Inequality Solver

Solve linear, quadratic, and absolute value inequalities step by step. Visualize the solution set on a number line, get interval notation, and learn the logic behind each transformation.

Use standard algebraic notation (>, <, ≥, ≤)
Solving...
Solution
Your solution will appear here
Solution Set
Critical Points
Step-by-Step Solution
Graphical Representation

Examples

Try these examples to see how the inequality solver works:

Common Inequality Types
x + 3 > 5
Linear inequality
2x - 4 ≤ 6
Linear inequality with less than or equal
x² - 4 > 0
Quadratic inequality
x² - 5x + 6 ≤ 0
Quadratic inequality with interval solution
(x-1)/(x+2) > 0
Rational inequality
|x - 3| < 2
Absolute value inequality
x³ - x ≥ 0
Cubic inequality
x(x-1)(x+2) < 0
Polynomial inequality
√(x-2) > 3
Radical inequality

Understanding Inequalities: A Complete Guide

An inequality is a mathematical statement that relates two expressions using symbols such as <, >, <=, or >=. Solving inequalities means finding all values of the variable that make the inequality true. Unlike equations, solutions are often intervals or unions of intervals on the real number line. Our solver handles three fundamental types: linear, quadratic, and absolute value inequalities.

The solution set can be expressed in interval notation (e.g., (a, b), [c, ∞)) or graphically on a number line.

How the Solver Works: Algorithmic Approach

Our inequality solver uses symbolic parsing and algebraic manipulation. It normalizes the inequality to bring all terms to one side (f(x) ⋛ 0), then identifies the type:

  • Linear: Isolates x while respecting sign reversal when multiplying/dividing by negative numbers.
  • Quadratic: Finds roots via the quadratic formula, then performs sign analysis based on the leading coefficient.
  • Absolute value: Uses definition |A| ⋛ c and splits into compound inequalities (|A| <= c -> -c <= A <= c; |A| >= c -> A <= -c or A >= c).

Solution intervals are displayed both in interval notation and on an interactive number line with proper open/closed endpoints.

Step-by-Step Validation: Real Examples

Inequality Type Solution (Interval) Graph Interpretation
2x - 5 <= 7 Linear (-∞, 6] All x <= 6, closed dot at 6
x^2 - 9 > 0 Quadratic (-∞, -3) ∪ (3, ∞) Two open intervals
|x + 1| < 4 Absolute (-5, 3) Open interval between -5 and 3
3 - 2x >= 7 Linear (-∞, -2] Closed dot at -2, extends left
-x^2 + 4x >= 0 Quadratic [0, 4] Closed interval between roots
Real-World Application: Profit & Cost Analysis

A manufacturing company determines profit P(x) = -2x^2 + 120x - 1000 (in thousands) where x is thousands of units. The company wants profit > 800. Solving -2x^2 + 120x - 1800 > 0 simplifies to x^2 - 60x + 900 < 0 -> (x-30)^2 < 0, which has no real solution. This tells management that profit never exceeds 800; they must adjust costs or pricing. Inequalities are critical in optimization, quality control, and engineering tolerance design.

Frequently Asked Questions

Equations give specific values (e.g., x = 3), while inequalities provide a range of values. Inequality solutions are intervals, often infinite, and require special care with negative multiplication (reverses the inequality sign).

Currently the tool focuses on polynomial (linear/quadratic) and absolute value. For rational inequalities, we recommend simplifying to polynomial numerator/denominator sign charts; future updates may include full rational support.

The graph scales dynamically based on critical points. It uses exact rational/floating calculations and correctly differentiates open vs closed endpoints.

It summarizes key algebraic manipulations: isolating variable, finding roots, splitting absolute values, or sign analysis for quadratics. This supports learning and verification.

References: Stewart, J. "Precalculus" (7e); OpenStax "Algebra and Trigonometry"; Weisstein, E.W. "Inequality" from MathWorld.