Absolute Value Calculator
Absolute Value Equation Solver
Solve equations of the form a|bx + c| + d = e. Use vertical bars | | to denote absolute value.
Click an example to load it:
Enter equation
Use | for absolute value bars. Supported format: a|bx+c|+d=e Example: 3|2x+1|+4=25
Evaluate Absolute Value Expression
Evaluate an absolute value expression at a specific value of x. Enter the expression and the value of x.
Expression
Value of x
Absolute Value Inequality Solver
Solve inequalities of the form a|bx + c| + d < e. Fill in the coefficients to form the inequality.
This forms: a|bx + c| + d [sign] e
Basic Absolute Value Calculator
Find the absolute value of any real number, fraction, or decimal.
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Absolute Value Equation Reference Table
| Equation / Inequality Form | Condition | Solution |
|---|---|---|
| |x| = c, c > 0 | c is positive | x = c or x = -c |
| |x| = 0 | c equals zero | x = 0 (one solution) |
| |x| = c, c < 0 | c is negative | No solution |
| |ax + b| = c, c > 0 | standard form | x = (c-b)/a or x = (-c-b)/a |
| |x| < c, c > 0 | less than | -c < x < c |
| |x| <= c, c > 0 | less than or equal | -c <= x <= c |
| |x| > c, c > 0 | greater than | x < -c or x > c |
| |x| >= c, c > 0 | greater than or equal | x <= -c or x >= c |
| |x| < c, c <= 0 | c is zero or negative | No solution |
| |x| > c, c < 0 | c is negative | All real numbers |
What is Absolute Value?
The absolute value of a number is its distance from zero on the number line, regardless of direction. It is always non-negative. The absolute value of x is written as |x|.
|x| = x if x >= 0
|x| = -x if x < 0
How to Solve Absolute Value Equations
To solve an equation like a|bx + c| + d = e, follow these steps:
Step 1: Isolate the absolute value term.
a|bx + c| = e – d
|bx + c| = (e – d) / a
Step 2: Check the right side.
If right side < 0: No solution.
If right side = 0: One solution (bx + c = 0).
If right side > 0: Two solutions (split into two cases).
Step 3: Split into two cases.
Case 1: bx + c = (e-d)/a => x = ((e-d)/a – c) / b
Case 2: bx + c = -(e-d)/a => x = (-(e-d)/a – c) / b
Worked Example: 3|2x+1|+4 = 25
Step 1: 3|2x+1| = 25 – 4 = 21
Step 2: |2x+1| = 21 / 3 = 7
Step 3: 2x+1 = 7 or 2x+1 = -7
Step 4: 2x = 6 or 2x = -8
x = 3 or x = -4
How to Solve Absolute Value Inequalities
|x| < c means -c < x < c (and / intersection)
|x| > c means x < -c or x > c (or / union)
Key Properties of Absolute Value
Non-negativity: |x| >= 0 for all x
Identity: |x| = 0 only when x = 0
Symmetry: |-x| = |x|
Multiplicative: |x * y| = |x| * |y|
Triangle inequality: |x + y| <= |x| + |y|