Absolute Value Inequality Calculator
Solve absolute value inequalities with case splitting, interval notation, and step-by-step explanations.
Absolute value inequalities translate distance from zero into compound linear inequalities. This page shows when answers are intersections and when they are unions.
What this page covers
- Converts |A| <= k into a double inequality
- Converts |A| >= k into two outer branches
- Keeps endpoints open or closed based on the symbol
Realtime Solver
Linear, quadratic, absolute-value, rational, system, and coordinate-plane inequality support.
Supported Inputs
- 2x + 3 > 7 for linear inequalities
- x^2 - 5x + 6 < 0 for quadratic sign analysis
- |x - 2| <= 4 for absolute value
- (x + 1) / (x - 2) > 0 for rational sign charts
- x > 2; x <= 5 for one-variable systems
- 2x + 3y < 6 for coordinate-plane graphing
- x >= 0; y >= 0 for a two-variable system
Math Keyboard
Tap symbols if you do not want to use the keyboard.
Result
The absolute-value solution set is shown below in notation and on the number line.
Original
Interval Notation
Set Notation
Number Line
Drag to pan. Use the controls or mouse wheel to zoom the view.
Value Checker
Test a number against the computed solution set.
Steps
Move through the reasoning one step at a time or expand everything.
Isolate the absolute value expression
Step 1 / 10The absolute-value expression is already isolated on one side of the inequality.
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Guide
How this calculator helps
Use this page for |x - 2| <= 4, |3x + 1| > 7, and similar distance-based inequalities.
Try these examples
How to use this page
Enter an absolute value inequality such as |x - 2| <= 4 or |3x + 1| > 7.
Follow the case split to see whether the problem becomes an AND interval or an OR union.
Compare the interval notation with the graph so the distance interpretation stays clear.
Common mistakes to avoid
Using OR for a <= problem that should become a bounded AND interval.
Ignoring that a negative right-hand side can make the inequality impossible or automatic.
Dropping endpoint inclusion after rewriting the absolute value condition.
FAQ
What does an absolute value inequality mean?
It measures distance from zero. For example, |x - 2| <= 4 means x is within 4 units of 2.
When do absolute value inequalities use AND or OR?
|A| <= k gives an AND interval, while |A| >= k gives an OR answer with two outer intervals.
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Need more context?
Visit the learning library for explanations about interval notation, graphing, sign changes, and the rules that make inequalities different from equations.
Browse learning guides