What is the difference between strict and non-strict inequalities?

A strict inequality excludes the boundary value, so the number line uses an open point. A non-strict inequality includes the boundary value.

Why does the sign flip when dividing by a negative number?

Multiplying or dividing by a negative reverses the order of numbers, so the inequality sign must reverse too.

What if the variable cancels out?

If the remaining statement is true, all real numbers work. If it is false, there is no solution.

How do I write the answer as an interval?

Use parentheses for excluded boundaries and brackets for included boundaries. Infinity always uses parentheses.

Can I enter fractions and parentheses?

Yes. Numeric fractions and parentheses are supported. Variable denominators belong to rational inequalities and need interval analysis.

Why does this not solve quadratic inequalities?

Quadratic inequalities are not linear. Their solution is usually an interval or a union of intervals found with a different method.

Linear Inequality Solver

Solve one-variable linear inequalities with steps, sign-flip rule, interval notation, and a number line.

Enter an inequality

Supports parentheses, numeric fractions, and four inequality signs. The variable degree must be 1 or lower.

Quick input

Sign:

Examples

Inequality solution

x < 5

x ∈ (−∞; 5)

Number line solution

x < 5

−∞

+∞

x < 5

-2

-1

Legend

Solution interval

All x values satisfying the inequality, extending to -infinity.

Closed boundary (<=, >=)

The boundary is included in the solution.

Open boundary (<, >)

The boundary is not included in the solution.

Excluded value x != ...

There are no excluded denominator values.

zero / ticks· click a point for details

What the solution shows

After moving terms, there is one boundary on the number line: 5. The solution lies to the left of this boundary.

The boundary is excluded because the inequality is strict.

A linear inequality contains one variable only to the first power. It is solved like a linear equation, except that dividing by a negative number reverses the sign. Variable denominators require interval analysis.

How to solve linear inequalities

A linear inequality has one variable only to the first power. The solver expands parentheses, moves terms, divides by the coefficient of x, and shows whether the inequality sign must flip. The answer is shown as an inequality, interval notation, and a number line.

a x + b ◃ c

x is the unknown; the symbol can be <, <=, >, or >=.

A \cdot x + B ◃ 0

A is the coefficient of x after moving all terms to one side.

x ◃ - \frac{B}{A}

The direction depends on the sign of A.

Expand parentheses and combine like terms
Move variable terms and constants
Divide by the coefficient of x
Flip the inequality sign when dividing by a negative number
Write the answer in interval notation and on a number line

The sign-flip rule

Flip the inequality sign

When both sides are multiplied or divided by a negative number, the inequality sign reverses. Multiplying or dividing by an expression with x needs interval analysis because its sign can change.

Operation	Inequality sign
Add or subtract a number	Does not change
Multiply or divide by a positive number	Does not change
Multiply or divide by a negative number	Flips
Multiply or divide by zero	Not allowed

Interval notation and number line

The number line shows the boundary value, ray direction, and whether the boundary is included. Strict inequalities use an open point; non-strict inequalities use a closed point.

Answer type	Interval	Point
Strict sign and ray left	Boundary excluded	Open circle
Non-strict sign and ray left	Boundary included	Closed circle
Strict sign and ray right	Boundary excluded	Open circle
Non-strict sign and ray right	Boundary included	Closed circle

Special cases and limits

If the variable cancels out and the remaining numeric statement is true, every real number is a solution. If the remaining statement is false, there is no solution.

Not a general inequality solver

This tool solves linear inequalities only. Quadratic, rational, absolute-value, logarithmic, exponential, and system inequalities need other methods.

Frequently Asked Questions

Sources and References

Inequality
Wolfram MathWorld

Calculations are based on the listed reference sources. Links open in a new tab.

Updated: June 2, 2026

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