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Linear Inequality Calculator

This linear inequality calculator helps you solve, simplify, and understand inequalities involving addition, subtraction, multiplication, or division. Ideal for students, teachers, and anyone studying algebra, it shows step-by-step solutions while also explaining how inequality rules change based on the operations and values involved.

Linear Inequalities Calculator

Input Fields
If enabled, the result will update automatically when you change any value.

Rules of Linear Inequality Operations

Formula
\[\text{If } X > Y:\quad X + Z > Y + Z \quad X – Z > Y – Z \]\[\quad \text{If } Z > 0, \quad X \cdot Z > Y \cdot Z \]\[\quad \text{If } Z < 0, \quad X \cdot Z < Y \cdot Z \]\[\quad \text{If } Z > 0, \quad \frac{X}{Z} > \frac{Y}{Z} \]\[\quad \text{If } Z < 0, \quad \frac{X}{Z} < \frac{Y}{Z}\]
Explanation

  • Addition/Subtraction: The inequality remains the same.
  • Multiplication/Division: If you multiply or divide by a positive number, the inequality stays the same. If it’s a negative number, the inequality sign flips.

Linear inequalities describe the relationship between two expressions using inequality signs: $$>$$, $$<$$, $$≥$$, or $$≤$$. These are foundational in algebra and used in solving real-world problems such as budgeting, optimization, and constraints in programming. Solving them involves applying the same operations to both sides, keeping track of whether the direction of the inequality needs to be reversed—especially when multiplying or dividing by a negative number.

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