Linear Inequality Calculator

The fastest way to use the linear inequality calculator is to type the inequality exactly as you would write it in class. You can enter forms such as 2x + 3 > 7, 5 - x <= 2, 3(x + 2) >= 12, or x/2 - 3 > 1. The parser accepts common keyboard input, so you do not need special math markup just to get started.

After you enter a problem, the page keeps the result area tightly focused on the four views that matter most for linear work: algebraic steps, the graph, interval notation, and verification. That structure is intentional. Many calculator pages stop after they isolate x, but students usually need one more layer of interpretation. They want to know what the answer means on a number line and whether a test value really works.

If you are checking homework, begin with the Steps tab and compare each transformation to your own work. If you are learning the topic for the first time, switch to the Graph and Interval Notation tabs right after that. Seeing the algebra and the visual answer together is the quickest way to understand why x > 2 becomes an open endpoint with shading to the right, while x <= 5 becomes a closed endpoint with shading to the left.

A linear inequality compares two first-degree expressions. Instead of asking when two expressions are equal, it asks when one side is greater than, less than, or equal to the other in an inequality sense. That is why a problem like 2x + 3 > 7 usually has many correct values. Every real number larger than 2 works, so the answer is a set of values, not a single solution.

This is the key distinction between a linear equation and a linear inequality. An equation such as 2x + 3 = 7 has one exact answer, x = 2. The related inequality 2x + 3 > 7 has an entire ray of answers, x > 2. The variable still has degree one, the algebra rules are still familiar, but the final interpretation is different because the result describes a range.

You will most often see four symbols in linear inequality work: >, <, >=, and <=. Strict symbols, > and <, exclude the boundary value. Inclusive symbols, >= and <=, include the boundary value. That inclusion rule shows up everywhere: in the final isolated statement, in interval notation, and on the graph where the endpoint changes from an open circle to a closed circle.

A number line translates algebra into a picture. Once you isolate x, the graph answers two quick questions: where is the boundary, and which side should be shaded? For x > 2, the boundary is 2 and the shading extends to the right because larger values satisfy the inequality. For x < 2, the boundary is still 2, but the shading moves left because smaller values work instead.

The endpoint style matters just as much as the direction. Open circles match strict symbols, > and <, because the boundary itself does not belong to the solution. Closed circles match inclusive symbols, >= and <=, because the endpoint is part of the answer. If you can read a number line correctly, interval notation becomes much more intuitive because you can see why (2, infinity) excludes 2 while [2, infinity) includes it.

Interval notation is the compact form of a graph. Parentheses mean an endpoint is excluded, and brackets mean an endpoint is included. Infinity is never written with a bracket because infinity is not a reachable endpoint on the real line.

A good habit is to read interval notation out loud. For example, [3, infinity) means every real number from 3 upward, with 3 included. That verbal check helps you catch mistakes before you turn in homework or copy the final answer into another system.