Solve Absolute Value Inequalities Calculator

Master absolute inequalities with guided steps and intuitive controls for learning. Handle strict and inclusive cases, negatives, edge cases, and unions of intervals. See algebraic transformations, interval notation, and verification checks automatically for each step. Export results as CSV or PDF with one click.

Input

Example Data Table

A few sample inequalities and their solution sets.

# Inequality Solution Intervals
1|2x - 3| <= 5[-1, 4]
2|-4x + 1| < 2(-0.25, 0.75)
3|3x + 6| > 9(-∞, -5) ∪ (1, ∞)
4|1x - 2| >= 0ALL
5|0x + 5| > 3ALL

Solution & Steps

Enter values and click Solve to see the step-by-step solution.

Formulas Used

  • |E| < k with k > 0 ⇒ -k < E < k.
  • |E| ≤ k with k > 0 ⇒ -k ≤ E ≤ k.
  • |E| > k with k ≥ 0 ⇒ E < -k or E > k.
  • |E| ≥ k with k ≥ 0 ⇒ E ≤ -k or E ≥ k.
  • |E| < 0 has no solution; |E| ≤ 0 ⇒ E = 0.
  • If k < 0: |E| > k or ≥ k ⇒ all reals; |E| < k or ≤ k ⇒ none.
  • For E = a x + b: when dividing by a negative, reverse inequality signs.

How to Use

  1. Enter a, b, select an operator, and enter c.
  2. Use decimals or fractions like -3/4.
  3. Click Solve for intervals and detailed algebraic steps.
  4. Export results as CSV or use Download PDF to print to PDF.
  5. Review Formulas Used and FAQs for edge cases and guidance.

FAQs

Enter coefficients a, b, and c as decimals or fractions (e.g., 5/2). Choose an inequality operator: <, ≤, >, or ≥.

When dividing by a negative a, inequality directions flip. The tool computes endpoints and orders them correctly, so the final intervals are accurate.

The expression becomes constant |b|. The inequality is then either always true (all real numbers) or never true (no solution), depending on |b| and c.

|E| < 0: no solution. |E| ≤ 0: E = 0 (a single point). |E| > 0: all reals except where E = 0. |E| ≥ 0: all real numbers.

Yes. It logs each algebraic transformation, endpoint calculation, and quick test checks so you can follow or audit the solution path.

Click Download PDF to open the browser's print dialog for the results area. Choose "Save as PDF" to export a nicely formatted document.

Operator Behavior Summary

Quick reference for translating absolute value inequalities into equivalent linear forms and understanding interval shapes.

Form Rewrite Rule Typical Solution Shape Boundary Included? Notes
|E| < k, k > 0 -k < E < k Bounded interval (L, U) No Inside the endpoints; strict.
|E| ≤ k, k > 0 -k ≤ E ≤ k Bounded interval [L, U] Yes Inside the endpoints; inclusive.
|E| > k, k ≥ 0 E < -k or E > k Unbounded union (-∞, L) ∪ (U, ∞) No Outside the endpoints; strict.
|E| ≥ k, k ≥ 0 E ≤ -k or E ≥ k Unbounded union (-∞, L] ∪ [U, ∞) Yes Outside the endpoints; inclusive.
k < 0 special — All reals for >, ≥; none for <, ≤ — Because |E| is always ≥ 0.

Worked Examples: Endpoints & Interval Construction

Computed using the same solver; endpoints shown when meaningful (c > 0 and a ≠ 0).

# Inequality L U Solution Intervals Shape
1|5x - 10| < 31.42.6(1.4, 2.6)Bounded (between)
2|-2x + 4| <= 7-1.55.5[-1.5, 5.5]Bounded (between)
3|3x - 6| > 21.3333332.666667(-∞, 1.333333) ∪ (2.666667, ∞)Unbounded (outside)
4|-1x + 5| >= 0——ALLAll reals
5|0x + 7| > -1——ALLAll reals

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