Enter up to four linear inequalities
Each row solves a one-variable linear inequality in the form ax + b relation cx + d. Activate only the rows you want included.
Example data table
| Example | System | Reduced bounds | Final solution |
|---|---|---|---|
| 1 | 2x + 3 ≤ x + 9, -3x + 12 < 0, x - 1 ≥ -2 | x ≤ 6, x > 4, x ≥ -1 | (4, 6] |
| 2 | 4x - 8 ≥ 0, x + 5 < 9 | x ≥ 2, x < 4 | [2, 4) |
| 3 | x + 2 < 1, x - 3 ≥ 0 | x < -1, x ≥ 3 | ∅ |
Formula used
General row: ax + b ∘ cx + d
Move x-terms to one side: (a − c)x ∘ d − b
Divide by (a − c): x ∘ (d − b)/(a − c)
Important rule: Reverse the inequality sign whenever you divide by a negative value.
System solution: Intersect every active row’s valid region.
Contradiction test: The system is impossible when the lower bound exceeds the upper bound, or when equal bounds are both strict.
How to use this calculator
- Activate the rows you want in the system.
- Enter coefficients and constants for each inequality.
- Choose the correct relation sign for every row.
- Optionally enter a test value for x.
- Keep the integer summary enabled if you also want discrete results.
- Press Solve system to generate the interval, table, and graph.
- Use the CSV or PDF buttons to export the current result summary.
FAQs
1) What kind of inequalities does this page solve?
It solves one-variable linear systems written as ax + b compared with cx + d. Each active row becomes a bound, tautology, or contradiction.
2) Can it detect impossible systems?
Yes. If one row contradicts the others, or if the combined lower bound exceeds the upper bound, the solver returns the empty set.
3) Why does the inequality sign sometimes flip?
The sign reverses when both sides are divided by a negative coefficient. This preserves the correct order of the real numbers.
4) What does a single-point solution mean?
It means the lower and upper bounds meet at one included number. In that case, only one real value satisfies the full system.
5) What does the graph represent?
The graph is a number-line view. It highlights the feasible interval after every active inequality has been reduced and intersected.
6) Can I test a value of x quickly?
Yes. Enter any test value, and the calculator will show whether it satisfies each inequality and whether it belongs to the final solution set.
7) What is the integer summary for?
It translates the real-number interval into whole-number results. That is useful when only integer answers are meaningful in your problem.
8) Can I leave some rows unused?
Yes. Use the Active checkbox on each card. Inactive rows are ignored during solving, testing, exporting, and plotting.