An inequality solver finds the values of a variable that make an inequality true. Unlike an equation, which usually has one or more exact solutions, a linear inequality typically represents a range of values. This calculator solves one-variable linear inequalities and shows the answer in inequality notation, interval notation, and on a number line.
What Is an Inequality?
An inequality compares two values or expressions using symbols such as <, >, ≤, or ≥. For example, x < 5 means every value of x smaller than 5 is part of the solution set.
Inequality Symbols
| Symbol | Meaning | Example |
|---|---|---|
| < | Less than | x < 5 |
| > | Greater than | x > 5 |
| ≤ | Less than or equal to | x ≤ 5 |
| ≥ | Greater than or equal to | x ≥ 5 |
How to Solve Linear Inequalities
- Simplify both sides of the inequality.
- Combine like terms where possible.
- Move variable terms to one side.
- Move constants to the other side.
- Divide by the final variable coefficient.
- If you divide by a negative number, reverse the inequality sign.
- Write the result in inequality notation and interval notation.
- Check the solution using a test value.
When Do You Flip the Inequality Sign?
You must reverse the inequality symbol whenever both sides are multiplied or divided by a negative number.
Divide both sides by -3:
x < -2
The sign changes from > to < because the division was by a negative number.
Worked Inequality Examples
Example 1: Solve 2x + 5 ≤ 11
2x + 5 ≤ 11
Subtract 5 from both sides:
2x ≤ 6
Divide by 2:
x ≤ 3
Interval notation: (-∞, 3]
Number line: closed circle at 3, shaded to the left.
Example 2: Solve -3x + 4 > 10
-3x + 4 > 10
Subtract 4 from both sides:
-3x > 6
Divide by -3 and reverse the sign:
x < -2
Interval notation: (-∞, -2)
Interval Notation for Inequalities
Interval notation is another way to write a solution range. Parentheses mean the endpoint is excluded, while brackets mean the endpoint is included.
| Inequality | Interval Notation |
|---|---|
| x < a | (-∞, a) |
| x ≤ a | (-∞, a] |
| x > a | (a, ∞) |
| x ≥ a | [a, ∞) |
Graphing an Inequality on a Number Line
Number lines make the solution range easier to understand visually.
- Open circle: the boundary value is not included, as in x < 3.
- Closed circle: the boundary value is included, as in x ≤ 3.
- Shade left: values smaller than the boundary.
- Shade right: values greater than the boundary.
Common Inequality Mistakes
1. Forgetting to reverse the sign
When multiplying or dividing by a negative number, always reverse the inequality symbol.
2. Treating < and ≤ as the same
x < 3 excludes 3, while x ≤ 3 includes 3.
3. Using the wrong number-line endpoint
Strict inequalities use an open circle. Inclusive inequalities use a closed circle.
4. Moving terms incorrectly
Apply the same operation to both sides of the inequality, just as you would when solving an equation.
5. Not checking the solution
Substitute a value from the proposed solution range into the original inequality to verify it makes the statement true.
How to Check Your Inequality Solution
Choose a value inside the solution range and substitute it into the original inequality. For example, if your result is x < -2, try x = -3.
Also test a value outside the range, such as x = 0. The original inequality should be false for the outside value.
Checking both an inside and outside value is a quick way to catch an incorrect sign direction.
Related Algebra Calculators
Continue with related tools in our Algebra Calculators collection:
- Linear Equation Solver
- Quadratic Equation Solver
- Simplify Expression Calculator
- Polynomial Calculator
- Exponent Calculator
How This Inequality Solver Works
The solver parses each side of a one-variable linear inequality into a linear form of the type ax + b. It then moves variable terms and constants to opposite sides, combines like terms, and divides by the final coefficient.
If the final coefficient is negative, the solver reverses the inequality symbol. The resulting boundary is then converted into interval notation and displayed on a number line.
This version is intentionally limited to linear inequalities in one variable. It does not claim to solve quadratic, absolute-value, polynomial, rational, or multi-variable systems.
Prepared by the Utilixea Editorial Team.
Last updated: July 21, 2026.
Inequality Solver FAQs
An inequality solver finds the values of a variable that make an inequality true.
Reverse the inequality symbol whenever both sides are multiplied or divided by a negative number.
Interval notation represents a range of solutions. Parentheses exclude an endpoint, while brackets include it.
The symbol < excludes the boundary value, while ≤ includes the boundary value.
Yes. The solver handles negative constants and negative coefficients and automatically reverses the inequality when required.
Yes. Linear inequalities such as 5x - 7 < 2x + 8 are simplified by moving variable terms to one side and constants to the other.
Yes. It supports linear expressions with simple fraction-style division, such as x/2 + 3 ≥ 5, as long as the denominator does not contain the variable.
An open circle means the boundary value is excluded from the solution, as with x < 3 or x > 3.
A closed circle means the boundary value is included in the solution, as with x ≤ 3 or x ≥ 3.
Need another algebra calculation? Explore our Algebra Calculators for equations, expressions, polynomials, exponents, and more.
