Much-less-than sign

In mathematics, an inequality is a relation that holds between two values when they are different (see also: equality).

If the values in question are elements of an ordered set, such as the integers or the real numbers, they can be compared in size.

In contrast to strict inequalities, there are two types of inequality relations that are not strict:

In engineering sciences, a less formal use of the notation is to state that one quantity is "much greater" than another, normally by several orders of magnitude.

Inequalities are governed by the following properties. All of these properties also hold if all of the non-strict inequalities (≤ and ≥) are replaced by their corresponding strict inequalities (< and >) and (in the case of applying a function) monotonic functions are limited to strictly monotonic functions.

The transitive property of inequality states:

The relations ≤ and ≥ are each other's converse:

A common constant c may be added to or subtracted from both sides of an inequality:

i.e., the real numbers are an ordered group under addition.

The properties that deal with multiplication and division state:

More generally, this applies for an ordered field, see below.

The properties for the additive inverse state:

The properties for the multiplicative inverse state:

These can also be written in chained notation as:

Any monotonically increasing function may be applied to both sides of an inequality (provided they are in the domain of that function) and it will still hold. Applying a monotonically decreasing function to both sides of an inequality means the opposite inequality now holds. The rules for the additive inverse, and the multiplicative inverse for positive numbers, are both examples of applying a monotonically decreasing function.

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