Inequality Calculator

Category: Algebra II
- May 22, 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

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 >).

What is the Inequality Calculator?

The Inequality Calculator is a helpful tool that solves various types of inequalities. It can handle linear, quadratic, and rational inequalities. Users can easily input their expressions and check their solutions step by step. Additionally, the calculator provides a visual representation of the results on a number line, making understanding inequalities much simpler.

Types of Inequalities You Can Solve

This calculator lets you work with different inequality types. You can choose from the following:

  • Linear Inequalities: Simple inequalities like 2x + 3 > 7.
  • Quadratic Inequalities: More complex challenges such as x² - 4x + 3 ≤ 0.
  • Rational Inequalities: These include fractions, such as (x - 1)/(x + 2) > 0.
  • Systems of Inequalities: Solve multiple inequalities together, like 2x + y > 5 and x - y ≤ 3.

How to Enter Your Inequality

Using the calculator is straightforward. Start by selecting the type of inequality you want to solve. Input your expression in the provided box and tick the options to show steps and visualizations if you wish. Once you’re ready, press the solve button, and the calculator will provide the solution along with a visual display.

Understanding the Results

After solving an inequality, you’ll see the solution set and its interval notation. The solution set shows the values that satisfy the inequality, while interval notation represents this range more compactly. For example, if the solution set is x > 2, the interval notation would be (2, ∞).

Visualising Your Inequalities

One of the standout features of this calculator is its visualisation capability. After solving an inequality, you can see a number line that highlights the solution. You can also view a graph for systems of inequalities to better understand how the inequalities interact with each Other, providing a clear visual guide to the results.

Learning Through Step-by-Step Solutions

The calculator shows a complete breakdown of how it arrived at the solution. This step-by-step approach helps users learn and understand the process of solving inequalities. Each step is clearly laid out, making it easier to follow along and grasp the concepts involved.

Rules for Solving Inequalities

When working with inequalities, there are some important rules to keep in mind:

  • Addition/Subtraction: You can add or subtract the same number from both sides without changing the inequality.
  • Multiplication/Division: Multiplying or dividing by a positive number keeps the inequality the same. But be careful! If you multiply or divide by a negative number, you must flip the inequality sign.
  • Quadratic Inequalities: These often need you to find critical points and test intervals to see where the inequality holds true.

Using the Inequality Calculator Effectively

This calculator can be a great resource for students and anyone needing help with inequalities. It’s user-friendly and provides accurate solutions quickly. Whether you're studying for a test or brushing up on your Math skills, the Inequality Calculator makes learning about inequalities engaging and easy.