Inequality Solver - Solve Linear Inequalities Step by Step

Inequality Solver

Enter an inequality or choose a preset. Supports linear, quadratic, and compound inequalities.

Example Inequalities

Inequality Rules and Properties

When solving inequalities, most operations work like regular equations, but there is one critical exception:

About Linear Inequalities

A linear inequality is an expression that shows the relationship between a linear expression and a value using inequality operators (<, ≤, >, ≥). Unlike equations which have a single solution, inequalities have a range of solutions.

How to Solve Linear Inequalities

The process is nearly identical to solving linear equations:

1. Simplify both sides if needed
2. Isolate the variable term on one side
3. Divide or multiply to solve for the variable
4. Flip the inequality sign if you multiply or divide by a negative number

Example Walkthrough

Solve: -3x + 7 > 13

Compound Inequalities

A compound inequality combines two inequalities, showing that a variable falls within a range. For example, 1 < x < 5 means x is greater than 1 AND less than 5.

To solve compound inequalities, perform the same operation on all three parts simultaneously, maintaining the inequality relationships.

Absolute Value Inequalities

Absolute value inequalities involve the distance from zero. They split into two cases:

• |x| < a means -a < x < a (x is within distance a from zero)
• |x| > a means x < -a OR x > a (x is outside distance a from zero)

Interval Notation Guide

Solutions to inequalities are often written in interval notation, which provides a compact way to express ranges:

Note: Parentheses ( ) indicate the endpoint is NOT included (open interval), while brackets [ ] indicate the endpoint IS included (closed interval). Infinity always uses parentheses.

Frequently Asked Questions

Why do we flip the inequality sign when dividing by a negative?

When you multiply or divide by a negative number, you reverse the order of numbers on the number line. For example, 2 < 3, but -2 > -3. To maintain the truth of the inequality, we must flip the sign.

What is the difference between < and ≤?

The symbol < means "strictly less than" (the value is not included), while ≤ means "less than or equal to" (the value is included). On a number line, < is shown with an open circle, ≤ with a closed circle.

How do I graph an inequality on a number line?

Use an open circle for < or > (endpoint not included) and a closed circle for ≤ or ≥ (endpoint included). Draw an arrow in the direction of the solution: left for less than, right for greater than.

Can inequalities have no solution?

Yes. For example, |x| < -3 has no solution because absolute value is always non-negative. Similarly, a compound inequality like x > 5 AND x < 2 has no solution.

What is the difference between AND and OR in compound inequalities?

AND (intersection) means both conditions must be true simultaneously, like 1 < x < 5. OR (union) means at least one condition must be true, like x < 1 OR x > 5.

Does this calculator store my data?

No. All calculations run entirely in your browser. No data is sent to any server, and nothing is stored.

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Inequality Solver FAQ