Solve An Inequality with Interval Notation Calculator

This calculator helps you solve inequalities and express the solutions in interval notation. Whether you're a student learning algebra or a professional working with mathematical expressions, this tool provides a clear, step-by-step solution to your inequality problems.

How to Use This Calculator

Using our inequality solver is simple. Follow these steps:

Enter your inequality in the input field. For example, you might enter x > 5 or 2x - 3 < 7.
Select the type of inequality you're solving (linear, quadratic, etc.).
Click the "Calculate" button to solve the inequality.
Review the solution in interval notation and the number line visualization.

The calculator will display the solution in interval notation, such as (5, ∞) or (-∞, 3), and provide a visual representation of the solution on a number line.

What Is Interval Notation?

Interval notation is a way to represent a set of real numbers using parentheses and brackets. It's a concise and precise method for describing ranges of numbers.

Interval Notation Symbols

(a, b) - All numbers between a and b, not including a and b
[a, b] - All numbers between a and b, including a and b
(a, b] - All numbers between a and b, not including a but including b
[a, b) - All numbers between a and b, including a but not including b
(-∞, a) - All numbers less than a
(a, ∞) - All numbers greater than a

Interval notation is commonly used in algebra, calculus, and other branches of mathematics to describe the domain and range of functions, as well as the solutions to inequalities.

Solving Inequalities

Solving inequalities involves finding all the values of the variable that satisfy the given condition. The process is similar to solving equations, but with some important differences.

Key Differences

When you multiply or divide both sides of an inequality by a negative number, you must reverse the inequality sign.
When solving compound inequalities, you must satisfy both conditions simultaneously.
When solving absolute value inequalities, you must consider both the positive and negative cases.

After solving the inequality, you can express the solution in interval notation. For example, if you solve x > 5, the solution in interval notation is (5, ∞).

Example Problems

Here are a few example problems to help you understand how to use the calculator and interpret the results.

Example 1: Simple Linear Inequality

Solve x > 5 and express the solution in interval notation.

The solution is (5, ∞). This means all real numbers greater than 5 satisfy the inequality.

Example 2: Quadratic Inequality

Solve x² - 4x < 0 and express the solution in interval notation.

The solution is (0, 4). This means all real numbers between 0 and 4 satisfy the inequality.

Example 3: Compound Inequality

Solve -3 < 2x + 1 < 5 and express the solution in interval notation.

The solution is (-2, 2). This means all real numbers between -2 and 2 satisfy the inequality.

Frequently Asked Questions

What is interval notation?

Interval notation is a way to represent a set of real numbers using parentheses and brackets. It's a concise and precise method for describing ranges of numbers.

How do I solve an inequality?

Solving inequalities involves finding all the values of the variable that satisfy the given condition. The process is similar to solving equations, but with some important differences, such as reversing the inequality sign when multiplying or dividing by a negative number.

What is the difference between parentheses and brackets in interval notation?

Parentheses ( ) indicate that the endpoint is not included in the interval, while brackets [ ] indicate that the endpoint is included. For example, (a, b) includes all numbers between a and b, not including a and b, while [a, b] includes all numbers between a and b, including a and b.