In mathematics, an inequation is a statement that two objects or expressions are not the same, or do not represent the same value. This relation is written with a crossed-out equal sign, like

x ≠ y.

(In programming languages and electronic communications, the notations x != y and x <> y are used instead.)

Inequations should not be confused with mathematical inequalities, which express numerical relations such as 3 < 5 ('3 is less than 5'). In a linearly ordered set, any inequation implies an inequality: if $x\; \backslash neq\; y$, then or $x\; >\; y$ by the trichotomy law.

Properties

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Some useful properties of inequations in algebra are:

1. Any quantity can be added to both sides.
2. Any quantity can be subtracted from both sides.
3. Any nonzero quantity can be multiplied to both sides.
4. Any nonzero quantity can divide both sides.
5. Generally, any injective function can be applied to both sides.

Property (5) is somewhat of a tautology, since injective functions may be defined as functions that always preserve inequations.

If a function that is not injective is applied to both sides of an inequation, the resulting statement may be false. For an extreme example, if f is a constant function, such as multiplication by zero, then the statement "f(x)≠f(y)" is always false. This consideration explains why one must use a nonzero quantity in property (3) above.

See also

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• Equation

Elementary algebra

Ungleichung | 부등식 | Ongelijkheid | 不等式 | Nierówność | Olikhet | Bất phương trình | 不等