Inequality Solver: How to Solve Inequalities Step by Step

An Inequality Solver helps you find every value that makes an inequality true, not just a single answer like an equation gives. If you have ever stared at a sign like < or ≥ and wondered how to isolate the variable without flipping the wrong symbol, you are in the right place.

What an Inequality Solver Does

An inequality solver is a tool that takes an inequality, such as 3x + 5 < 14, and finds every value of the variable that makes the statement true. You type in the expression, and it returns a solution set, usually written as an inequality like x < 3 or in interval notation like (-∞, 3). For a linear inequality, the output is a single range of values. For a compound inequality, such as -2 < 2x + 1 ≤ 9, the solver returns the range that satisfies both parts at once and often shows it on a number line. Many Inequality Solver tools also display the steps, so you can check your own work.

How to Solve a Linear Inequality by Hand

Solving a linear inequality works much like solving a linear equation. Your goal is to isolate the variable on one side, using the same inverse operations you already know. The key difference is the inequality sign, which can flip. Work through each step carefully, and you can reach a clear answer every time.

1. Clear the parentheses and fractions. Distribute any multiplication and multiply through to remove fractions so you work with whole numbers.
2. Collect the variable terms. Add or subtract the variable terms so they sit on one side of the inequality.
3. Isolate the variable. Undo the constant by adding or subtracting, then divide or multiply to leave the variable alone.
4. Flip the sign if needed. When you multiply or divide both sides by a negative number, reverse the inequality sign. For example, -2x < 6 becomes x > -3.
5. Check your answer. Substitute a number from your solution into the original inequality. It should make the statement true.

Inequality Solver Tools Compared

Several free and paid tools can check your work or show each step. Each one trades off differently between step-by-step detail, graphing, and cost. Pricing and features change often, so confirm current details on each tool's official site before you rely on it.

Graphing and Writing Your Solution

Once you have your answer, you can show it visually on a number line. Start by drawing a horizontal line and marking the boundary number from your solution. Use an open circle for a strict inequality like , since that value is not included. Use a closed circle for ≤ or ≥, because the boundary is part of the answer. Then shade toward the side where the solution lies.

To write the same answer in interval notation, use a parenthesis for an open end and a bracket for a closed end. For example, x ≥ 3 becomes [3, ∞), and x < 5 becomes (-∞, 5). Infinity symbols always get a parenthesis.

• Open circle or parenthesis: the endpoint is not included.
• Closed circle or bracket: the endpoint is included.
• Shade toward the solution, then check one test number inside that region.

Related guide: System of Equations Solver

Related guide: Simplify Fractions

Related guide: Fraction to Decimal

Key Takeaways

• An inequality solver finds the range of values that make an inequality true, not just a single answer.
• Isolate the variable the same way you would with an equation, and flip the sign whenever you multiply or divide by a negative number.
• Use a tool such as Desmos, Symbolab, or Wolfram Alpha to check your work, and graph your answer on a number line.
• Write the final result in interval notation or set notation, and test a value inside and outside the solution to confirm it.

Frequently Asked Questions About Inequality Solver

Can an inequality solver show the steps for solving an inequality?

Many inequality solvers do show steps, especially tools like Symbolab and Wolfram Alpha, which break the process into isolating the variable, simplifying, and writing the final interval. Free options vary, and some show steps only with a paid plan. Check the tool's features page before you rely on it for homework.

Why does the inequality sign flip when you divide by a negative number?

Dividing or multiplying by a negative number reverses the order of values on the number line, so a larger number becomes smaller. For example, 2 is less than 4, but -2 is greater than -4. To keep the statement true, the inequality sign must flip.

How do I check if my inequality solution is correct?

Pick a test number from inside your solution range and plug it into the original inequality. Then pick a number outside the range and test it too. If the inside value makes the statement true and the outside value makes it false, your solution is very likely correct.

How do you write the solution to an inequality in interval notation?

Interval notation lists the lower and upper bounds of the solution set inside brackets or parentheses. A square bracket means the endpoint is included, while a parenthesis means it is excluded. Infinity symbols always get a parenthesis. For example, x is greater than or equal to 3 is written [3, ∞).

What is the difference between a strict inequality and a compound inequality?

A strict inequality uses < or > and does not include the boundary value, while a non-strict one uses ≤ or ≥ and does. A compound inequality joins two inequalities with "and" or "or," such as -2 < x and x ≤ 5. Solve each part separately, then combine the results using the connecting word.

References & Further Reading

• Khan Academy – Khan Academy
• Desmos – Desmos
• Symbolab – Symbolab

About This Article

This guide was written with AI assistance from publicly available information. Prices, nutrition values and availability can change, so confirm important details with the official source before you decide.