About this inequality calculator
This inequality calculator solves one-variable linear and quadratic inequalities from their coefficients. Choose less than, less than or equal to, greater than, or greater than or equal to. The result is a real solution set in interval notation, with boundary roots and working that matches the actual inputs.
The form intentionally asks for coefficients rather than accepting arbitrary algebra text. For a linear comparison, enter ax + b on the left and cx + d on the right. For a quadratic comparison, first write ax² + bx + c compared with zero. If you want a single equality rather than a range of solutions, use the Linear Equation Calculator or Quadratic Formula Calculator.
How to solve an inequality with coefficients
- Choose Linear or Quadratic and select the comparison sign from the menu.
- Enter all visible coefficients. A missing term needs a zero: for x² − 9 > 0, enter a = 1, b = 0 and c = −9.
- For a quadratic with a nonzero right side, subtract that side first. For example, x² > 9 becomes x² − 9 > 0.
- Calculate and read the interval notation together with its endpoint rule. Check the working to see whether the comparison reversed or whether the polynomial changes sign.
Changing the form clears the old answer. Reset clears coefficients and restores the initial mode. Copy, Download and Print include the solution and explanation. To simplify arithmetic while preparing a fraction coefficient, you can use the Fraction Calculator.
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For a separate arithmetic check, you may browse scientific calculators (paid link). Check the model’s supported functions; choose a model with the log, ln and arithmetic functions you need; ordinary calculators do not necessarily solve symbolic inequalities. No purchase is required.
Linear inequalities and quadratic sign charts
A linear comparison ax + b compared with cx + d reduces to (a − c)x + (b − d) compared with zero. Dividing by a positive coefficient preserves the comparison, while dividing by a negative coefficient reverses it. If the coefficient of x is zero, the remaining constant comparison is either true for every real x or false for every real x.
For a quadratic, D = b² − 4ac identifies the real boundary roots. With two distinct real roots, the polynomial has the sign of a outside the roots and the opposite sign between them. A repeated root does not cause a sign change. With no real roots, the sign is the sign of a everywhere. The tool combines that sign chart with the chosen strict or inclusive comparison.
Worked linear and quadratic inequality examples
For −3x + 2 ≤ 8, use Linear mode with a = −3, b = 2, c = 0 and d = 8. Subtracting 2 gives −3x ≤ 6. Dividing by −3 reverses the sign, giving x ≥ −2 and the interval [−2, ∞).
For x² − 5x + 6 < 0, use Quadratic mode with coefficients 1, −5 and 6. The boundary roots are 2 and 3, and the solution is (2, 3). Changing the comparison to ≤ includes those roots, giving [2, 3]. The Polynomial Factoring Calculator can provide a separate factorisation check for supported expressions.
The comparison (x − 1)² < 0 has no real solution; a square cannot be negative. The same expression with ≤ 0 has only x = 1. These repeated-root cases demonstrate why selecting the endpoint rule matters as much as finding the roots.
Exact endpoints, approximations and supported limits
Integers, decimals with up to six places and simple fractions such as −2/3 are represented using exact rational arithmetic. Each coefficient must be within −10,000 to 10,000; fraction numerators and denominators must also be within that range, with a nonzero denominator. Inputs are limited to 30 characters. Do not enter scientific notation, commas or a full expression into a coefficient field.
The discriminant and its sign are determined exactly. Rational roots are displayed as reduced fractions. Irrational roots keep exact radical expressions in the solution interval, alongside separate decimal approximations rounded to 12 significant digits. Use the exact endpoint, not the rounded decimal, when an endpoint is close to a proposed value. The Square Root Calculator can help you explore a positive radicand separately.
If a = 0 in Quadratic mode, the remaining linear or constant comparison is handled directly. This page does not solve cubic, rational, absolute-value, compound or multi-variable inequalities. It also does not interpret units or decide which inequality represents a real-world situation. Select a specialised tool if your problem falls outside this scope.
Reading interval notation and checking your answer
Parentheses exclude an endpoint and square brackets include it. Infinity always uses a parenthesis because it is not an attainable endpoint. The symbol ∪ joins separate intervals; ∅ indicates no real solution; (−∞, ∞) includes every real number. A singleton such as {1} represents just one solution.
For practice, pick a point well inside each proposed interval and substitute it into the original inequality. Then test a point outside the solution. For an inclusive comparison, check that each listed finite boundary gives equality. A Scientific Calculator is useful for those numerical substitutions, but an approximate test alone should not replace the sign-chart argument or exact endpoint.
Inequality Calculator FAQs
Sources and methodology
These educational references explain the mathematical ideas. This independently implemented calculator is not affiliated with their publisher. Its supported methods, exact checks and limits are documented above.