Our linear inequalities calculator solves inequalities using standard algebraic techniques. The process involves isolating the variable by performing inverse operations, with special attention to flipping the inequality sign when multiplying or dividing by negative numbers. The calculator provides complete solutions in multiple formats including interval notation and number line descriptions.

Mathematical Foundation

Linear inequalities are solved using properties of inequalities: addition, subtraction, and multiplication/division (with sign flipping for negatives). Unlike equations that have point solutions, inequalities have solution sets representing ranges of values. The solution set can be represented algebraically, in interval notation, or graphically on a number line.

• Adding/subtracting the same value from both sides preserves the inequality
• Multiplying/dividing by positive numbers preserves the inequality direction
• Multiplying/dividing by negative numbers REVERSES the inequality direction
• Solution sets are intervals: rays (half-lines) or line segments
• Inequalities with "or equal to" (≥, ≤) use closed intervals [ ]
• Strict inequalities (>, <) use open intervals ( )

Common Examples and Patterns

Applications of Linear Inequalities

Linear inequalities model real-world constraints and limitations:

• Budget Constraints: Spending ≤ Income
• Time Management: Work hours ≤ 40 hours/week
• Speed Limits: Speed ≤ 65 mph
• Minimum Requirements: Grade ≥ 60 to pass
• Capacity Limits: Passengers ≤ Seats available
• Optimization: Finding feasible regions in linear programming

Sources & References