# Inequality Calculator

Solve linear inequalities like ax + b > c step by step. See the solution in interval notation and number line form. Free inequality solver.

## What this calculates

Solve any linear inequality of the form ax + b > c (or <, ≥, ≤). This calculator shows step-by-step work, interval notation, and describes the number line representation.

## Inputs

- **Coefficient a (before x)** — The coefficient of x. Must not be zero.
- **Constant b (added to ax)** — The constant term on the left side.
- **Inequality Sign** — options: Greater than (>), Less than (<), Greater than or equal (≥), Less than or equal (≤) — Select the inequality operator.
- **Right side value c** — The value on the right side of the inequality.

## Outputs

- **Original Inequality** — formatted as text — The inequality as entered.
- **Solution** — formatted as text — The solved inequality for x.
- **Interval Notation** — formatted as text — The solution in interval notation.
- **Number Line Description** — formatted as text — How to represent the solution on a number line.
- **Steps** — formatted as text — Step-by-step solution process.

## Details

A linear inequality is like a linear equation, but instead of an equals sign it uses an inequality symbol: >, <, ≥, or ≤.

**Solving Linear Inequalities:**

The process is the same as solving equations, with one critical rule: **if you multiply or divide both sides by a negative number, flip the inequality sign**.

**Worked Example:**

Solve -3x + 7 ≥ 1.

1. Subtract 7 from both sides: -3x ≥ -6
2. Divide both sides by -3 (flip the sign): x ≤ 2
3. Interval notation: (-∞, 2]
4. Number line: closed dot at 2, shade to the left.

**Interval Notation:**

- (a, b) means a < x < b (endpoints not included)
- [a, b] means a ≤ x ≤ b (endpoints included)
- (-∞, a) means x < a
- [a, ∞) means x ≥ a
- Always use parentheses (not brackets) next to infinity.

**Number Line Representation:**

- Open dot = endpoint not included (strict inequality)
- Closed/filled dot = endpoint included (≤ or ≥)
- Shading indicates which direction contains the solutions.

## Frequently Asked Questions

**Q: When do I flip the inequality sign?**

A: Flip the sign whenever you multiply or divide both sides by a negative number. For example, solving -2x > 6: divide by -2 and flip to get x < -3. If you multiply or divide by a positive number, the sign stays the same. This is the most common mistake in inequality problems.

**Q: What does interval notation mean?**

A: Interval notation describes a set of numbers between two endpoints. Parentheses ( ) mean the endpoint is not included, brackets [ ] mean it is. For x > 3, write (3, ∞). For x ≤ 5, write (-∞, 5]. Infinity always gets a parenthesis because it is not a number you can reach.

**Q: How do I graph an inequality on a number line?**

A: Draw a number line, mark the critical value, and place either an open dot (for > or  3: open dot at 3, shade right. For x ≤ -1: filled dot at -1, shade left.

**Q: What if the coefficient of x is zero?**

A: If a = 0, the inequality reduces to b > c (or  3 simplifies to 5 > 3, which is true for all x.

**Q: Can I solve compound inequalities with this calculator?**

A: This calculator handles single linear inequalities. For compound inequalities like 2 < 3x + 1 ≤ 10, split it into two separate inequalities (2 < 3x + 1 and 3x + 1 ≤ 10), solve each, and take the intersection of the solutions.

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Source: https://vastcalc.com/calculators/math/inequality
Category: Math
Last updated: 2026-04-08