Inequality to Interval Notation Calculator

Solve a one-variable linear inequality and express the solution set in interval notation.

How the Math Works

The Inequality to Interval Notation Calculator solves linear inequalities of the form ax + b ≷ c by systematically isolating the variable x. First, the calculator subtracts b from both sides to obtain ax ≷ c - b. Then it divides both sides by a, carefully checking the sign of a: if a is positive, the inequality direction remains unchanged, but if a is negative, the inequality sign flips (for example, ≤ becomes ≥). This critical sign-flipping rule ensures the solution set remains mathematically correct. Once x is isolated, the calculator converts the resulting inequality into interval notation, representing all possible solutions as a continuous range on the real number line.

Practical Applications

To use this calculator, simply input the coefficients a, b, and c along with the inequality symbol (≷) from your linear inequality expression. The calculator processes these values through the isolation steps: first subtracting the constant term, then dividing by the coefficient while automatically checking whether sign flipping is required. For instance, entering the inequality 3x + 5 > 11 would yield the interval (2, ∞), indicating all real numbers greater than 2. The tool handles all four inequality types (>, <, ≥, ≤) and provides the precise interval notation solution instantly.

Day-to-Day Use

This calculator proves valuable in everyday scenarios involving constraints and limitations. Whether determining minimum spending requirements (spending at least $50 for a sale), calculating break-even points in small business finances, establishing study time boundaries for exam preparation, or analyzing physical constraints like weight limits and speed restrictions, the calculator transforms complex inequality problems into clear, actionable intervals. Instead of manually solving and potentially making sign-flipping errors, users can quickly verify their manual calculations or directly apply results to practical decision-making in budgeting, scheduling, and resource allocation tasks.

Worked example

2x + 1 < 9 → x < 4 → (−∞, 4).

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