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Inequality Calculator

Choose the inequality type and enter your expression (e.g., x + 1 > 3). The calculator will provide a step-by-step solution and a visual number line representation of the solution.

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Our inequality calculator allows you to solve linear, quadratic, and rational inequalities quickly and accurately. It provides step-by-step solutions and graphical representations, making complex inequalities easy to understand and visualize.

What is an Inequality in Mathematics?

An inequality is a statement that compares two expressions using symbols like greater than, less than, greater than or equal to, or less than or equal to.

Examples:

3 + 5 < 10, 2y - 4 > 6

Common Inequality Symbols:

Understanding these symbols is essential for solving inequalities:

> : Greater than
>= : Greater than or equal to
< : Less than
<= : Less than or equal to

The symbols > and < indicate strict inequalities, while >= and <= allow equality as well.

Rules for Solving Inequalities:

These rules are applied automatically by our calculator for linear, quadratic, and rational inequalities.

Rule 1:

Multiplying both sides by a negative number reverses the inequality.

If a > b and c < 0, then a·c < b·c

Rule 2:

Multiplying both sides by a positive number preserves the inequality.

If a > b and c > 0, then a·c > b·c

Rule 3:

Dividing both sides by a negative number reverses the inequality.

If a > b and c < 0, then a/c < b/c

Rule 4:

Dividing both sides by a positive number preserves the inequality.

If a > b and c > 0, then a/c > b/c

Rule 5:

Adding the same number to both sides does not change the inequality.

If a > b and c ∈ R, then a + c > b + c

Rule 6:

Subtracting the same number from both sides does not change the inequality.

If a > b and c ∈ R, then a - c > b - c

Rule 7:

Squaring both sides with positive numbers preserves the inequality.

Rule 8:

Squaring both sides with negative numbers reverses the inequality.

Rule 9:

Taking the reciprocal of non-zero values reverses the inequality.

How to Solve a Linear Inequality:

For example, to solve 5x > 10, divide both sides by 5 to get x > 2. A linear inequality calculator can also provide a graph of the solution.

Linear Inequality

How to Solve a Quadratic Inequality:

Quadratic inequalities like x² + 5x + 6 < 0 are solved by finding the roots and analyzing the intervals. A quadratic inequality calculator provides step-by-step solutions and graphs. Here, the roots are x = -2 and x = -3, and the inequality holds between the roots.

Quadratic Inequality

Solving a Rational Inequality:

Rational inequalities like (x + 2)/(x - 1) > 3 are solved using critical points and test intervals. A rational inequality calculator also provides graphical representation for clarity.

Rational Inequality