Enter inequality values
Use the responsive input grid below. Large screens show three columns, smaller screens show two, and phones show one.
Example data table
These examples show how different operators and coefficients change the final interval and boundary behavior.
| Example inequality | Rearranged form | Solution | Interval notation |
|---|---|---|---|
| 3x - 5 ≤ x + 7 | 2x ≤ 12 | x ≤ 6 | (-∞, 6] |
| 4x + 8 > 2x - 6 | 2x > -14 | x > -7 | (-7, ∞) |
| -2x + 3 ≥ x + 12 | -3x ≥ 9 | x ≤ -3 | (-∞, -3] |
| 2x + 4 ≤ 2x + 9 | 0 ≤ 5 | All real numbers | (-∞, ∞) |
| 5x - 1 < 5x - 4 | 0 < -3 | No solution | ∅ |
Formula used
The calculator solves one-variable linear inequalities of the form:
ax + b ? cx + d
Core transformation
- Subtract cx from both sides.
- Subtract b from both sides.
- Get (a - c)x ? (d - b).
- Divide by (a - c).
Important rule
If you divide or multiply by a negative number, reverse the inequality sign. That is the key rule this calculator checks automatically.
Result interpretation
- Strict inequality uses an open endpoint: < or >.
- Inclusive inequality uses a closed endpoint: ≤ or ≥.
- All real numbers appears when variable terms cancel and the statement remains true.
- No solution appears when variable terms cancel and the statement becomes false.
How to use this calculator
- Enter the left expression as ax + b.
- Select the inequality symbol between both expressions.
- Enter the right expression as cx + d.
- Choose the variable letter and decimal precision.
- Optionally enter a test value to verify one number.
- Set graph limits for the number-line style plot.
- Press Solve inequality to view the result above the form.
- Download a CSV or PDF summary after solving.
FAQs
1) What types of inequalities does this calculator solve?
It solves one-variable linear inequalities written as ax + b compared with cx + d. It supports less than, less than or equal, greater than, and greater than or equal signs.
2) Why does the inequality sign sometimes flip?
The sign flips only when both sides are divided or multiplied by a negative number. This preserves the correct ordering of values on the number line.
3) What does interval notation mean here?
Interval notation shows the full solution set compactly. Parentheses mean the endpoint is excluded. Brackets mean the endpoint is included.
4) What happens if the variable terms cancel out?
The calculator checks the remaining constant statement. If it is true, every real number works. If it is false, there is no solution.
5) Can I use decimal coefficients and constants?
Yes. All coefficient, constant, graph, and test-value fields accept decimals. You can also control how many decimal places appear in the output.
6) What does the test value feature do?
It substitutes a chosen value into both sides of the original inequality. The result confirms whether that specific number satisfies the inequality.
7) Why is the graph useful?
The graph shows the boundary point and shaded solution region on a number line. Open circles represent excluded boundaries. Filled circles represent included boundaries.
8) Can this handle quadratic or absolute-value inequalities?
No. This page is designed for linear one-variable inequalities only. More complex expressions need different solving rules and graphing logic.