Solve The Following Inequality Calculator

This inequality solver helps you find all possible values of x that satisfy a given inequality. Whether you're studying algebra, preparing for exams, or solving real-world problems, this tool provides step-by-step solutions with clear explanations.

How to Use This Calculator

Enter your inequality in the input field below. The calculator will solve for x and display the solution set. You can also choose to see the step-by-step solution process.

Tip: For best results, enter inequalities in standard form (e.g., 2x + 3 > 7). The calculator handles both linear and quadratic inequalities.

Inequality Basics

An inequality is a mathematical statement that compares two expressions using symbols other than the equals sign (=). The main inequality symbols are:

< - Less than
> - Greater than
≤ - Less than or equal to
≥ - Greater than or equal to

The solution to an inequality is the set of all values that make the inequality true. This set is often expressed in interval notation.

Methods for Solving Inequalities

1. Isolating the Variable

The most common method involves isolating the variable on one side of the inequality. This requires performing the same operation on both sides while maintaining the inequality's direction.

Example: Solve 3x - 5 > 10

Add 5 to both sides: 3x > 15
Divide both sides by 3: x > 5

2. Reversing the Inequality

When multiplying or dividing both sides by a negative number, the inequality symbol must be reversed.

Example: Solve -2x + 4 < 10

Subtract 4 from both sides: -2x < 6
Divide both sides by -2 (reverse inequality): x > -3

Common Types of Inequalities

Linear Inequalities

These involve linear expressions and can be solved using basic algebraic operations. Example: 2x + 3 < 7

Quadratic Inequalities

These involve quadratic expressions and often require finding critical points and testing intervals. Example: x² - 4x > 4

Absolute Value Inequalities

These involve absolute value expressions and can be solved by considering the definition of absolute value. Example: |x - 5| < 3

Graphical Method

For visual learners, you can solve inequalities graphically by plotting the corresponding equation and determining where the inequality holds true.

Note: The graphical method is particularly useful for quadratic inequalities where the solution set is often a combination of intervals.

Practical Examples

Example 1: Simple Linear Inequality

Solve: 4x - 7 > 5

Add 7 to both sides: 4x > 12
Divide by 4: x > 3

Solution: All real numbers greater than 3 (x > 3)

Example 2: Compound Inequality

Solve: -2 < 3x + 4 < 10

Subtract 4 from all parts: -6 < 3x < 6
Divide by 3: -2 < x < 2

Solution: All real numbers between -2 and 2 (-2 < x < 2)

Frequently Asked Questions

What types of inequalities can this calculator solve?

This calculator can solve linear, quadratic, and absolute value inequalities. It handles both simple and compound inequalities.

How do I enter inequalities with fractions?

Enter fractions in the form "a/b" (e.g., 1/2). The calculator will handle them appropriately in the solution process.

Can I solve inequalities with variables on both sides?

Yes, the calculator can handle inequalities with variables on both sides, though you may need to combine like terms first.

What if my inequality has no solution?

The calculator will indicate when an inequality has no solution (e.g., 2x + 3 > 2x + 5 has no solution).

How accurate are the solutions?

The calculator uses precise mathematical algorithms to ensure accurate solutions. However, always verify critical results with a teacher or calculator.