Calculator Inputs
Formula Used
This calculator solves a linear inequality written in the form:
ax + b ? cx + d
It rearranges the expression using these rules:
- Subtract cx from both sides.
- Subtract b from both sides.
- Simplify to (a − c)x ? (d − b).
- Divide by (a − c) to isolate the variable.
- If you divide by a negative value, reverse the inequality sign.
Final form: x ? (d − b) / (a − c), with the sign reversed whenever (a − c) < 0.
How to Use This Calculator
- Enter the coefficient and constant for the left side.
- Select the inequality sign you want to solve.
- Enter the coefficient and constant for the right side.
- Choose a variable name and your preferred domain.
- Optionally set graph limits and a custom test value.
- Click Solve Inequality to view the result above the form.
- Review the steps, interval notation, number line, and test check.
- Use the export buttons to download CSV or PDF output.
Example Data Table
| Example | Reduced Form | Solution | Interval |
|---|---|---|---|
| 2x + 3 ≤ 5x + 15 | -3x ≤ 12 | x ≥ -4 | [-4, ∞) |
| 4x - 7 > x + 8 | 3x > 15 | x > 5 | (5, ∞) |
| 6x + 9 ≥ 2x - 3 | 4x ≥ -12 | x ≥ -3 | [-3, ∞) |
| 3x + 2 < 3x + 8 | 2 < 8 | All real numbers | (-∞, ∞) |
| 5x - 1 > 5x + 6 | -1 > 6 | No solution | ∅ |
Frequently Asked Questions
1. What does this calculator solve?
It solves one-variable linear inequalities of the form ax + b ? cx + d. It shows the rearranged form, final answer, interval notation, a number line, and a truth check using a test value.
2. Why does the inequality sign sometimes reverse?
The sign reverses only when both sides are divided by a negative number. That rule preserves the correct order of values on the number line and keeps the solution mathematically valid.
3. What happens when the variable cancels out?
If the variable disappears, the result becomes a constant statement. A true statement means all numbers work. A false statement means no value can satisfy the inequality.
4. What is interval notation?
Interval notation describes the complete real-number solution set compactly. Parentheses exclude endpoints, while square brackets include them. The calculator automatically converts the solved inequality into interval form.
5. Can I use decimals or negative numbers?
Yes. The inputs accept integers, decimals, and negative values. This makes the tool useful for classroom exercises, exam practice, worksheets, and quick checks involving non-whole-number coefficients.
6. What does the integer-only option do?
It keeps the real-number boundary but also reports the equivalent integer form. For example, x < 4.2 becomes x ≤ 4 when the solution domain is restricted to integers.
7. Why is a test value useful?
A test value helps verify the solution. By substituting a sample number into the original inequality, you can quickly confirm whether the proposed region truly satisfies the problem.
8. What do the CSV and PDF exports include?
The exports capture the main result details, including the original inequality, classification, solution form, interval notation, boundary value, test check, and the solution steps shown on the page.