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Calculator Answers

Inequality Calculator

Blake Boege
Written by Blake Boege · Founder, Calculator Answers

An inequality calculator is an algebraic tool that determines the set of values that satisfy a mathematical inequality. This one handles linear inequalities in a single variable, applying the rule that the sign reverses when both sides are multiplied or divided by a negative number. Quadratic, polynomial, rational and absolute-value inequalities are not supported. It outputs the solution set in inequality notation, in interval notation, and as a written number-line description; the line itself is not drawn. Algebra students use this utility to verify homework and visualize solution intervals.

Type a linear inequality in one variable. The calculator moves the variable to one side, divides by its coefficient (flipping the direction if that coefficient is negative), and reports the solution as both an inequality and an interval.

Quick Answer

Solve a one-variable linear inequality. Enter the coefficients to get the solution set, interval notation, and a written number-line description saying which endpoint is open and which way to shade.

Inequality

Linear inequality in one variable. Use <, <=, >, or >= between the two sides.

e.g. 2x + 3 < 11

Single letter. · e.g. x

What is supported

  • One-variable linear inequalities, any side mix of variable and constant terms.
  • All four comparison operators, including the ≤ and ≥ characters.
  • Automatic flip when both sides are divided by a negative coefficient.

Quadratic and compound inequalities are not supported. For nonlinear cases, solve each piece and intersect the intervals by hand.

Solution

Solution

x < 4, so x ∈ (−∞, 4)

StepMove terms to one side and divide by 2: x < 4.
Number lineopen circle at x = 4, shading to the left.

For an inequality that compares ratios, see the proportion calculator. For checking the resulting bound against a numeric test value, use the scientific calculator.

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Examples

2x + 3 < 11

x < 4

5 − x ≥ 2x − 4

x ≤ 3

−3x > 9

x < −3 (flip)

x + 1 > x

Always true

How it works

The calculator rewrites the inequality as A · x + B ◊ 0 by subtracting one side from the other, then divides both sides by A.

Solve · x ◊ −B / A (flip ◊ when A < 0)

If A = 0, the variable cancels: the inequality is always true or has no solution depending on B and the operator.

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More tools from Education

Compound Inequality CalculatorSolve compound linear inequalities of the form a ◊ mx + b ◊ c (AND) or mx + b ◊ c OR px + q ◊ r. Returns the solution interval with intersection-or-union reading.

Frequently asked questions

An inequality where both sides are linear expressions in the same variable, like 2x + 3 < 11 or 5 − x ≥ 2x − 4. Solving means describing every value of x that makes the inequality true.

Whenever you multiply or divide both sides by a negative number. For example, −2x > 6 divided by −2 becomes x < −3 (the > flipped to <). The calculator does this automatically when the coefficient of x is negative.

Both as an inequality like x < 4 and as an interval like (−∞, 4). Closed brackets like [ or ] mean the endpoint is included (≤ or ≥); open parentheses like ( or ) mean the endpoint is excluded (< or >).

When the variable cancels out, the inequality reduces to a comparison of constants. If the constants make the comparison true, every real number satisfies the inequality. If they make it false, no real number does.

Not directly. Solve each piece on its own and intersect the intervals by hand.

Decimals and negatives are fine. Fractions written with the / character are not parsed; convert to decimals first or use the fraction-to-decimal calculator.