Solving inequality...
Choose an inequality type, enter the coefficients, and solve it over the real numbers.
What Is an Inequality Solver?
An inequality solver finds every real value that makes a comparison true. Unlike an equation, which usually asks for isolated values, an inequality often describes a continuous interval, two separate rays, every real number, or an empty set. This linear and quadratic inequality solver reports the answer in interval notation and displays the same solution on a number line.
Enter coefficients in standard form and select one of the four comparison operators. The calculator handles strict inequalities using < or > and inclusive inequalities using ≤ or ≥. Quadratic answers include the discriminant, critical roots, and a sign chart showing exactly which regions satisfy the comparison.
How Endpoint Symbols Change an Inequality Solution
A strict inequality excludes its boundary, while an equality bar includes it. The diagram connects each symbol to its endpoint circle, shading direction, and interval notation.
Strict operators
The symbols < and > use an open circle because the boundary value is excluded.
Inclusive operators
The symbols ≤ and ≥ use a filled circle because equality includes the boundary.
Infinity endpoints
Infinity is not a reachable number, so interval notation always uses a parenthesis beside ±∞.
How to Solve Linear Inequalities
Solve a linear inequality by isolating x with the same inverse operations used in a linear equation. Add or subtract the same quantity on both sides, combine like terms, and divide by the coefficient of x. The essential extra rule is that multiplying or dividing both sides by a negative number reverses the inequality symbol.
For example, −2x + 6 ≥ 0 becomes −2x ≥ −6. Dividing by −2 reverses ≥ to ≤, giving x ≤ 3. The interval is (−∞, 3], and its number line has a filled point at 3 with shading to the left.
If the x coefficient becomes zero, the remaining comparison is either always true or always false. A true constant statement gives all real numbers; a false statement gives no solution.
How to Solve Quadratic Inequalities
Place the quadratic inequality in the form ax² + bx + c compared with zero. Find the real roots of the related equation ax² + bx + c = 0. Those roots divide the number line into regions where the quadratic is positive or negative.
When an upward-opening quadratic has two distinct roots, it is positive outside the roots and negative between them. A downward-opening quadratic has the opposite pattern. Select the regions matching the requested sign, then include roots only when the operator has an equality bar.
The discriminant b² − 4ac determines whether there are two critical roots, one repeated root, or no real root. A repeated root does not change the sign. With no real roots, the quadratic keeps the sign of its leading coefficient across the entire real line.
How to Write Interval Notation
Interval notation lists endpoints from left to right. Parentheses indicate excluded endpoints, while brackets indicate included finite endpoints. A union symbol joins separate solution regions, such as (−∞, 2] ∪ [3, ∞).
All real numbers are written (−∞, ∞), and no solution is written ∅. A single included root is most clearly displayed as a set such as 2. Infinity always receives a parenthesis because it represents an unbounded direction rather than a value that can be included.
| Symbol | Meaning | Endpoint | Example interval |
|---|---|---|---|
| < | Less than | Open | (-∞, 3) |
| ≤ | Less than or equal | Closed | (-∞, 3] |
| > | Greater than | Open | (3, ∞) |
| ≥ | Greater than or equal | Closed | [3, ∞) |
Inequality Solver FAQ
When do you flip an inequality sign?
Reverse an inequality sign only when multiplying or dividing both sides by a negative number. Adding or subtracting a negative number does not by itself reverse the comparison.
How do you solve a quadratic inequality?
Find the real roots of the related quadratic equation, then use them to split the number line into sign regions. Select the regions where the polynomial has the requested sign and include roots only with ≤ or ≥.
What is interval notation for inequalities?
Interval notation is a compact way to describe continuous ranges of real numbers. Parentheses exclude an endpoint, brackets include a finite endpoint, and a union joins disjoint ranges.
What do open and closed circles mean on a number line?
An open circle means the boundary is not part of the solution, as with < or >. A closed circle means the boundary is included, as with ≤ or ≥.
Can a quadratic inequality have no solution?
Yes. For example, x² + 1 < 0 has no real solution because x² + 1 is always positive. The calculator displays this outcome as the empty set ∅.
Can an inequality solution be all real numbers?
Yes. The inequality x² + 1 > 0 is true for every real x. Constant inequalities and repeated-root quadratics can also produce an all-real solution.
How does the discriminant affect a quadratic inequality?
A positive discriminant creates two real boundaries, zero creates one repeated boundary, and a negative value creates no real boundary. This determines how many sign regions must be considered.
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