Inequality Calculator

Category: Algebra II
- April 01, 2025
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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

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

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

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

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

Display Options

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 >).

Linear Inequality: \( ax + b \; \{<, >, \leq, \geq\} \; c \)

Quadratic Inequality: \( ax^2 + bx + c \; \{<, >, \leq, \geq\} \; 0 \)

Rational Inequality: \( \frac{P(x)}{Q(x)} \; \{<, >, \leq, \geq\} \; 0 \)

System of Inequalities (2D): \( ax + by \; \{<, >, \leq, \geq\} \; c \)

What Is the Inequality Calculator?

The Inequality Calculator is a step-by-step tool that helps you solve and understand linear, quadratic, rational, and system inequalities. Whether you're working with simple expressions like 2x + 3 > 7 or more complex rational forms, this calculator offers clear solutions and visual support.

It works as a math helper for high school and college students, educators, or anyone looking to improve their problem-solving skills. It's ideal for reinforcing concepts in algebra, pre-calculus, and calculus courses.

Key Features

  • Supports four types of inequalities: linear, quadratic, rational, and systems of inequalities.
  • Provides step-by-step solutions to guide your understanding.
  • Visualizes results on a number line or 2D graph for better clarity.
  • Includes real examples for quick reference.
  • Gives results in multiple formats: solution set and interval notation.

How to Use the Calculator

  1. Select the type of inequality from the tabs: Linear, Quadratic, Rational, or System.
  2. Enter your inequality in the input field provided. Use symbols like <, >, ≤, and ≥.
  3. (Optional) Choose whether to display step-by-step solutions and visualizations by checking the boxes.
  4. Click Solve Inequality to view your solution.
  5. To start over, click Reset.

Why It’s Useful

This calculator helps you practice and verify your algebra skills with immediate feedback. It also gives insights into inequality behavior by showing the direction of solutions and key values like critical points.

Visual learners benefit from seeing graphs and number lines, while those preparing for exams can follow step-by-step logic. It’s also a quick alternative when solving equations by hand takes too long.

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FAQs

Can I enter multiple inequalities at once?

Yes. Use the “System of Inequalities” tab to enter two 2D inequalities for simultaneous solution and graphing.

What symbols should I use for inequalities?

Use >, <, ≥, and ≤ for proper inequality formatting. The calculator also supports the textual format like “>=” or “<=”.

Does it show all steps in the solution?

Yes. If you check the "Show step-by-step solution" option, you'll see a clear breakdown of each step taken to reach the answer.

Can I see a graph or number line?

Yes. The calculator provides a number line for 1-variable inequalities and a coordinate graph for 2D systems, helping you understand the range of solutions visually.

What formats does the solution use?

You’ll get results in both solution set form (e.g., x ≥ 3) and interval notation (e.g., [3, ∞)), making it easier to study and compare formats.

Get More Out of Your Math Practice

The Inequality Calculator simplifies the process of solving, checking, and learning inequalities. Whether you're looking to graph inequalities or understand algebraic concepts in more depth, it's a great resource to boost your skills.

Try pairing it with other helpful tools like the Logarithm Calculator for exponential equations or the Midpoint Calculator when working with coordinates. You can even use the Simplify Expressions Calculator to tidy up before solving.

Use this tool to solve inequalities with confidence—fast, clear, and visual.