Inequalities Calculator Interval Notation

This inequalities calculator helps you solve mathematical inequalities and convert between inequality notation and interval notation. Whether you're a student studying algebra or a professional working with mathematical expressions, this tool provides a clear, step-by-step solution to your inequality problems.

Introduction

Inequalities are mathematical statements that compare two expressions using inequality symbols. The most common inequality symbols are:

< - Less than
> - Greater than
≤ - Less than or equal to
≥ - Greater than or equal to

Solving inequalities involves finding all values of the variable that make the inequality true. The solution can be expressed in inequality notation (using the inequality symbol) or in interval notation (using brackets and parentheses).

How to Use This Calculator

To use the inequalities calculator:

Enter your inequality in the input field. For example, you might enter x < 5 or 2x + 3 ≥ 7.
Select the type of notation you want to convert to (inequality or interval).
Click the "Calculate" button to see the solution.
Review the step-by-step solution and the graphical representation of the solution set.

The calculator will display the solution in the requested notation and provide a visual representation of the solution set on the number line.

Inequality Basics

When solving inequalities, you must perform the same operations on both sides of the inequality to maintain the inequality's validity. Here are the basic rules:

Addition and subtraction: If you add or subtract the same number from both sides, the inequality sign remains the same.
Multiplication and division: If you multiply or divide both sides by a positive number, the inequality sign remains the same. If you multiply or divide both sides by a negative number, you must reverse the inequality sign.

Example: Solve 3x + 2 < 11 Subtract 2 from both sides: 3x < 9 Divide both sides by 3: x < 3

Interval Notation

Interval notation is a way to represent a set of real numbers using brackets and parentheses. The most common types of intervals are:

(a, b) - Open interval, does not include a and b
[a, b] - Closed interval, includes a and b
(a, b] - Half-open interval, does not include a but includes b
[a, b) - Half-open interval, includes a but does not include b
(a, ∞) - All numbers greater than a
(-∞, b) - All numbers less than b
(-∞, ∞) - All real numbers

Interval notation is particularly useful when graphing solutions on a number line.

Conversion Examples

Here are some examples of converting between inequality notation and interval notation:

Inequality Notation	Interval Notation
x > 2	(2, ∞)
x ≤ 5	(-∞, 5]
-3 < x < 4	(-3, 4)
x ≥ -2 and x ≤ 7	[-2, 7]

Common Pitfalls

When working with inequalities, there are several common mistakes to avoid:

Forgetting to reverse the inequality sign when multiplying or dividing by a negative number.
Incorrectly interpreting interval notation, especially with open and closed brackets.
Solving compound inequalities incorrectly by not considering the intersection of the two inequalities.

Remember: When multiplying or dividing both sides of an inequality by a negative number, you must reverse the inequality sign. For example, if you have -2x > 6, dividing both sides by -2 gives x < -3.

FAQ

What is the difference between inequality notation and interval notation?

Inequality notation uses inequality symbols (<, >, ≤, ≥) to represent a range of values. Interval notation uses brackets and parentheses to represent the same range of values. For example, x > 2 is equivalent to the interval (2, ∞).

How do I solve compound inequalities?

To solve compound inequalities, solve each part of the inequality separately and then find the intersection of the two solution sets. For example, to solve -3 < x < 4 and x ≥ -1, the solution is -1 ≤ x < 4.

What does it mean when an interval is open or closed?

An open interval does not include the endpoints, while a closed interval includes the endpoints. For example, (2, 5) means all numbers greater than 2 and less than 5, but not including 2 and 5. [2, 5] means all numbers from 2 to 5, including 2 and 5.

How do I graph an inequality on a number line?

To graph an inequality on a number line, first convert the inequality to interval notation. Then, use open circles for parentheses and closed circles for brackets to indicate whether the endpoints are included. Draw a line connecting the circles to represent the solution set.