Calculator Inputs
Use the form below to identify where the inequality is defined. Domain is checked before any actual inequality solving.
Example Data Table
| Mode | Example inequality | Domain rule | Domain result |
|---|---|---|---|
| Rational | (x + 3) / (x - 5) > 0 | x - 5 ≠ 0 | (-∞, 5) ∪ (5, ∞) |
| Square root | √(2x - 8) ≥ 1 | 2x - 8 ≥ 0 | [4, ∞) |
| Logarithmic | log base 2 (3x + 6) ≤ 4 | 3x + 6 > 0 | (-2, ∞) |
| Mixed radical-rational | √(x + 9) / (x - 1) ≥ 0 | x + 9 ≥ 0 and x - 1 ≠ 0 | [-9, 1) ∪ (1, ∞) |
Formula Used
The calculator checks only the definition conditions of the inequality expression. It does not test whether the inequality is true over those values.
- Rational form: For
(ax+b)/(cx+d), requirecx+d ≠ 0. - Square-root form: For
√(ax+b), requireax+b ≥ 0. - Logarithmic form: For
log base k(ax+b), requirek > 0,k ≠ 1, andax+b > 0. - Mixed form: For
√(ax+b)/(cx+d), require bothax+b ≥ 0andcx+d ≠ 0. - Boundary value: Solve the linear expression equal to zero, then keep or exclude that value depending on the rule.
- Interval notation: Endpoints use
[ ]when included and( )when excluded.
How to Use This Calculator
- Choose the inequality family that matches your expression.
- Enter the coefficients shown in the form fields.
- Use the relation and right-side value to keep your expression readable.
- For logarithms, enter a valid base greater than zero and not equal to one.
- Click Calculate Domain to view interval notation and set-builder form.
- Download the result as CSV or PDF for homework, revision, or recordkeeping.
Frequently Asked Questions
1. What does domain mean in an inequality?
Domain is the set of x-values that make every part of the inequality defined. It comes before solving the inequality itself.
2. Is domain the same as solution set?
No. Domain tells where the expression exists. The solution set is the subset of that domain where the inequality statement is actually true.
3. Why are denominator zeros excluded?
A fraction is undefined when its denominator equals zero. Those values must be removed from the domain before any sign testing or interval solving starts.
4. Why can square roots include zero?
For real numbers, a square root is defined whenever the radicand is zero or positive. That is why the calculator uses the condition radicand ≥ 0.
5. Why must logarithm arguments be positive?
Real logarithms are defined only for positive arguments. Zero and negative inputs are excluded, even if the inequality sign itself looks harmless.
6. What makes a logarithm base invalid?
A real logarithm base must be positive and cannot equal one. Bases like -2, 0, or 1 do not produce a valid real logarithmic function.
7. Can the calculator handle decimals and negatives?
Yes. You can enter decimal coefficients, negative coefficients, and decimal right-side values. The interval output is rounded using your selected precision.
8. When should I use the mixed mode?
Use mixed mode when the same inequality contains both a square root and a denominator. The calculator intersects both restrictions automatically.