Inequality Solver

MiniWebtool’s Inequality Solver finds solution sets for supported linear, quadratic, polynomial, and rational inequalities. Algebraic steps, interval notation, and a number-line display make boundaries and excluded values easier to inspect.

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Help me understand MiniWebtool's Inequality Solver: https://miniwebtool.com/inequality-solver/. What can I use it for, what inputs or settings do I need, and how do I use it and understand the output? Walk me through one simple example. Use this page as your source and answer in English.

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Roots split the number line; each piece keeps one sign

Enter a linear, quadratic, polynomial or rational inequality below to get the solution in interval notation, on a number line, with the steps.

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 ≥ Compound: -3 < 2x + 1 ≤ 7 Absolute value: |x - 3| < 5

📖 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

Compound & Absolute Value -3 < 2x+1 <= 7
|x-3| < 5 or abs(x-3) < 5

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 != · compound: -3 < 2x+1 <= 7 · absolute value: \|x-3\| < 5)

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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.

At a Glance

Inequality Solver finds the solution set of linear, quadratic, polynomial and rational inequalities in one variable. It shows the algebraic steps, interval notation and a number line with boundaries and excluded values marked.

Inequality Solver: features at a glance
Feature	What you get	Good for
Four inequality types	Solves linear, quadratic, polynomial (degree 3 or higher) and rational inequalities. The type is detected automatically from what you enter.	Algebra homework and checking solutions worked by hand
Input syntax	Use <, >, <=, >= or != with +, -, * and /. Write exponents as ^ or **, group terms with parentheses, and use a single variable.	Typing fractions such as (x+1)/(x-1) > 0
Interval notation	The solution is written in standard interval notation, with the union symbol joining separate intervals.	Writing answers in the form textbooks expect
Number line	Filled circles mark included points, open circles mark excluded points, orange open circles mark discontinuities, and green shading marks intervals that satisfy the inequality.	Seeing boundaries and excluded values at a glance
Steps and critical points	Steps cover moving terms to one side, factoring, finding zeros and discontinuities, and testing each interval. Factored forms are shown when they apply.	Learning the method behind an answer

Key Features of Our Inequality Solver

Multiple Inequality Types: Solve linear, quadratic, polynomial, and rational inequalities
Visual Number Line: See your solution graphically represented on an interactive number line
Interval Notation: Results displayed in standard mathematical interval notation
Step-by-Step Solutions: Understand each step involved in solving the inequality
Critical Points Analysis: Identify zeros and discontinuities automatically
Automatic Type Detection: The calculator identifies whether your inequality is linear, quadratic, polynomial, or rational
Factored Forms: See factored representations when applicable
Educational Insights: Learn mathematical principles through detailed explanations
LaTeX-Formatted Output: Beautiful mathematical rendering using MathJax

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

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
Click Solve: Process your inequality and view the results.
Review Step-by-Step Solution: Learn from detailed explanations of each solving step.
View Number Line: See the solution visualized on a number line with critical points marked.
Check Interval Notation: Read your solution in standard interval notation.

Inequality Input Guidelines

For best results, follow these input conventions:

Inequality Symbols: Use < for less than, > for greater than, <= for less than or equal to, >= for greater than or equal to
Multiplication: Use * or simply write variables together (e.g., 2*x or 2x)
Exponents: Use ^ or ** (e.g., x^2 or x**3)
Parentheses: Use parentheses for grouping (e.g., (x+1)/(x-1) > 0)
Single Variable: The calculator works with single-variable inequalities only

Understanding Inequality Solutions

Number Line Representation

The number line shows:

Filled circles (●): Points included in the solution (for ≤ or ≥)
Open circles (○): Points excluded from the solution (for < or >)
Orange open circles: Discontinuities where the expression is undefined
Green shaded regions: Intervals where the inequality is satisfied

Interval Notation

Solutions are expressed using interval notation:

(a, b): All numbers between $a$ and $b$, excluding endpoints
[a, b]: All numbers between $a$ and $b$, including endpoints
(a, b]: All numbers between $a$ and $b$, excluding $a$ but including $b$
(-∞, a): All numbers less than $a$
(a, ∞): All numbers greater than $a$
∪: Union symbol, combines multiple intervals

Methods for Solving Inequalities

For Linear Inequalities

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

For Quadratic and Polynomial Inequalities

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

For Rational Inequalities

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

Applications of Inequalities

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

Economics: Profit and loss analysis, budget constraints, optimization problems
Physics: Velocity ranges, acceleration bounds, energy constraints
Engineering: Safety margins, tolerance specifications, design constraints
Statistics: Confidence intervals, hypothesis testing, probability ranges
Computer Science: Algorithm complexity, resource allocation, optimization
Business: Break-even analysis, pricing strategies, capacity planning
Chemistry: Reaction rate conditions, concentration ranges, pH levels

Common Mistakes to Avoid

Not Reversing the Inequality: When multiplying or dividing both sides by a negative number, you must reverse the inequality sign
Forgetting Domain Restrictions: For rational inequalities, points where the denominator equals zero must be excluded
Incorrect Test Points: When testing intervals, choose points that actually lie within each interval
Misinterpreting Interval Notation: Remember that parentheses ( ) exclude endpoints while brackets [ ] include them
Combining Inequalities Incorrectly: You cannot perform the same operations on compound inequalities as you would with equations

Why Choose Our Inequality Solver?

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

Accuracy: Powered by SymPy, a robust symbolic mathematics library
Visual Learning: Number line representations make solutions intuitive
Comprehensive Solutions: Step-by-step explanations for every type of inequality
Educational Value: Learn mathematical concepts while solving problems
Speed: Instant results for even complex inequalities
Versatility: Handles linear, quadratic, polynomial, and rational inequalities
Free Access: No registration or payment required

Tips for Working with Inequalities

Always move all terms to one side before solving
Factor expressions when possible to identify critical points easily
Remember to check for domain restrictions in rational inequalities
Use test points to verify which intervals satisfy the inequality
Draw a number line to visualize the solution
Double-check whether endpoints should be included or excluded
Verify your solution by substituting test values back into the original inequality

Additional Resources

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

Inequality (Mathematics) - Wikipedia

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

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