Calculator Input
Enter integers, decimals, or simple fractions like 3/4. The calculator solves inequalities of the form ax + b □ cx + d.
Example Data Table
| Inequality | Key Rearrangement | Solution | Interval |
|---|---|---|---|
| 3x + 5 ≤ x + 9 | 2x ≤ 4 | x ≤ 2 | (-∞, 2] |
| 4x - 7 > 2x + 1 | 2x > 8 | x > 4 | (4, ∞) |
| -2x + 3 ≥ x - 6 | -3x ≥ -9 | x ≤ 3 | (-∞, 3] |
| 5x + 2 < 5x + 8 | 2 < 8 | All real numbers | (-∞, ∞) |
Formula Used
Move variable terms to one side and constants to the other:
Then divide both sides by (a - c):
If you divide by a negative number, reverse the inequality sign. That sign change is the most important rule in linear inequality solving.
How to Use This Calculator
- Enter the left coefficient and left constant.
- Choose the correct inequality sign.
- Enter the right coefficient and right constant.
- Optionally change the variable symbol and decimal precision.
- Click Solve Inequality to view the answer, interval notation, graph, and algebra steps.
- Use the CSV and PDF buttons to export the solved result.
FAQs
1. What kinds of inequalities can this calculator solve?
This calculator solves linear inequalities in one variable, such as ax + b < cx + d or ax + b ≥ cx + d. It supports integers, decimals, and simple fractions for every coefficient and constant field.
2. Why does the inequality sign sometimes reverse?
The sign reverses only when you divide or multiply both sides by a negative number. This preserves the true order of values on the number line and keeps the final solution mathematically correct.
3. What does interval notation mean?
Interval notation shows the full set of valid answers. Parentheses mean the endpoint is not included, while brackets mean it is included. For example, x ≤ 2 becomes (-∞, 2].
4. Can I use fractions and decimals together?
Yes. You can enter values like 1/2, -3, or 2.75 in any field. The solver works with exact fraction arithmetic first, then also shows a decimal approximation for the boundary when needed.
5. What happens when the variable terms cancel out?
If the x terms cancel, the inequality becomes a constant statement such as 2 < 8 or 9 ≥ 12. That means the answer is either all real numbers or no solution.
6. Why might there be no solution?
No solution appears when the final statement is always false. For instance, if solving leads to 7 < 3, no real value of the variable can make the original inequality true.
7. How should I read the graph?
The graph is a number line. A filled point means the endpoint is included. An open point means it is excluded. The shaded side shows all values that satisfy the inequality.
8. How can I check the solution manually?
Pick a test value from the shaded region and substitute it into the original inequality. Then try a value outside the region. One should make the statement true and the other false.