What is interval notation?

Interval notation represents a set of real-number solutions using parentheses, brackets and infinity symbols.

Linear Inequality Solver

Inequality Calculator

Solve simple and compound linear inequalities with step-by-step working, interval notation, set-builder notation and an interactive number-line result.

Instant results Automatic sign reversal Number-line graph

Linear Inequality Calculator

Choose a simple or compound inequality

Ready

Enter the Inequality

ax + b < c

X Coefficient

Expression Constant

Operator

Right-Side Constant

Left Boundary

Left Operator

X Coefficient

Expression Constant

Right Operator

Right Boundary

Inequality Preview

ax + b ≤ c

Optional Test Value

Check whether a specific number belongs to the final solution set.

Valid Solution Interval

The inequality solution will appear here.

Solution

x ≤ 0

Interval Notation

(−∞, 0]

Set-Builder Form

{x | x ≤ 0}

Number-Line Solution

Visual Interval

Solution Steps

Test-Value Result

Enter an optional test value to check whether it belongs to the solution set.

Try an example

What Is an Inequality Calculator?

An inequality calculator solves mathematical statements that compare two values using less than, greater than, less than or equal to, or greater than or equal to.

This calculator handles simple linear inequalities and compound inequalities containing two connected conditions. It supports positive, negative, decimal and zero coefficients.

Results are displayed as a standard inequality, interval notation, set-builder notation and a number-line graph.

Simple inequalities Solve expressions such as 3x + 5 ≤ 20.

Compound inequalities Solve two connected conditions and find their shared interval.

Automatic sign reversal The inequality direction changes correctly when dividing by a negative value.

Important Inequality Rules

Most equation operations work normally, but negative multiplication and division require special attention.

Add or subtract The inequality direction stays the same.

Multiply or divide by positive The inequality direction stays the same.

Multiply or divide by negative Reverse < with > and ≤ with ≥.

Compound inequalities Solve both conditions and intersect the results.

Simple Process

How the Inequality Calculator Works

Solve and visualize a linear inequality in three simple steps.

Select the Inequality Type

Choose a simple inequality or a compound inequality with two connected comparisons.

Enter the Values

Add the x coefficient, expression constant, comparison operators and required boundary values.

Review the Solution

View the inequality, interval notation, set-builder form, number line and calculation steps.

Solving Rules

How to Solve Linear Inequalities

Linear inequalities are solved by isolating the variable while preserving or correctly reversing the comparison.

Add or Subtract

The same value may be added to or subtracted from both sides without changing the inequality direction.

x + 4 < 10 → x < 6

Divide by a Positive Number

Dividing both sides by a positive value keeps the original comparison sign.

3x ≥ 12 → x ≥ 4

Divide by a Negative Number

Dividing or multiplying by a negative value reverses the comparison direction.

−2x < 8 → x > −4

Compound Inequalities

Solve both sides as separate inequalities and keep only values satisfying both conditions.

−5 ≤ 2x + 3 < 13

Worked Examples

Linear Inequality Examples

Compare common inequality types and their solution notations.

Inequality	Solution	Interval Notation
2x + 3 ≤ 11	x ≤ 4	(−∞, 4]
−4x + 5 > 17	x < −3	(−∞, −3)
−5 ≤ 2x + 3 < 13	−4 ≤ x < 5	[−4, 5)
0x + 4 < 10	All real numbers	(−∞, ∞)
0x + 12 ≤ 5	No solution	∅

Helpful Answers

Inequality Calculator FAQs

Find answers about signs, interval notation, compound inequalities and number-line endpoints.

What is a linear inequality?

A linear inequality compares two linear expressions using <, >, ≤ or ≥. Its solution is usually an interval containing multiple real numbers.

When should the inequality sign be reversed?

Reverse the inequality sign whenever both sides are multiplied or divided by a negative number.

Can this calculator solve compound inequalities?

Yes. Select Compound Inequality and enter the left boundary, middle expression and right boundary.

What do brackets mean in interval notation?

A square bracket means the endpoint is included. A parenthesis means the endpoint is excluded. Infinity always uses a parenthesis.

What does an open circle mean on a number line?

An open circle means the boundary value is not included. It corresponds to a strict inequality such as x < 4 or x > 4.

What does a closed circle mean?

A closed circle means the boundary is included. It corresponds to ≤ or ≥.

Why can an inequality have all real numbers?

When the x coefficient is zero, the inequality may reduce to a constant statement that is always true, such as 4 < 10.

Why can an inequality have no solution?

A constant comparison may be false, or the two conditions in a compound inequality may have no shared values.

Solve Linear Inequalities Instantly

Enter your inequality above to receive a complete solution, notation formats, steps and number-line graph.

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