Solve The Compound Inequality Write Answer in Interval Notation Calculator

How to Use This Calculator

To solve a compound inequality using this calculator:

1. Enter your inequality in the input field. For example, you might enter -3 < x + 2 < 5.
2. Select the type of inequality (less than, greater than, or between).
3. Click the "Calculate" button to solve the inequality.
4. The calculator will display the solution in interval notation and explain the steps taken.

This tool is designed to handle both simple and complex inequalities, providing clear, step-by-step solutions.

How to Solve Compound Inequalities

Compound inequalities are inequalities that combine two or more simple inequalities with the word "and" or "or." To solve them, follow these steps:

1. Identify the type of compound inequality: Determine whether it's a conjunction (using "and") or a disjunction (using "or").
2. Solve each simple inequality separately: Treat each part of the compound inequality as a separate equation and solve for the variable.
3. Combine the solutions: For a conjunction, the solution is the intersection of the two individual solutions. For a disjunction, it's the union.
4. Express the solution in interval notation: Use parentheses for strict inequalities and brackets for inclusive inequalities.

Understanding Interval Notation

Interval notation is a shorthand way of writing subsets of real numbers. It's commonly used in algebra and calculus to describe the domain and range of functions, as well as the solution sets of inequalities.

Here are the basic symbols used in interval notation:

• (a, b): All numbers greater than a and less than b.
• [a, b]: All numbers greater than or equal to a and less than or equal to b.
• (a, b]: All numbers greater than a and less than or equal to b.
• [a, b): All numbers greater than or equal to a and less than b.
• (-∞, a): All numbers less than a.
• (a, ∞): All numbers greater than a.
• (-∞, ∞): All real numbers.

Interval notation is concise and easy to read, making it ideal for mathematical writing and communication.

Worked Examples

Example 1: Solving a Simple Compound Inequality

Solve the inequality -3 < x + 2 < 5 and express the solution in interval notation.

1. Subtract 2 from all parts of the inequality: -3 - 2 < x < 5 - 2 becomes -5 < x < 3.
2. The solution in interval notation is (-5, 3).

Example 2: Solving a Compound Inequality with "And"

Solve the inequality x > -2 and x < 4 and express the solution in interval notation.

1. The solution is all x values greater than -2 and less than 4.
2. The interval notation is (-2, 4).

Example 3: Solving a Compound Inequality with "Or"

Solve the inequality x < -1 or x > 3 and express the solution in interval notation.

1. The solution is all x values less than -1 or greater than 3.
2. The interval notation is (-∞, -1) ∪ (3, ∞).