Solve The Inequality Calculator Interval Notation

How to Use This Calculator

Using our inequality solver is simple:

1. Enter your inequality in the input field. For example, you might enter x + 3 < 7 or x² - 5x + 6 > 0.
2. Select the type of inequality (linear or quadratic) from the dropdown menu.
3. Click the "Calculate" button to solve the inequality.
4. Review the solution in interval notation and the step-by-step explanation.
5. If needed, use the "Reset" button to clear the form and start over.

The calculator will display the solution in interval notation, such as (-∞, 4) or (2, ∞), and provide a detailed explanation of how the solution was derived.

What Is Interval Notation?

Interval notation is a way to represent a set of real numbers using parentheses and brackets. It's commonly used in algebra and calculus to describe the range of values that satisfy an inequality.

Here are the basic symbols used in interval notation:

• ( ) - Parentheses indicate that the endpoint is not included in the interval.
• [ ] - Brackets indicate that the endpoint is included in the interval.
• -∞ - Represents negative infinity.
• ∞ - Represents positive infinity.

For example, the interval notation (2, 5) represents all real numbers greater than 2 and less than 5, not including 2 and 5. The notation [1, 4] represents all real numbers greater than or equal to 1 and less than or equal to 4, including 1 and 4.

Solving Linear Inequalities

Linear inequalities are inequalities that can be written in the form ax + b < c, where a, b, and c are constants. To solve a linear inequality:

1. Isolate the variable term on one side of the inequality.
2. Isolate the constant term on the other side.
3. Divide or multiply both sides by the coefficient of the variable to solve for the variable.
4. Reverse the inequality sign if you multiply or divide by a negative number.

For example, to solve the inequality 3x + 2 > 11:

1. Subtract 2 from both sides: 3x > 9.
2. Divide both sides by 3: x > 3.

The solution in interval notation is (3, ∞).

Solving Quadratic Inequalities

Quadratic inequalities are inequalities that can be written in the form ax² + bx + c < 0, where a, b, and c are constants. To solve a quadratic inequality:

1. Find the roots of the corresponding quadratic equation by setting the inequality to zero and solving for x.
2. Plot the roots on a number line and divide the number line into intervals.
3. Test a point from each interval in the original inequality to determine where the inequality holds true.
4. Write the solution in interval notation based on the test results.

For example, to solve the inequality x² - 5x + 6 > 0:

1. Find the roots by solving x² - 5x + 6 = 0, which gives x = 2 and x = 3.
2. Divide the number line into three intervals: (-∞, 2), (2, 3), and (3, ∞).
3. Test a point from each interval:
   • For x = 0: 0 - 0 + 6 = 6 > 0 (true)
   • For x = 2.5: 6.25 - 12.5 + 6 = -0.25 > 0 (false)
   • For x = 4: 16 - 20 + 6 = 2 > 0 (true)
4. The inequality holds true for x < 2 and x > 3, so the solution in interval notation is (-∞, 2) ∪ (3, ∞).

Common Mistakes to Avoid

When solving inequalities, it's easy to make mistakes. Here are some common errors to watch out for:

• Forgetting to reverse the inequality sign when multiplying or dividing by a negative number. For example, if you have x/2 < 3 and you multiply both sides by 2, you should get x < 6, not x > 6.
• Incorrectly identifying the critical points when solving quadratic inequalities. Make sure to find all the roots of the corresponding quadratic equation.
• Misinterpreting interval notation. Remember that parentheses indicate that the endpoint is not included, while brackets indicate that the endpoint is included.
• Not testing all intervals when solving quadratic inequalities. Always test a point from each interval to determine where the inequality holds true.