Inequalities compare sizes without forcing equality
In mathematics an inequality relates two expressions by a non-equal comparison, usually of size on the number line. Strict forms a < b and a > b forbid equality; they say one quantity is cleanly less or greater than the other.
Beside the strict signs sit two relaxed ones: a ≤ b says a is at most b, and a ≥ b says a is at least b, each also printed with slanted or doubled bars. The symbols have a history. In 1670 John Wallis drew his bar above the angle rather than beneath it, and in 1734 Pierre Bouguer introduced versions with two horizontal bars under the angle, which later writers trimmed to one bar or a slanted stroke. A slash through a sign negates it, giving not greater than and not less than.
The sign ≠ merely says two things differ. It claims neither is bigger and does not even need the values to come from an ordered set, though some authors still class it as strict. Engineers add ≪ and ≫ for much less and much greater, usually meaning several orders of magnitude, so the smaller quantity can be dropped from an approximation, as in the ultrarelativistic limit in physics. Mirror-image signs are interchangeable: a < b says exactly what b > a says.
Several rules govern manipulation. Inequalities are transitive, and if either link in the chain is strict, so is the conclusion. Adding or subtracting the same constant on both sides keeps the direction, which makes the reals an ordered group under addition. Multiplying or dividing by a positive number preserves the direction, but a negative multiplier flips it. For two positive numbers, or two negatives, taking reciprocals also reverses the comparison.
More generally, any increasing function can be applied to both sides safely, while a decreasing one turns the relation around; negation and reciprocals are just examples of the latter. Because the natural logarithm is strictly increasing, logging two positive quantities keeps a strict comparison strict. In the abstract, a partial order is any relation that is reflexive, antisymmetric and transitive, and a set equipped with one is called a partially ordered set.
Source: Inequality (mathematics)
