Solve The Compound Inequality and Write in Interval Notation Calculator

This calculator helps you solve compound inequalities and express the solution in interval notation. Whether you're a student studying algebra or a professional needing to verify your work, this tool provides a clear, step-by-step solution.

How to Use This Calculator

Enter your compound inequality in the input field. The calculator will:

Parse the inequality to identify the variables and operators
Solve each part of the compound inequality separately
Combine the solutions according to the logical operator (AND or OR)
Express the final solution in interval notation

The calculator handles inequalities with variables on both sides, multiple operators, and different types of inequalities (>, <, ≥, ≤).

What Is a Compound Inequality?

A compound inequality is a mathematical statement that combines two or more inequalities with logical operators (AND or OR). For example:

-1 < x + 3 < 5

This is a compound inequality with an AND operator between the two inequalities.

Compound inequalities are commonly used in algebra to express ranges of values that satisfy multiple conditions simultaneously.

How to Solve Compound Inequalities

To solve a compound inequality:

Identify the logical operator (AND or OR)
Solve each inequality separately
Combine the solutions based on the operator:
- For AND: The solution is the intersection of both individual solutions
- For OR: The solution is the union of both individual solutions
Express the final solution in interval notation

Remember that when solving inequalities, you must perform the same operation on all parts of the inequality to maintain the inequality's direction.

Interval Notation Explained

Interval notation is a way to represent a set of real numbers using parentheses and brackets:

(a, b) - All numbers between a and b, not including a and b
[a, b] - All numbers between a and b, including a and b
(a, b] - All numbers between a and b, not including a but including b
[a, b) - All numbers between a and b, including a but not including b
(-∞, a) - All numbers less than a
(a, ∞) - All numbers greater than a
(-∞, ∞) - All real numbers

For compound inequalities, the solution is typically expressed as a union of intervals when using OR, or as a single interval when using AND.

Worked Examples

Example 1: AND Compound Inequality

Solve: -1 < x + 3 < 5

Subtract 3 from all parts: -1 - 3 < x < 5 - 3
Simplify: -4 < x < 2
In interval notation: (-4, 2)

Example 2: OR Compound Inequality

Solve: x < -2 or x > 3

First part: x < -2 → (-∞, -2)
Second part: x > 3 → (3, ∞)
Combined solution: (-∞, -2) ∪ (3, ∞)

Frequently Asked Questions

What is the difference between AND and OR in compound inequalities?

AND means both conditions must be true simultaneously, while OR means either condition can be true. The solution for AND is the intersection of both individual solutions, and for OR it's the union.

How do I handle inequalities with variables on both sides?

First, isolate the variable on one side by performing the same operation on all parts of the inequality. Then solve as a simple inequality.

What if the inequality has a variable in the denominator?

You must consider the denominator's sign when solving. The denominator cannot be zero, and the inequality sign may change depending on whether the denominator is positive or negative.

How do I express no solution in interval notation?

Use the empty set symbol: ∅ or {}.