Enter any linear, quadratic, polynomial, or rational inequality — such as 2x+3>7 or x²-4≤0 — and the Inequality Solver will return the solution set, interval notation, and a number line visualization. Choose your inequality type from the dropdown, type your expression into the inequality input field, and get a fully worked solution with all critical steps shown.
Solution Set
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Interval Notation
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Critical Point(s)
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Solution Type
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An inequality is a mathematical statement that compares two expressions using symbols like > (greater than), < (less than), ≥ (greater than or equal to), or ≤ (less than or equal to). Unlike equations, inequalities typically have a range of solutions rather than a single value.
This solver handles four main types: linear inequalities (e.g. 2x + 3 > 7), quadratic inequalities (e.g. x² - 4 ≤ 0), absolute value inequalities (e.g. |2x - 1| < 5), and rational inequalities (e.g. (x+1)/(x-2) > 0). Each type uses a different solution strategy.
When you multiply or divide both sides of an inequality by a negative number, you must flip the inequality sign. For example, -2x > 6 becomes x < -3 after dividing by -2. This is one of the most common sources of error when solving inequalities manually.
The symbol < means 'strictly less than', so the boundary value is NOT included in the solution set. The symbol ≤ means 'less than or equal to', so the boundary value IS included. In interval notation, strict inequalities use parentheses ( ) while ≤ or ≥ use square brackets [ ].
Yes — most inequalities have infinitely many solutions expressed as a range or interval. Some inequalities (like x² < -1) have no real solution because no real number satisfies the condition. Others (like x² ≥ 0) are true for all real numbers, giving a solution of all reals.
Interval notation describes a range of values. For example, (2, ∞) means all numbers greater than 2 (not including 2), while [2, ∞) means all numbers greater than or equal to 2. A union symbol ∪ is used when the solution consists of two separate intervals, such as (-∞, -2) ∪ (2, ∞).
To solve a quadratic inequality like x² - 4 ≤ 0, first find the critical points by solving x² - 4 = 0, giving x = ±2. Then test each interval between and beyond these points to determine which regions satisfy the inequality. For this example, the solution is [-2, 2].
Critical points are the values of x where the inequality expression equals zero or is undefined. They divide the number line into intervals, and you test each interval to see whether the inequality holds. For rational inequalities, values that make the denominator zero are also critical points.