Calculator
Example data table
| Inequality | Interpretation | Interval notation |
|---|---|---|
| |2x - 6| ≤ 8 | Inside a closed distance band | [-1, 7] |
| |x + 4| > 3 | Outside two open endpoints | (-∞, -7) ∪ (-1, ∞) |
| |3x| < 9 | Strict inside interval | (-3, 3) |
| |x - 5| = 2 | Two symmetric points | [3, 3] ∪ [7, 7] |
| |2x + 1| ≥ 0 | Always true case | (-∞, ∞) |
Formula used
|ax + b| < c becomes -c < ax + b < c.
|ax + b| ≤ c becomes -c ≤ ax + b ≤ c.
|ax + b| > c becomes ax + b < -c or ax + b > c.
|ax + b| ≥ c becomes ax + b ≤ -c or ax + b ≥ c.
|ax + b| = c becomes ax + b = -c or ax + b = c.
Endpoint form uses h = -b / a and r = c / |a|. Then solve around the center h using the radius r.
How to use this calculator
- Enter the coefficient a and constant b.
- Choose the comparison sign for the inequality or equality.
- Enter the right side value c.
- Pick the variable symbol and decimal precision.
- Press Solve inequality to see the result above the form.
- Read the interval notation, set-builder form, endpoints, and steps.
- Use the CSV or PDF button if you want a saved copy.
Understanding Absolute Value Inequalities
Absolute value inequalities compare distance with a limit. They show how far an expression sits from zero. This calculator solves linear forms quickly. It converts the answer into interval notation, endpoint values, and readable inequality statements. That saves time during practice, homework, and exam review.
Why Interval Notation Matters
Interval notation condenses long solution sets into a standard format. Open parentheses mark excluded endpoints. Square brackets mark included endpoints. Unions join separate parts of a solution. This makes graphing, checking, and comparing answers much easier. It is also the format most textbooks and tests expect.
How This Calculator Helps
Enter the coefficient of the variable, the constant term, a comparison sign, and the right side value. The tool handles less than, less than or equal to, greater than, greater than or equal to, and equality cases. It also explains impossible cases and all real number cases when the setup demands them.
How the Math Works
For an inequality like |ax + b| < c, the expression represents a distance. If the result must stay below c, the solution lies between two endpoints. If it must exceed c, the solution lies outside them. When c is negative, many cases have no solution because absolute value never becomes negative.
Useful for Practice and Checking
This page is useful for algebra students, tutors, and test takers. It supports verification after hand solving. It also helps reveal mistakes with signs, open intervals, and unions. Use the example table, formula section, and steps to build confidence before moving to harder inequality problems.
When to Use It
Use this calculator when a worksheet asks for interval notation or endpoints. It is helpful when coefficients are decimals or negatives, because those details often cause errors. The tool shows whether the final answer is a bounded interval, two rays, one point, no solution, or every real number.
Better Accuracy, Faster Review
You can also export the result for notes or class discussion. The summary table is easy to scan later. That makes this page practical for revision sessions, tutoring reviews, and classroom demonstrations. Clear structure improves checking speed, and repeated use strengthens intuition about algebra rules.
FAQs
1. What does an absolute value inequality measure?
It measures distance from zero or from a center point after rewriting the expression. The final answer shows where the variable can stay inside or outside a distance limit.
2. Why do some answers use parentheses and others use brackets?
Parentheses mean an endpoint is excluded. Brackets mean an endpoint is included. Strict inequalities use open endpoints, while inclusive inequalities keep the boundary points.
3. When do I get two separate intervals?
You get two intervals for greater than or greater than or equal to cases. Those conditions describe values outside a distance band, so the answer splits into left and right rays.
4. Why can a negative right side create no solution?
Absolute value cannot be negative. So expressions like |ax + b| < -2 or |ax + b| = -2 have no real solution. The calculator detects that immediately.
5. Can this calculator handle equality too?
Yes. When you choose the equality sign, the tool returns one point or two symmetric points. It also writes them in interval notation and set-builder notation.
6. What happens when the coefficient a is zero?
The inside expression becomes constant. Then the problem is no longer variable-based. The statement is either always true for all real numbers or never true at all.
7. What is the center and radius method?
The center is h = -b / a. The radius is r = c / |a|. Together they convert the inequality into a distance form that makes endpoints easier to find.
8. Is interval notation enough for school assignments?
Usually yes, but some teachers also want compound or set-builder notation. This page gives all three forms, so you can match the format your class requires.