Mathematics
Solving inequalities
Enter the coefficients and choose the direction: you get the solution as intervals, with the discriminant and roots it follows from. Leave a = 0 and it becomes a linear inequality.
How an inequality is solved
A linear inequality is solved like an equation, with one difference that changes everything: when you multiply or divide by a negative number, the direction flips. From −2x + 4 > 0 you get −2x > −4 and then, dividing by −2, x < 2 and not x > 2. It is the rule most often forgotten, and the wrong answer is perfectly symmetric to the right one, so nothing looks amiss.
In the quadratic case the logic changes: you do not isolate the unknown, you study the sign of a parabola. Find the roots of the associated equation and then reason about the shape of the graph. With a positive the parabola opens upwards, so it is positive outside the interval between the roots and negative inside it. With a negative everything flips. The sentence to remember is that the trinomial has the sign of a for values outside the roots.
The special cases follow from the same reasoning. If the discriminant is negative the parabola never meets the axis, so it keeps the sign of a everywhere: the inequality is satisfied either by every real number or by none. If the discriminant is zero the parabola grazes the axis at one point, and that point is the only place the trinomial vanishes — which is why it matters only in non-strict inequalities.
Common mistakes
- Not flipping the direction when dividing by a negative coefficient: the classic linear error, and it produces exactly the opposite interval.
- Applying the linear method to a quadratic by isolating x: a quadratic inequality is solved by studying signs, not by moving terms.
- Forgetting the negative-discriminant cases: x² + 1 > 0 holds for every real x, and x² + 1 < 0 for none. Having no roots does not mean having no solution.
Frequently asked questions
When does the direction of an inequality flip?
Every time you multiply or divide both sides by a negative number. Adding or subtracting never flips it, and multiplying by a positive number does not either.
How do you solve a quadratic inequality?
Find the roots of the associated equation and study the sign of the trinomial. It has the same sign as a for values outside the interval between the roots, and the opposite sign inside.
What happens when the discriminant is negative?
The trinomial never vanishes and keeps the sign of the coefficient a everywhere. The inequality is therefore satisfied by all real numbers or by none, depending on the direction asked for.
How is the solution written in interval notation?
A round bracket excludes the endpoint, a square bracket includes it. A solution such as x < −1 or x > 3 is written (−∞; −1) ∪ (3; +∞), where ∪ marks the union of the two intervals.
How this calculation works
Linear, bx + c > 0: the solution is x > −c/b if b > 0, and x < −c/b if b < 0 — this is where the direction flips. With b = 0 the inequality holds either always or never, depending on the sign of c. Quadratic, ax² + bx + c: compute Δ = b² − 4ac. If Δ > 0 the roots are x = (−b ± √Δ)/(2a) and the trinomial has the sign of a outside the interval between them and the opposite sign inside. If Δ = 0 there is a double root and the trinomial has the sign of a everywhere except at that point, where it is zero. If Δ < 0 the trinomial has the sign of a for every x. Non-strict inequalities include the roots, strict ones exclude them.