Calculator Inputs
Use the expression family that matches your inequality, then enter coefficients for the restriction part.
Example Data Table
These examples show the kind of domain restrictions the calculator evaluates.
| Expression Family | Example Expression | Restriction Rule | Domain |
|---|---|---|---|
| Linear | 3x - 7 | No restriction | (-∞, ∞) |
| Rational | (x + 2) / (x - 5) | x - 5 ≠ 0 | (-∞, 5) ∪ (5, ∞) |
| Square Root | √(2x - 8) | 2x - 8 ≥ 0 | [4, ∞) |
| Reciprocal Root | 1 / √(5 - x) | 5 - x > 0 | (-∞, 5) |
| Logarithm | log base 10 of (x + 1) | x + 1 > 0 | (-1, ∞) |
| Radical Rational | √((x - 4) / (x - 1)) | (x - 4)/(x - 1) ≥ 0 and x ≠ 1 | (-∞, 1) ∪ [4, ∞) |
Formula Used
The calculator focuses on domain restrictions that must be satisfied before solving an inequality.
- Linear:
ax + bis defined for all real numbers. - Rational:
(ax + b)/(cx + d)requirescx + d ≠ 0. - Square root:
√(ax + b)requiresax + b ≥ 0. - Reciprocal square root:
1/√(ax + b)requiresax + b > 0. - Logarithm:
log base k(ax + b)requiresax + b > 0,k > 0, andk ≠ 1. - Radical rational:
√((ax + b)/(cx + d))requires the fraction to be nonnegative and the denominator to stay nonzero.
How to Use This Calculator
- Select the expression family that appears inside your inequality.
- Enter the coefficients for the restriction expression.
- Use the log base field only for logarithmic expressions.
- Click Find Domain to calculate interval notation and set-builder notation.
- Review the step list, critical points table, and Plotly graph.
- Use the CSV or PDF buttons to export the result summary.
Frequently Asked Questions
1) What does this calculator find?
It finds the real-number domain of the expression inside an inequality. That means it lists the x-values where the expression is valid before any solving steps begin.
2) Does it solve the entire inequality?
No. It handles the domain only. You still solve the inequality after restricting x to values where the expression is actually defined.
3) Why is denominator zero excluded?
Division by zero is undefined in real-number algebra. Any x-value that makes a denominator zero must be removed from the domain immediately.
4) Why does a square root use a nonnegative rule?
For real-number work, the quantity inside an even root must be zero or positive. Negative radicands would produce non-real values.
5) Why does 1/√ use a strict positive rule?
Because zero inside the root would make the denominator equal zero after taking the square root. That turns the whole expression undefined.
6) Why must logarithm arguments be positive?
Real logarithms are defined only for positive inputs. Zero and negative arguments do not produce valid real logarithm values.
7) What if the log base is one or negative?
That base is invalid. A real logarithm base must be positive and cannot equal one, so the expression has no valid real domain.
8) Can the graph and interval notation help each other?
Yes. The graph highlights valid x-regions visually, while interval notation states them exactly. Using both is a good way to verify algebra work.