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Home Page > Math > Algebra Calculators

Inequality Solver

Solve linear, quadratic, polynomial, and rational inequalities with step-by-step solutions. Display results graphically on a number line and in interval notation.

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Inequality SolverTry it now — free ▼

Try the following examples:

Linear: 2x - 5 > 3 Quadratic: x² - 5x + 6 > 0 Polynomial: x³ - 4x < 0 Rational: (x+2)/(x-1) > 0 Quadratic ≤ Rational ≥
📖 Input Guide - How to Enter Inequalities
Variables Use a single variable: x, y, or z
Inequality Symbols < less than
> greater than
<= less or equal
>= greater or equal
Multiplication 2*x or simply 2x
(x+1)*x or (x+1)x
Division (Fractions) (x+1)/(x-1) > 0
x/2 <= 5
Exponents x^2 > 4 or x**3 < 8
For powers: (x+1)^2 >= 0
Parentheses (2x+3)/(x-1) < 0
(x+1)(x-1) > 0
Smart Input: Type x^2-5x+6>0 or x^2 - 5*x + 6 > 0. Both work! Always use parentheses for complex fractions.
Inequality:
(Use <, >, <=, >=, or != operators)

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About Inequality Solver

Welcome to our Inequality Solver, a comprehensive online tool designed to help students, teachers, and mathematics enthusiasts solve linear, quadratic, polynomial, and rational inequalities with detailed step-by-step solutions. Our calculator provides visual representations on a number line and displays results in interval notation, making it easy to understand and verify your solutions.

Key Features of Our Inequality Solver

What is an Inequality?

An inequality is a mathematical statement that compares two expressions using inequality symbols. Unlike equations that use equals signs, inequalities use symbols like greater than, less than, greater than or equal to, or less than or equal to. The solution to an inequality is typically a range or set of values rather than a single number.

Types of Inequalities Supported

1. Linear Inequalities

Inequalities of the form $ax + b < 0$ where $a$ and $b$ are constants.

Example: $2x - 5 > 3$ or $-3x + 7 \le 1$

2. Quadratic Inequalities

Inequalities involving a quadratic expression of the form $ax^2 + bx + c < 0$.

Example: $x^2 - 5x + 6 > 0$ or $-x^2 + 4x - 3 \le 0$

3. Polynomial Inequalities

Inequalities involving polynomial expressions of degree 3 or higher.

Example: $x^3 - 4x > 0$ or $x^4 - 5x^2 + 4 \le 0$

4. Rational Inequalities

Inequalities involving rational expressions (fractions with polynomials).

Example: $\frac{x+2}{x-1} > 0$ or $\frac{x^2-4}{x^2+1} \le 1$

How to Use the Inequality Solver

  1. Enter Your Inequality: Type your inequality in the input field. You can use:
    • Variables: x, y, z, etc. (single variable only)
    • Operators: +, -, *, / for arithmetic
    • Inequality symbols: <, >, <=, >=, !=
    • Exponents: ^ or ** (e.g., x^2 or x**3)
    • Parentheses: ( ) for grouping
  2. Click Solve: Process your inequality and view the results.
  3. Review Step-by-Step Solution: Learn from detailed explanations of each solving step.
  4. View Number Line: See the solution visualized on a number line with critical points marked.
  5. Check Interval Notation: Read your solution in standard interval notation.

Inequality Input Guidelines

For best results, follow these input conventions:

Understanding Inequality Solutions

Number Line Representation

The number line shows:

Interval Notation

Solutions are expressed using interval notation:

Methods for Solving Inequalities

For Linear Inequalities

  1. Isolate the variable on one side
  2. Perform the same operations on both sides
  3. Reverse the inequality sign when multiplying or dividing by a negative number
  4. Express the solution in interval notation

For Quadratic and Polynomial Inequalities

  1. Move all terms to one side (set to zero on the other)
  2. Factor the polynomial if possible
  3. Find critical points (zeros of the polynomial)
  4. Test intervals between critical points
  5. Determine which intervals satisfy the inequality

For Rational Inequalities

  1. Move all terms to one side
  2. Combine into a single fraction
  3. Find zeros of the numerator (included in solution for ≤ or ≥)
  4. Find zeros of the denominator (always excluded - discontinuities)
  5. Test intervals between critical points
  6. Determine which intervals satisfy the inequality

Applications of Inequalities

Inequalities are fundamental in mathematics and have numerous real-world applications:

Common Mistakes to Avoid

Why Choose Our Inequality Solver?

Solving inequalities can be challenging, especially for complex polynomial and rational expressions. Our calculator offers:

Tips for Working with Inequalities

Additional Resources

To deepen your understanding of inequalities and algebra, explore these resources:

Reference this content, page, or tool as:

"Inequality Solver" at https://MiniWebtool.com/inequality-solver/ from MiniWebtool, https://MiniWebtool.com/

by miniwebtool team. Updated: Dec 08, 2025

You can also try our AI Math Solver GPT to solve your math problems through natural language question and answer.

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