Solve The Inequality in Terms of Intervals Calculator

This calculator helps you solve linear and quadratic inequalities and express the solutions in interval notation. Whether you're a student learning algebra or a professional needing quick reference, this tool provides clear, step-by-step solutions to help you understand the process.

How to Use This Calculator

Using our inequality solver is simple:

Enter your inequality in the input field. For example, you might enter x^2 - 5x + 6 < 0.
Select the type of inequality (linear or quadratic).
Click "Calculate" to see the solution in interval notation.
Review the detailed steps and graph (if available) to understand how the solution was derived.

The calculator will display the solution set in interval notation, such as (-2, 3), and provide a visual representation of the solution on the number line.

What Is an Inequality?

An inequality is a mathematical statement that compares two expressions using symbols other than the equal sign (=). Common inequality symbols include:

< (less than)
> (greater than)
≤ (less than or equal to)
≥ (greater than or equal to)

For example, x + 3 < 7 is an inequality that asks, "What values of x satisfy this condition?"

Solving Inequalities

Solving inequalities involves finding all values of the variable that make the inequality true. Here are the general steps:

Isolate the variable on one side of the inequality.
Perform the same operation on both sides to maintain the inequality's balance.
Consider the direction of the inequality when multiplying or dividing by negative numbers.
Express the solution in interval notation.

When solving quadratic inequalities, remember to factor the quadratic expression and identify critical points where the expression equals zero. These points divide the number line into intervals that you test to determine where the inequality holds true.

Interval Notation

Interval notation is a way to represent sets of real numbers using parentheses and brackets. Here are the common symbols:

(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.

For example, the solution to x > 2 in interval notation is (2, ∞).

Example Problems

Example 1: Linear Inequality

Solve 2x - 5 > 9.

Add 5 to both sides: 2x > 14.
Divide both sides by 2: x > 7.
Solution in interval notation: (7, ∞).

Example 2: Quadratic Inequality

Solve x^2 - 4x - 12 ≤ 0.

Factor the quadratic: (x - 6)(x + 2) ≤ 0.
Find critical points: x = -2 and x = 6.
Test intervals: The inequality holds for [-2, 6].
Solution in interval notation: [-2, 6].

Frequently Asked Questions

What types of inequalities can this calculator solve?: This calculator can solve linear and quadratic inequalities. For more complex inequalities, consult a math textbook or online resource.
How do I interpret the interval notation?: Interval notation represents the solution set of an inequality. Parentheses indicate that the endpoint is not included, while brackets indicate that the endpoint is included. For example, (3, 7) means all numbers greater than 3 and less than 7.
Can I solve inequalities with absolute value?: This calculator does not currently support absolute value inequalities. For these, you may need to solve the inequality manually or use a different tool.
Why does the calculator show different results for similar inequalities?: The calculator follows strict mathematical rules when solving inequalities. If you're getting unexpected results, double-check the inequality you entered and ensure you've selected the correct inequality type.
Is there a mobile app version of this calculator?: Currently, this calculator is available as a web application. We are working on a mobile app version that will be available soon.