Inequality Calculator

- May 10, 2025
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This inequalities calculator can solve linear, quadratic, and rational inequalities, display step-by-step solutions, and visualise results on a number line. Select the inequality type, enter your expression, and click "Solve" to get started.

Inequality Type

Examples: 2x + 3 > 7, 4x - 2 ≤ 10, -3x + 4 ≥ -8, 5 < x + 2

Examples: x² - 4x + 3 > 0, x² - 9 ≤ 0, 2x² + 5x - 3 ≥ 0, x² + 6x + 9 < 0

Examples: (x - 1)/(x + 2) > 0, (x² + 1)/x ≥ 3, 1/(x - 3) < 2, (x + 4)/(x² - 4) ≤ 0

Examples for 2D systems: 2x + y > 5, x - y ≤ 3, y ≥ 2x, x + 2y < 6

Display Options

Understanding Inequalities

Inequalities express relationships between quantities where one is greater than (>), less than (<), greater than or equal to (≥), or less than or equal to (≤) another. Solving inequalities involves finding all values that make the inequality true.

Rules for Solving Inequalities
  • Addition/Subtraction: Adding or subtracting the same value from both sides preserves the inequality.
  • Multiplication/Division: Multiplying or dividing both sides by a positive number preserves the inequality. Multiplying or dividing by a negative number reverses the inequality sign.
  • Quadratic Inequalities: Typically solved by finding critical points, analysing sign behaviour, and using intervals.
  • Rational Inequalities: Require finding values where the expression equals zero or is undefined, then testing intervals.
Notation
  • Interval Notation: (a, b) represents a < x < b, [a, b] represents a ≤ x ≤ b, and combinations like [a, b) or (a, b] represent mixed inequality boundaries.
  • Set Builder Notation: {x | x > a} represents "the set of all x such that x is greater than a".
  • Number Line: Solid dots (●) represent included endpoints (≤ or ≥), while hollow dots (○) represent excluded endpoints (< or >).