Inequation

In mathematics, an inequation is a statement that two objects or expressions are not the same, or do not represent the same value.^[1] This relation is written with a crossed-out equal sign ("≠") as in

$x \neq y.$

Verbally it may be spoken as "does not equal," or "is unequal to." In programming languages and electronic communications, the notations

x != y

x /= y

x <> y

and others, are used instead.

Some sources^[2] use the term inequation synonymously with inequality. Others^[3] include ≠ as a type of inequality. In a linearly ordered set, any inequation implies an inequality: if x ≠ y then x < y or x > y by the trichotomy law.

Properties

Some useful properties of inequations in algebra are:

1. Any quantity can be added or subtracted to both sides.
2. Both sides can be multiplied or divided by any nonzero quantity. When multiplying or dividing both sides by a negative number, a "<" sign must be changed to a ">" sign and vice-versa in order to retain the truth value. For example 4 < 5 (true), (-1)(4)>(-1)(5), -4 > -5 (still true).
3. Generally, any injective function can be applied to both sides. (This is something of a tautology, since injective functions may be defined as functions that always preserve inequations.)

See also

• Equation
• Equals sign
• Inequality (mathematics)
• Relational operator

References

1. ^ Weisstein, Eric W., "Inequation" from MathWorld.
2. ^ For example Sinha, K.C.. A Text Book of Mathematics XI. Rastogi. ISBN 8171339123. 
3. ^ For example Hazewinkel, Michiel, ed. (2001), "Inequality", Encyclopaedia of Mathematics, Springer, ISBN 978-1556080104, http://eom.springer.de/I/i050790.htm