Inequality Solver
Free online inequality solver for linear, quadratic, polynomial, and rational inequalities with step-by-step solutions, number line visualization, and interval notation.
Free Online Inequality Solver
Our Inequality Solver helps you solve linear, quadratic, polynomial, and rational inequalities with step-by-step solutions. Whether you are studying algebra, preparing for exams, or working on advanced mathematics, this tool provides instant results with interval notation and number line visualization.
What is an Inequality?
An inequality is a mathematical statement that compares two expressions using inequality symbols. Unlike equations that use an equals sign, inequalities show that one expression is less than, greater than, less than or equal to, or greater than or equal to another. The solution to an inequality is typically a range or set of values rather than a single number.
Types of Inequalities Supported
Linear Inequalities
Inequalities of the form $ax + b < 0$ where $a$ and $b$ are constants. Example: $2x - 5 > 3$ or $-3x + 7 \le 1$.
Quadratic Inequalities
Inequalities involving a quadratic expression of the form $ax^2 + bx + c < 0$. Example: $x^2 - 5x + 6 > 0$ or $-x^2 + 4x - 3 \le 0$.
Polynomial Inequalities
Inequalities involving polynomial expressions of degree 3 or higher. Example: $x^3 - 4x > 0$ or $x^4 - 5x^2 + 4 \le 0$.
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
- Enter your inequality in the input field. Use standard mathematical notation with operators like
<,>,<=,>=. - Use exponents with the
^symbol (e.g.,x^2for $x^2$). - Use parentheses for grouping, especially in rational inequalities (e.g.,
(x+2)/(x-1) > 0). - Click an example or type your own inequality to see instant results.
- Review the solution displayed in interval notation with a number line visualization and detailed step-by-step breakdown.
Methods for Solving Inequalities
Linear Inequalities
Isolate the variable on one side by performing the same operations on both sides. Remember to reverse the inequality sign when multiplying or dividing by a negative number.
Quadratic and Polynomial Inequalities
Move all terms to one side, factor the polynomial if possible, find critical points (zeros), test intervals between critical points, and determine which intervals satisfy the inequality.
Rational Inequalities
Move all terms to one side, combine into a single fraction, find zeros of the numerator (included in solution for ≤ or ≥) and denominator (always excluded as discontinuities), test intervals, and determine which intervals satisfy the inequality.
Understanding the Solution
The solution is displayed in interval notation on a number line:
- Filled circles (●) indicate points included in the solution (for ≤ or ≥).
- Open circles (○) indicate points excluded from the solution (for < or >).
- Green shaded regions show intervals where the inequality is satisfied.
Common Mistakes to Avoid
- Not reversing the inequality when multiplying or dividing by a negative number.
- Forgetting domain restrictions - points where the denominator equals zero must be excluded.
- Incorrectly combining intervals - use the union symbol (∪) for disconnected solution sets.
Frequently Asked Questions
What types of inequalities can this solver handle?
This solver handles linear inequalities (e.g., $2x - 5 > 3$), quadratic inequalities (e.g., $x^2 - 5x + 6 > 0$), polynomial inequalities of degree 3 or higher (e.g., $x^3 - 4x < 0$), and rational inequalities (e.g., $\frac{x+2}{x-1} > 0$).
How do I enter exponents in the inequality solver?
Use the caret symbol (^) for exponents. For example, type x^2 for $x^2$ or x^3 for $x^3$. Parentheses work automatically, so (x+1)^2 is also supported.
What does interval notation mean?
Interval notation is a way to describe sets of real numbers. Parentheses ( ) exclude endpoints while brackets [ ] include them. For example, $(2, 5)$ means all numbers between 2 and 5 (exclusive), while $[2, 5]$ includes 2 and 5. The symbol $\cup$ represents the union of two intervals, and $\infty$ (infinity) represents an unbounded direction.
What happens when there is no solution?
When no real number satisfies the inequality, the solver displays the empty set symbol $\emptyset$, meaning there is no solution. For example, $x^2 < 0$ has no real solution because a square can never be negative.
What if every real number satisfies the inequality?
When all real numbers satisfy the inequality, the solver displays "All real numbers" or $(-\infty, \infty)$ in interval notation. For example, $x^2 > 0$ is true for all real numbers except $x = 0$, so the solution would be $(-\infty, 0) \cup (0, \infty)$.
Can this solver handle rational inequalities with variables in the denominator?
Yes, the solver handles rational inequalities where the variable appears in the denominator. The solver automatically identifies discontinuities (points where the denominator equals zero) and excludes them from the solution set.
Why do I need to reverse the inequality sign when multiplying by a negative number?
Multiplying or dividing both sides of an inequality by a negative number reverses the order relationship. For example, $2 < 5$ is true, but multiplying both sides by $-1$ gives $-2 > -5$. This fundamental property of inequalities must be applied to maintain a correct solution.
For more math tools, try the Absolute Value Inequality Solver, Absolute Value Equation Solver.
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