Absolute Value Inequality Solver Calculator

Enter coefficients and choose an operator to instantly compute solution sets with clear steps. Supports compound and disjoint intervals, including degenerate, universal, and empty cases with inclusive bounds. Automatically normalizes inequalities when coefficients are negative, preserving proper direction across both sides consistently. Set precision and variable. Export results to CSV and PDF easily today.

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Example Data Table

abcOperatorInequalitySolution (interval form)
2-35<|2x - 3| < 5( -1, 4 )
-143≤|-x + 4| ≤ 3[ 1, 7 ]
020>|2| > 0( -∞, ∞ )
1-20≤|x - 2| ≤ 0[ 2, 2 ]
314≥|3x + 1| ≥ 4( -∞, -5/3 ] ∪ [ 1, ∞ )

Formula Used

How to Use

  1. Enter coefficients a, b, and c to model |ax + b| compared with c.
  2. Select an operator: <, ≤, >, or ≥.
  3. Optionally change the variable symbol and rounding precision.
  4. Click Solve to compute intervals and step-by-step reasoning.
  5. Use Download CSV or Download PDF to export results.
  6. Reset to clear inputs and start a fresh computation.

FAQs

If c < 0, then |X| < c or |X| ≤ c has no solution, while |X| > c or |X| ≥ c holds for all real numbers.

Open bounds use parentheses ( ), and closed bounds use brackets [ ]. Strict inequalities give open bounds; inclusive ones give closed bounds.

The expression becomes constant |b|. The inequality either holds for all x or for none, depending on |b| versus c and the operator.

Yes. For c = 0 with a ≠ 0 and the ≤ operator, the unique solution is x = -b/a, shown as [x₀, x₀].

When dividing by a negative a, the inequality direction reverses. We account for this automatically when computing bounds.

Displayed endpoints are rounded to the chosen precision. Internally, full-precision calculations are used before formatting.
This tool provides educational guidance for algebraic inequalities. Always verify with course or exam conventions.

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