Interval Inequality Solver Calculator

Example	Inequality	Solved Form	Interval Notation
1	2x + 3 ≤ 5x + 15	x ≥ -4	[-4, ∞)
2	4x - 7 > x + 8	x > 5	(5, ∞)
3	3x + 9 < 6	x < -1	(-∞, -1)
4	-2x + 1 ≥ 7	x ≤ -3	(-∞, -3]

Formula Used

General form: a x + b ≤ c x + d, or any other linear inequality sign.

Rearrangement: a x - c x ≤ d - b, which becomes (a - c)x ≤ d - b.

Division rule: x ≤ (d - b)/(a - c), unless a - c is negative. When dividing by a negative value, reverse the inequality sign.

Interval conversion: open endpoints use parentheses. Inclusive endpoints use brackets. Infinite sides always use parentheses.

This calculator automates rearrangement, sign reversal, interval notation, and number line plotting for one-variable linear inequalities.

How to Use This Calculator

Enter the left coefficient and constant for the expression on the left side.
Select the inequality sign: less than, less than or equal, greater than, or greater than or equal.
Enter the right coefficient and constant for the expression on the right side.
Choose the variable symbol and decimal precision.
Enable steps if you want the algebra shown line by line.
Click Solve Inequality to display the result above the form.
Review the interval notation, set-builder form, and graph.
Use the CSV and PDF buttons to export the current result.

FAQs

1. What does this interval inequality solver calculate?

It solves one-variable linear inequalities, simplifies them, and converts the answer into interval notation. It also shows set-builder form and a number line graph.

2. Why does the inequality sign sometimes reverse?

When you divide or multiply both sides by a negative number, the inequality direction must flip. This preserves the correct order relationship between the two sides.

3. What is interval notation?

Interval notation is a compact way to write solution sets on the real number line. Parentheses show excluded endpoints, while brackets show included endpoints.

4. What happens when the variable terms cancel out?

The inequality becomes a constant statement. If that statement is true, every real number is a solution. If false, there is no solution.

5. Can this calculator solve strict and inclusive inequalities?

Yes. It supports <, ≤, >, and ≥. The selected sign determines whether the endpoint is open or closed in interval notation.

6. What does the graph represent?

The graph places the boundary value on a number line and shades the side that satisfies the inequality. Closed markers mean inclusive endpoints.

7. Can I export the solution for reports or homework notes?

Yes. The calculator includes CSV and PDF export buttons, so you can save the inequality inputs, solved form, and interval notation.

8. Is this useful for checking algebra homework?

Yes. The step option helps verify rearrangement, sign changes, and interval conversion. It is especially helpful for avoiding mistakes with negative divisors.