Inequality Calculator
Solve inequality problems step-by-step with our free calculator. See formulas, worked examples, and clear explanations.
Reviewed by Manoj Kumar, Mathematics Educator
Formula
ax + b < cx + d (solve for x)
Enter the coefficients and constants for both sides of the inequality. The calculator moves all variable terms to one side and constants to the other, then divides by the coefficient. If dividing by a negative number, the inequality sign is automatically flipped.
Worked Examples
Example 1: Basic Linear Inequality
Problem:Solve the inequality: 3x - 7 < 5x + 2
Solution:Move all x terms to left side: 3x - 5x < 2 + 7\nSimplify: -2x < 9\nDivide by -2 (flip the sign): x > -4.5\nInterval notation: (-4.5, Infinity)\nTest: x = 0 gives 3(0) - 7 = -7 < 5(0) + 2 = 2, which is true.
Result:x > -4.5 or (-4.5, Infinity)
Example 2: Inequality with Equal Coefficients
Problem:Solve: 4x + 3 <= 4x + 8
Solution:Subtract 4x from both sides: 3 <= 8\nThis is always true.\nThe x terms cancel out, leaving a true numerical statement.\nTherefore the solution is all real numbers.\nInterval notation: (-Infinity, Infinity)
Result:All real numbers: (-Infinity, Infinity)
Frequently Asked Questions
What is an inequality and how does it differ from an equation?
An inequality is a mathematical statement that compares two expressions using inequality symbols such as less than, greater than, less than or equal to, or greater than or equal to. Unlike equations which have specific solutions, inequalities typically have a range of solutions. For example, x > 3 means any value greater than 3 satisfies the inequality, giving infinitely many solutions. Equations use the equals sign and find exact values, while inequalities define regions on the number line. Understanding this distinction is crucial because many real-world problems involve constraints and ranges rather than exact values.
References
Reviewed by Manoj Kumar, Mathematics Educator · Editorial policy