Inequality Calculator

Solve linear, quadratic, and complex algebraic inequalities instantly. This calculator helps you determine the exact range of values that satisfy an inequality statement, simplified into clear mathematical notation and step-by-step solutions. It is a valuable tool for checking homework, reducing calculation errors, and mastering algebraic relationships.

If you are working on multi-step equations, an Online Math Solver can guide you through every step of the process.

What Is an Inequality?

Unlike standard equations that use an equals sign (=) to show two expressions are identical, an inequality compares two values to show whether one is greater than, less than, or not equal to another.

Because inequalities express relative size, they often have an infinite range of solutions rather than a single fixed answer.

Types of Inequality Symbols

Inequalities rely on four primary comparison symbols:

| Symbol | Name | Meaning | Example | | :---: | :--- | :--- | :--- | | < | Less Than | Left side is strictly smaller than right side | x < 5 | | > | Greater Than | Left side is strictly larger than right side | y > 3 | | <= | Less Than or Equal To | Left side is smaller than or equal to right side | a <= 7 | | >= | Greater Than or Equal To | Left side is larger than or equal to right side | b >= 6 |

How to Use the Inequality Calculator

1. Enter the Inequality: Type your expression (e.g., 3x + 4 > 16) into the input field.
2. Review Your Inputs: Verify that all coefficients, variables, fractions, and inequality signs are entered correctly.
3. Calculate: Click the Solve button to process the expression.
4. Examine Results: Review the final solution set along with the step-by-step breakdown.

Step-by-Step Example

Problem: Solve the inequality 3x + 4 > 16.

• Step 1: Isolate the variable term. Subtract 4 from both sides of the inequality: 3x + 4 - 4 > 16 - 4 3x > 12

• Step 2: Solve for x. Divide both sides by 3: 3x / 3 > 12 / 3 x > 4

Result: Any number greater than 4 satisfies this inequality.

Essential Rules for Solving Inequalities

To solve inequalities accurately by hand or using a tool, keep these fundamental algebraic rules in mind:

1. Addition & Subtraction Rule: Adding or subtracting the same number on both sides does not change the inequality sign.
2. Positive Multiplication/Division: Multiplying or dividing both sides by a positive number keeps the inequality direction the same.
3. Negative Sign Flip Rule: Whenever you multiply or divide both sides by a negative number, you MUST reverse (flip) the inequality symbol (e.g., If -2x < 8, then x > -4).
4. Simplification First: Expand brackets and combine like terms on each side before attempting to isolate variables.

Benefits of Using an Inequality Calculator

• Accuracy: Minimizes sign errors, especially when dealing with negative coefficients.
• Instant Verification: Great for self-checking algebra assignments and exam preparation.
• Range Insights: Provides solutions clearly expressed as interval notation or inequalities.