Introduction
An inequality is a math statement that compares two values using symbols like < (less than), > (greater than), ≤ (less than or equal to), and ≥ (greater than or equal to). Instead of finding one exact answer like an equation, an inequality gives you a range of values that make the statement true.
This inequality calculator solves linear, quadratic, absolute value, rational, radical, logarithmic, exponential, and compound inequalities. Type your inequality into the input field and press Calculate. The tool will show your answer in inequality notation, interval notation, and set-builder notation. It also gives you a step-by-step solution, a number line graph, and a coordinate graph so you can see exactly where your solution set lies.
You can use the built-in math keyboard to enter special symbols, test values with the evaluate tab, or check your own work with the verify tab. Whether you are studying for a test or working through homework, this calculator breaks down each problem into clear steps so you can learn how to solve inequalities on your own.
How to Use Our Inequality Calculator
Enter any inequality and this calculator will solve it for you. It shows the answer in multiple forms, gives step-by-step work, and draws a number line and graph of the solution.
Inequality Type: Pick the type of inequality from the dropdown menu. Choose "Auto-Detect" if you are not sure. The calculator will figure out the type for you.
Inequality: Type your inequality into the input box. Use > for greater than, < for less than, >= for greater than or equal to, and <= for less than or equal to. Use ^ for exponents, sqrt() for square roots, and | | for absolute value. A live preview shows your inequality in math notation as you type.
Math Keyboard: Click "Show Keyboard" if you need help typing math symbols. It has tabs for Algebra, Trigonometry, and Calculus symbols you can insert with one click.
Quick Operations: Click Solve to find the solution. You can also click Simplify, Factor, Expand, Find Domain, Find Intercepts, or Find Vertex for more details about your expression.
Example Inequalities: Click any example tile to load a sample problem into the calculator. Click "Show More Examples" to see additional options.
Upload a File: You can upload a .txt, .pdf, .doc, .png, or .jpg file that contains an inequality. Text files will fill the input box automatically.
Solve For: When your inequality has more than one variable, use this dropdown to pick which variable to solve for.
Variable Sliders: Turn on "Show variable sliders" to drag sliders and test different values. The calculator tells you if the inequality is true or false at each value.
Results: After you click Calculate, the solution summary shows your answer in inequality notation, interval notation, and set-builder notation. Use the tabs below to view the step-by-step solution, number line, graph, evaluate tool, algebraic details, or the answer checker.
Verify Your Answer: Go to the Verify tab, type in your own answer, and click "Check My Answer" to see if it matches the correct solution.
What Is an Inequality?
An inequality is a math statement that compares two values using symbols like < (less than), > (greater than), ≤ (less than or equal to), and ≥ (greater than or equal to). Unlike an equation, which says two things are equal, an inequality says one side is bigger or smaller than the other.
For example, 2x + 1 > 5 asks: what values of x make the left side larger than 5? The answer is not just one number. It is a whole range of numbers. In this case, every number greater than 2 works.
Types of Inequalities
There are several common types of inequalities you will see in algebra:
- Linear inequalities have a variable with no exponent, like 3x − 4 < 8. These are solved much like a basic solve for x problem.
- Quadratic inequalities have a squared variable, like x² − 9 > 0. You can find the boundary points using the quadratic formula.
- Compound inequalities join two inequalities together, like −3 < 2x + 1 ≤ 7. Both parts must be true at the same time.
- Absolute value inequalities use absolute value bars, like |x − 2| < 5. These measure distance from a point on the number line.
- Rational inequalities have a variable in the denominator, like 1/(x − 2) > 0. You may also need a fraction calculator when working with complex rational expressions.
How to Solve an Inequality
Solving an inequality is a lot like solving an equation. You add, subtract, multiply, or divide both sides to get the variable alone. There is one important rule to remember: if you multiply or divide both sides by a negative number, you must flip the inequality sign. For example, if you divide both sides by −2, a < sign becomes a > sign.
How to Show the Answer
The solution to an inequality is usually a range of numbers. You can write it in three main ways:
- Inequality notation — uses symbols, like x > 2.
- Interval notation — uses brackets and parentheses, like (2, ∞). A parenthesis means the endpoint is not included. A bracket means it is included.
- Number line — a drawing that shades the part of the line where the answer lives. An open circle means the point is not included. A filled circle means it is. You can find specific points on the line using a midpoint calculator or a distance calculator.
Inequalities are used in everyday life more than you might think. They help set speed limits, budget ranges, age requirements, and safe temperatures. Any time you see a rule that says "at least," "no more than," or "between," you are looking at an inequality. For related algebra tools, try our slope calculator, polynomial calculator, or system of equations calculator.