Inequality Solver

About Inequality Solver

Welcome to our Inequality Solver, a comprehensive online tool designed to help students, teachers, and mathematics enthusiasts solve linear, quadratic, polynomial, and rational inequalities with detailed step-by-step solutions. Our calculator provides visual representations on a number line and displays results in interval notation, making it easy to understand and verify your solutions.

Key Features of Our Inequality Solver

• Multiple Inequality Types: Solve linear, quadratic, polynomial, and rational inequalities
• Visual Number Line: See your solution graphically represented on an interactive number line
• Interval Notation: Results displayed in standard mathematical interval notation
• Step-by-Step Solutions: Understand each step involved in solving the inequality
• Critical Points Analysis: Identify zeros and discontinuities automatically
• Automatic Type Detection: The calculator identifies whether your inequality is linear, quadratic, polynomial, or rational
• Factored Forms: See factored representations when applicable
• Educational Insights: Learn mathematical principles through detailed explanations
• LaTeX-Formatted Output: Beautiful mathematical rendering using MathJax

What is an Inequality?

An inequality is a mathematical statement that compares two expressions using inequality symbols. Unlike equations that use equals signs, inequalities use symbols like greater than, less than, greater than or equal to, or less than or equal to. The solution to an inequality is typically a range or set of values rather than a single number.

Types of Inequalities Supported

1. Linear Inequalities

Inequalities of the form $ax + b < 0$ where $a$ and $b$ are constants.

Example: $2x - 5 > 3$ or $-3x + 7 \le 1$

2. Quadratic Inequalities

Inequalities involving a quadratic expression of the form $ax^2 + bx + c < 0$.

Example: $x^2 - 5x + 6 > 0$ or $-x^2 + 4x - 3 \le 0$

3. Polynomial Inequalities

Inequalities involving polynomial expressions of degree 3 or higher.

Example: $x^3 - 4x > 0$ or $x^4 - 5x^2 + 4 \le 0$

4. Rational Inequalities

Inequalities involving rational expressions (fractions with polynomials).

Example: $\frac{x+2}{x-1} > 0$ or $\frac{x^2-4}{x^2+1} \le 1$

How to Use the Inequality Solver

1. Enter Your Inequality: Type your inequality in the input field. You can use:
   • Variables: x, y, z, etc. (single variable only)
   • Operators: +, -, *, / for arithmetic
   • Inequality symbols: <, >, <=, >=, !=
   • Exponents: ^ or ** (e.g., x^2 or x**3)
   • Parentheses: ( ) for grouping
2. Click Solve: Process your inequality and view the results.
3. Review Step-by-Step Solution: Learn from detailed explanations of each solving step.
4. View Number Line: See the solution visualized on a number line with critical points marked.
5. Check Interval Notation: Read your solution in standard interval notation.

Inequality Input Guidelines

For best results, follow these input conventions:

• Inequality Symbols: Use < for less than, > for greater than, <= for less than or equal to, >= for greater than or equal to
• Multiplication: Use * or simply write variables together (e.g., 2*x or 2x)
• Exponents: Use ^ or ** (e.g., x^2 or x**3)
• Parentheses: Use parentheses for grouping (e.g., (x+1)/(x-1) > 0)
• Single Variable: The calculator works with single-variable inequalities only

Understanding Inequality Solutions

Number Line Representation

The number line shows:

• Filled circles (●): Points included in the solution (for ≤ or ≥)
• Open circles (○): Points excluded from the solution (for < or >)
• Orange open circles: Discontinuities where the expression is undefined
• Green shaded regions: Intervals where the inequality is satisfied

Interval Notation

Solutions are expressed using interval notation:

• (a, b): All numbers between $a$ and $b$, excluding endpoints
• [a, b]: All numbers between $a$ and $b$, including endpoints
• (a, b]: All numbers between $a$ and $b$, excluding $a$ but including $b$
• (-∞, a): All numbers less than $a$
• (a, ∞): All numbers greater than $a$
• ∪: Union symbol, combines multiple intervals

Methods for Solving Inequalities

For Linear Inequalities

1. Isolate the variable on one side
2. Perform the same operations on both sides
3. Reverse the inequality sign when multiplying or dividing by a negative number
4. Express the solution in interval notation

For Quadratic and Polynomial Inequalities

1. Move all terms to one side (set to zero on the other)
2. Factor the polynomial if possible
3. Find critical points (zeros of the polynomial)
4. Test intervals between critical points
5. Determine which intervals satisfy the inequality

For Rational Inequalities

1. Move all terms to one side
2. Combine into a single fraction
3. Find zeros of the numerator (included in solution for ≤ or ≥)
4. Find zeros of the denominator (always excluded - discontinuities)
5. Test intervals between critical points
6. Determine which intervals satisfy the inequality

Applications of Inequalities

Inequalities are fundamental in mathematics and have numerous real-world applications:

• Economics: Profit and loss analysis, budget constraints, optimization problems
• Physics: Velocity ranges, acceleration bounds, energy constraints
• Engineering: Safety margins, tolerance specifications, design constraints
• Statistics: Confidence intervals, hypothesis testing, probability ranges
• Computer Science: Algorithm complexity, resource allocation, optimization
• Business: Break-even analysis, pricing strategies, capacity planning
• Chemistry: Reaction rate conditions, concentration ranges, pH levels

Common Mistakes to Avoid

• Not Reversing the Inequality: When multiplying or dividing both sides by a negative number, you must reverse the inequality sign
• Forgetting Domain Restrictions: For rational inequalities, points where the denominator equals zero must be excluded
• Incorrect Test Points: When testing intervals, choose points that actually lie within each interval
• Misinterpreting Interval Notation: Remember that parentheses ( ) exclude endpoints while brackets [ ] include them
• Combining Inequalities Incorrectly: You cannot perform the same operations on compound inequalities as you would with equations

Why Choose Our Inequality Solver?

Solving inequalities can be challenging, especially for complex polynomial and rational expressions. Our calculator offers:

• Accuracy: Powered by SymPy, a robust symbolic mathematics library
• Visual Learning: Number line representations make solutions intuitive
• Comprehensive Solutions: Step-by-step explanations for every type of inequality
• Educational Value: Learn mathematical concepts while solving problems
• Speed: Instant results for even complex inequalities
• Versatility: Handles linear, quadratic, polynomial, and rational inequalities
• Free Access: No registration or payment required

Tips for Working with Inequalities

• Always move all terms to one side before solving
• Factor expressions when possible to identify critical points easily
• Remember to check for domain restrictions in rational inequalities
• Use test points to verify which intervals satisfy the inequality
• Draw a number line to visualize the solution
• Double-check whether endpoints should be included or excluded
• Verify your solution by substituting test values back into the original inequality

Additional Resources

To deepen your understanding of inequalities and algebra, explore these resources:

• Inequality (Mathematics) - Wikipedia
• Inequalities - Khan Academy
• Inequality - Wolfram MathWorld
• Polynomial Inequalities - Paul's Online Math Notes