Solve The Inequality Using Interval Notation Calculator

This calculator helps you solve mathematical inequalities and express the solutions using interval notation. Whether you're a student learning algebra or a professional needing quick reference, this tool provides step-by-step guidance and instant results.

How to Use This Calculator

Using the inequality solver is straightforward:

Enter your inequality in the input field (e.g., x² - 5x > 6)
Select the variable you're solving for (usually x)
Click "Calculate" to see the solution in interval notation
Review the step-by-step solution and graph if available

The calculator will:

Solve the inequality algebraically
Express the solution in interval notation
Provide a graphical representation when possible
Show all critical points and test intervals

Understanding Interval Notation

Interval notation is a concise way to represent sets of real numbers. The basic symbols are:

( ) - Parentheses indicate that an endpoint is not included
[ ] - Brackets indicate that an endpoint is included
(∞ - Indicates the interval extends to positive infinity
-∞) - Indicates the interval extends to negative infinity

Example

The solution to -2 < x < 5 is written as (-2, 5) in interval notation.

If the inequality were -2 ≤ x ≤ 5, the notation would be [-2, 5].

Solving Inequalities

The process of solving inequalities follows these general steps:

Move all terms to one side to set the inequality to zero
Factor the expression if possible
Find the critical points by setting each factor to zero
Test intervals between critical points to determine where the inequality holds true
Express the solution in interval notation

Key Formulas

For quadratic inequalities: ax² + bx + c > 0

Find roots: x = [-b ± √(b² - 4ac)] / (2a)
Plot critical points and test intervals
Determine where the expression is positive

Common Inequality Types

Here are some common inequality patterns and their solutions:

Inequality Type	Example	Solution in Interval Notation
Linear	3x - 5 > 10	(5, ∞)
Quadratic	x² - 4x < 4	(0, 4)
Rational	(x + 2)/(x - 3) ≥ 0	[-2, 3)
Absolute Value	\|2x - 3\| ≤ 5	[-1, 4]

Frequently Asked Questions

What is interval notation?: Interval notation is a shorthand method of describing sets of real numbers using parentheses and brackets to indicate whether endpoints are included in the set.
How do I solve inequalities with absolute value?: For inequalities like |x - a| < b, solve the compound inequality -b < x - a < b by adding a to all parts to get a - b < x < a + b.
What does it mean when an inequality has no solution?: An inequality has no solution when the resulting statement is always false, such as x > x + 5, which simplifies to 0 > 5.
How do I graph the solution to an inequality?: Graph the critical points on a number line and use open or closed circles to indicate whether endpoints are included, then shade the appropriate intervals.
Can I solve inequalities with more than one variable?: This calculator currently solves inequalities with one variable. For systems of inequalities, you would need to solve each inequality separately and find the intersection of solutions.