Inequality Calculator

How to Solve a Linear Inequality

This inequality calculator solves linear and quadratic inequalities — and a linear inequality has the form ax + b ≤ c(or with <, >, ≥). To solve it, isolate x using the same algebraic steps as a linear equation: subtract b from both sides, then divide by a. The key rule that sets inequalities apart from equations: if you divide or multiply both sides by a negative number, you must reverse the inequality sign. For example, −2x < 6 becomes x > −3 after dividing by −2. The calculator above handles this automatically — when a is negative, the sign is flipped and the step-by-step panel notes the flip.

Writing the Solution in Interval Notation

Once you solve for x, express the solution as an interval. A strict inequality (< or >) uses a round parenthesis to exclude the endpoint; a non-strict inequality (≤ or ≥) uses a square bracket to include it. Infinity always takes a parenthesis. Examples:

• x < 3 → (−∞, 3)
• x ≤ 3 → (−∞, 3]
• x > −2 → (−2, +∞)
• x ≥ −2 → [−2, +∞)

How to Solve a Quadratic Inequality

A quadratic inequality has the form ax² + bx + c < 0(or ≤, >, ≥). The solution process has three steps. First, find the roots of ax² + bx + c = 0 using the quadratic formula calculator. Second, the two roots split the number line into three regions — pick a test point in each region and check the sign of the expression. Third, identify which regions satisfy the inequality and express the result as a union of intervals.

The Role of the Discriminant

The discriminant (Δ = b² − 4ac) determines how many real roots exist:

• Δ > 0 — two distinct real roots; full sign analysis applies
• Δ = 0 — one repeated root; the parabola touches the axis at one point
• Δ < 0 — no real roots; the expression is always positive (a > 0) or always negative (a < 0)

Interval Notation Reference

Interval notation is the standard way to describe solution sets for inequalities. Every endpoint uses either a parenthesis (open, excluded) or a bracket (closed, included). Compound solutions — such as those from quadratic inequalities with two roots — use the union symbol ∪ to join two intervals. For instance, x < −3 or x > 5 is written (−∞, −3) ∪ (5, +∞). The empty set (no solution) is written ∅, and all real numbers is written (−∞, +∞).

Special Cases

• a = 0 in a linear inequality — reduces to a comparison between two constants. True for all x or false for all x.
• Repeated root (Δ = 0) with ≤ — a single point, e.g. {2}
• Repeated root (Δ = 0) with < — and a > 0: no solution (expression is always ≥ 0)
• Repeated root (Δ = 0) with > — and a > 0: x ≠ root, which is (−∞, root) ∪ (root, +∞)

Quadratic Inequality Examples

Use these worked examples alongside the calculator to verify your understanding. For more work on the underlying equation, see the polynomial calculator.

• x² − x − 6 < 0— roots are x = −2 and x = 3; a > 0 so negative between roots → solution: (−2, 3)
• x² − 4x + 4 ≤ 0— discriminant = 0, repeated root x = 2; solution: {2}
• x² + 1 < 0— discriminant < 0, a > 0 so always positive → no solution (∅)
• −x² + 4 > 0— roots x = −2 and x = 2; a < 0 so positive between roots → solution: (−2, 2)

Why Does the Sign Flip When Dividing by a Negative?

When you multiply or divide both sides of an inequality by a negative number, the relative order of the two sides reverses. Consider 2 < 4. Multiply both sides by −1: −2 and −4. On the number line, −2 is to the right of −4, so −2 > −4. The original < became >. This rule holds for all inequalities — any negative divisor or multiplier flips the direction. Equations do not have this issue because equality is symmetric and does not depend on order.

Graphing Inequalities on a Number Line

A number line graph is the visual complement to interval notation. For x < 3, draw an open circle at 3 and shade left to −∞. For −2 ≤ x ≤ 5, draw a closed circle at −2, a closed circle at 5, and shade the segment between them. Open circles correspond to parentheses in interval notation; closed (filled) circles correspond to brackets. Union solutions — like (−∞, −1) ∪ (4, +∞) — appear as two separate shaded rays on the number line.