Inequality Calculator

Category: Algebra II

- April 16, 2025

| |

This inequalities calculator can solve linear, quadratic, and rational inequalities, display step-by-step solutions, and visualize results on a number line. Select the inequality type, enter your expression, and click "Solve" to get started.

Inequality Type

Enter Linear Inequality:

Examples: 2x + 3 > 7, 4x - 2 ≤ 10, -3x + 4 ≥ -8, 5 < x + 2

Enter Quadratic Inequality:

Examples: x² - 4x + 3 > 0, x² - 9 ≤ 0, 2x² + 5x - 3 ≥ 0, x² + 6x + 9 < 0

Enter Rational Inequality:

Examples: (x - 1)/(x + 2) > 0, (x² + 1)/x ≥ 3, 1/(x - 3) < 2, (x + 4)/(x² - 4) ≤ 0

First Inequality:

Second Inequality:

Examples for 2D systems: 2x + y > 5, x - y ≤ 3, y ≥ 2x, x + 2y < 6

Display Options

Show step-by-step solution

Show number line/graph visualization

Understanding Inequalities

Inequalities express relationships between quantities where one is greater than (>), less than (<), greater than or equal to (≥), or less than or equal to (≤) another. Solving inequalities involves finding all values that make the inequality true.

Rules for Solving Inequalities

Addition/Subtraction: Adding or subtracting the same value from both sides preserves the inequality.
Multiplication/Division: Multiplying or dividing both sides by a positive number preserves the inequality. Multiplying or dividing by a negative number reverses the inequality sign.
Quadratic Inequalities: Typically solved by finding critical points, analyzing sign behavior, and using intervals.
Rational Inequalities: Require finding values where the expression equals zero or is undefined, then testing intervals.

Notation

Interval Notation: (a, b) represents a < x < b, [a, b] represents a ≤ x ≤ b, and combinations like [a, b) or (a, b] represent mixed inequality boundaries.
Set Builder Notation: {x | x > a} represents "the set of all x such that x is greater than a".
Number Line: Solid dots (●) represent included endpoints (≤ or ≥), while hollow dots (○) represent excluded endpoints (< or >).

Understanding the Inequality Calculator

The Inequality Calculator is designed to help solve and simplify mathematical inequalities, including absolute value inequalities, compound inequalities, and quadratic inequalities. Whether you’re a student, teacher, or just someone exploring algebra, this tool makes the process easier and provides step-by-step explanations for better comprehension.

What is an Inequality?

An inequality is a mathematical statement that compares two expressions using symbols like:

<: Less than
>: Greater than
≤: Less than or equal to
≥: Greater than or equal to

Inequalities can involve absolute values, compound expressions, and quadratic equations. For instance:

Compound Inequality: \( -3 \leq x + 2 < 5 \)
Absolute Value Inequality: \( |x - 1| > 4 \)
Quadratic Inequality: \( x^2 - 4 > 0 \)

How to Use the Inequality Calculator

To use the Inequality Calculator effectively, follow these steps:

Select an Example: Use the dropdown menu to pick a predefined inequality example, such as \( 2 < |x - 1| \leq 4 \), and it will automatically fill the input field.
Enter a Custom Inequality: If you prefer, you can type your own inequality in the input field. Use the absolute value function as abs() for inequalities that involve absolute values.
Click "Calculate": The calculator will solve the inequality and provide a clear step-by-step explanation.
View Results: The solution will be displayed, along with all the steps taken to solve the inequality, making it easy to follow.
Clear Input: Use the "Clear" button to reset the fields and start over.

Features of the Inequality Calculator

Solves absolute value inequalities with clear steps.
Handles compound inequalities like \( 4 < 2x + 5 \leq 7 \).
Solves quadratic inequalities such as \( x^2 - 4 > 0 \).
Displays results in a neat, organised manner.
Provides step-by-step explanations to improve understanding.

Why Use the Inequality Calculator?

Solving inequalities can be tough, especially when they involve absolute values or complex expressions. This tool simplifies the process and gives a clear breakdown of each step, making it easier for users to:

Understand the solution process.
Check homework or practice problems.
Enhance algebra skills with detailed explanations.

FAQs

What types of inequalities does this calculator solve?: It solves absolute value inequalities, compound inequalities, and quadratic inequalities.
How do I enter absolute values?: Use the abs() function. For example, \( |2x - 4| \) should be entered as abs(2x-4).
Can I enter custom inequalities?: Yes, you can manually enter your own inequality in the input field.
What does the step-by-step explanation show?: It shows how the inequality is solved, starting with the input expression and breaking it down into simplified steps.
What if I receive an "Invalid Input" error?: Make sure the inequality is entered correctly, and use abs() for absolute values. If you’re unsure, try one of the predefined examples from the dropdown menu.

Example Inequalities

Here are some examples you can try with the calculator:

Absolute Value Inequality: \( 2 < |x - 1| \leq 4 \)
Compound Inequality: \( -3 \leq x + 2 < 5 \)
Quadratic Inequality: \( x^2 - 4 > 0 \)

Conclusion

The Inequality Calculator is a versatile tool for quickly and accurately solving various types of inequalities. Whether you’re tackling homework problems, learning algebra concepts, or verifying answers, this calculator provides clear solutions and step-by-step explanations to enhance your understanding.