Calculator Inputs
Enter two linear inequalities. The calculator solves each one, then finds the shared range that satisfies both.
Example Data Table
| Example | Inequality 1 | Inequality 2 | Final Range | Interval |
|---|---|---|---|---|
| Sample A | 2x + 3 ≥ 11 | 3x - 2 < 13 | 4 ≤ x < 5 | [4, 5) |
| Sample B | -2x + 1 ≤ 7 | x + 4 > 1 | x > -3 | (-3, ∞) |
| Sample C | x - 2 ≥ 5 | x + 1 ≤ 10 | 7 ≤ x ≤ 9 | [7, 9] |
| Sample D | 4x + 8 < 0 | 2x - 1 ≥ 5 | No overlap | ∅ |
Formula Used
General form for each inequality: ax + b [operator] c
Rearranged form: ax [operator] (c - b)
Solved form when a ≠ 0: x [operator] (c - b) / a
Important rule: If you divide by a negative coefficient, reverse the inequality sign.
After solving both inequalities separately, the final range is the intersection of both solution sets. The intersection keeps only values of x that satisfy both statements together.
Inclusive operators (≤, ≥) produce closed endpoints. Strict operators (<, >) produce open endpoints. The interval notation and graph follow that endpoint behavior exactly.
How to Use This Calculator
- Enter the coefficients and constants for the first linear inequality.
- Choose the correct inequality operator for that first expression.
- Repeat the same process for the second inequality.
- Set your preferred decimal precision and graph padding.
- Press Submit to solve both inequalities and compute the shared valid range.
- Review the interval notation, set-builder notation, solved forms, and graph.
- Use the CSV button for spreadsheet-friendly output or the PDF button for a printable report.
FAQs
1. What does this calculator solve?
It solves two linear inequalities in x, then finds the overlap between their individual solution sets. That overlap becomes the final valid range.
2. What happens when the coefficient of x is zero?
The inequality becomes a constant statement, such as 5 ≤ 8. That statement is either always true or never true, which changes the final overlap.
3. Why does the inequality sign reverse sometimes?
When you divide both sides by a negative number, the inequality direction must reverse. This rule preserves the correct order of values on the number line.
4. What do open and closed endpoints mean?
Closed endpoints include the boundary value and come from ≤ or ≥. Open endpoints exclude the boundary value and come from < or >.
5. Can the result be all real numbers?
Yes. If both inequalities are always true or place no limiting boundary on x, the final solution set can be every real number.
6. Can the result be a single number?
Yes. That happens when both inequalities meet at the same boundary and both endpoints are inclusive, so exactly one value satisfies both statements.
7. Why might there be no solution?
No solution appears when the two solved ranges do not overlap. One inequality may force x above a value while the other forces x below a smaller value.
8. What is the difference between interval and set-builder notation?
Interval notation shows boundaries using brackets and parentheses. Set-builder notation describes the same solution as a condition that x must satisfy.