + b = cx + d (same sign)
  2. ax + b = -(cx + d) = -cx - d (opposite signs)

Solve both equations and check for extraneous solutions.

Example: |x + 2| = |2x - 1| → Case 1: x + 2 = 2x - 1 → x = 3; Case 2: x + 2 = -(2x - 1) → x + 2 = -2x + 1 → 3x = -1 → x = -1/3. Solutions: x = 3, x = -1/3.

How do I solve absolute value inequalities?

For |ax + b| < c (c > 0): -c < ax + b < c

For |ax + b| ≤ c (c > 0): -c ≤ ax + b ≤ c

For |ax + b| > c (c > 0): ax + b < -c OR ax + b > c

For |ax + b| ≥ c (c > 0): ax + b ≤ -c OR ax + b ≥ c

Write the solution in interval notation.

What does absolute value mean geometrically?

Geometrically, |x - a| represents the distance between x and a on the number line. So |x| = |x - 0| is the distance from x to 0. |x - 3| = 5 means x is 5 units away from 3, so solutions are x = 8 and x = -2.

Can absolute value equations have no solution?

Yes. An absolute value equation has no solution if the isolated absolute value is set equal to a negative number. For example, |2x + 5| = -4 has no solution because absolute value cannot be negative. Also, |x - 2| = -|x + 1| has no solution because the left side is non-negative and the right side is non-positive (only zero is possible).

What are extraneous solutions?

Extraneous solutions are solutions that appear during the solving process but do not satisfy the original equation. They often occur when squaring both sides or when dealing with absolute values. Always check your solutions by plugging them back into the original equation.