When does an inequality sign flip?

When you multiply or divide both sides by a negative number, and only then. Adding or subtracting anything is always safe, and multiplying by a positive is safe too. The reason is visible in plain numbers: 2 < 6 is true, and multiplying both sides by −1 gives −2 < −6, which is false — the order has reversed because negation reflects the number line.

How do I solve a quadratic inequality?

Find the roots, then work out which side of them the expression is on. The roots are the only places the sign can change, so testing one point in each region settles it. Whether the answer lies between the roots or outside them depends on the leading coefficient: positive means the parabola opens upwards and is negative between the roots, negative means it opens downwards and is negative outside them.

What if the quadratic has no real roots?

Then the expression never crosses zero, so it keeps the same sign everywhere and the answer is either every real number or nothing at all. Which one depends on the leading coefficient and the direction: x² + 1 is always positive, so x² + 1 > 0 is satisfied by every real number and x² + 1 < 0 by none. The discriminant being negative is what tells you this is the situation.

Why is |x| c two rays?

Because of what the absolute value is measuring. |x − 3| 2 says the distance exceeds 2, which happens by being far below 1 or far above 5 — the conditions are alternatives, so it is an OR and the answer is two rays that never join.

What if an absolute value is compared with a negative number?

Why can I not multiply out a rational inequality?

Because the denominator’s sign depends on x, and multiplying by a negative would reverse the direction. For an equation the move is fine; for an inequality it silently assumes the denominator is positive. Take (x+1)/(x−2) > 0: multiplying gives x + 1 > 0 and so x > −1, which includes x = 0. But at x = 0 the real value is −0.5, and the inequality fails. The assumption was wrong for every x below 2.

How does a sign chart work?

Mark every point where the expression is zero or undefined — the zeros of the numerator and of the denominator. Those are the only places the sign can change, because a continuous expression cannot pass from positive to negative without either hitting zero or breaking. Then test one point in each region between them. One test settles the whole region, and no multiplication is ever needed.

Is the point where the denominator is zero ever included?

Never, whatever the direction and whether or not the inequality is strict. At that point the expression has no value at all, so it cannot satisfy any comparison. This catches people out with ≤ and ≥, where the instinct is that endpoints are included: the zero of the numerator does get included by a non-strict inequality, but the zero of the denominator never does. On the number line it stays a hollow circle regardless.

Can I square both sides of an inequality?

Not in general, and this catches people more often than it should. From −3 1/b follows from a < b only when a and b share a sign.

What do the brackets in interval notation mean?

A round bracket excludes the endpoint and a square one includes it. So (2, 3) is everything strictly between 2 and 3, while [2, 3] includes both ends, and (2, 3] includes only the upper one. A strict inequality always produces round brackets and a non-strict one square ones, except where a point has to be excluded for another reason. Infinity always takes a round bracket, since it is not a number that can be reached.

Inequality Solver

An inequality has a set of answers rather than a value, and the shape of that set is where the difficulty lives. It can be one interval, two separate rays, a single point, every real number, or nothing at all — and a solver that always hands back an interval gets the awkward cases wrong. Four kinds...

An inequality has a set of answers, not a value. That set can be an interval, two separate rays, a single point, every real number, or nothing at all — and a solver that always hands back an interval gets the awkward cases wrong. Each answer here is checked by sampling: the set is compared against the inequality itself at hundreds of points.

THE INEQUALITY

x²+x+0

x² − 5x + 6 < 0

(2, 3)

Checked at 401 sample points across the interesting range: the set and the inequality agreed at 401 of them.

The roots are 2 and 3, and a is positive so the parabola opens upwards — it is negative between the roots and positive outside.

Piece	From	To	Endpoints
(2, 3)	2	3	open / open

WHERE IT HOLDS

The circles carry real information. A hollow circle means the endpoint itself does not satisfy the inequality; a filled one means it does. A strict inequality always gives hollow endpoints, and an excluded denominator is hollow even when the inequality is not strict.

Solution verified by sampling · live calculation

Created with❤️byeaglecalculator.com

HOW TO USE

1
Pick the type first — linear, quadratic, absolute value or rational. Each is solved by a different method, and choosing the wrong one is the fastest way to a wrong answer.
2
Enter the coefficients and the direction. A missing term is a zero and needs writing in; setting the leading coefficient to zero is allowed and the page will tell you what it degenerates to.
3
Read the reasoning under the answer. It names which rule was applied — particularly whether the direction reversed and why — rather than just presenting a result.
4
Check the endpoints on the number line. A hollow circle is excluded and a filled one included, and an excluded denominator stays hollow even when the inequality is not strict.

REFERENCE FORMULAS

Rule	Formula	What it is for
Add or subtract	a < b ⟹ a + c < b + c	Always safe. Adding the same amount to both sides never changes the direction.
Multiply by positive	a < b, c > 0 ⟹ ac < bc	Safe. A positive multiplier preserves the direction.
Multiply by negative	a < b, c < 0 ⟹ ac > bc	The direction reverses. This is the single most common mistake in the subject.
Take reciprocals	0 < a < b ⟹ 1/a > 1/b	Reverses, but only when both sides share a sign. Across zero it fails entirely.
Square both sides	a < b ⇏ a² < b²	Not valid in general: −3 < 2 but 9 > 4. Only safe when both sides are non-negative.
Quadratic, a > 0	ax² + bx + c < 0 ⟹ r₁ < x < r₂	Opens upwards, so it is negative between the roots.
Quadratic, a < 0	ax² + bx + c < 0 ⟹ x < r₁ or x > r₂	Opens downwards, so it is negative outside the roots.
No real roots	b² − 4ac < 0	The expression never crosses zero, so the answer is all reals or nothing.
Absolute, less than	\|u\| < c ⟺ −c < u < c	One interval. The two conditions hold together, so this is an AND.
Absolute, greater than	\|u\| > c ⟺ u < −c or u > c	Two rays. The conditions are alternatives, so this is an OR.
Absolute, negative side	\|u\| < c with c < 0	No solution. An absolute value is never negative. With > it is instead all reals.
Rational	P(x)/Q(x) < 0	Never multiply by Q — its sign is unknown. Use a sign chart on the zeros of P and Q.
Excluded point	Q(x) = 0	Always excluded from the answer, whether the inequality is strict or not.
Interval notation	( ) excludes, [ ] includes	A strict inequality gives round brackets; ≤ and ≥ give square ones. ∞ is always round.

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Last updated: August 3, 2026 · Linear, quadratic, absolute value and rational · Every answer checked by sampling against the inequality itself.