In mathematics, an inequality is a statement about the relative size or order of two objects, or about whether they are the same or not (See also: equality).
In each statement above, a is not equal to b. These relations are known as strict inequalities. The notation a < b may also be read as "a is strictly less than b".
In contrast to strict inequalities, there are two types of inequality statements that are not strict:
An additional use of the notation is to show that one quantity is much greater than another, normally by several orders of magnitude.
If the sense of the inequality is the same for all values of the variables for which its members are defined, then the inequality is called an "absolute" or "unconditional" inequality. If the sense of an inequality holds only for certain values of the variables involved, but is reversed or destroyed for other values of the variables, it is called a conditional inequality.
Inequalities are governed by the following properties. Note that, for the transitivity, reversal, addition and subtraction, and multiplication and division properties, the property also holds if strict inequality signs (< and >) are replaced with their corresponding nonstrict inequality sign (≤ and ≥).
The trichotomy property states:
The transitivity of inequalities states:
The properties which deal with addition and subtraction state:
i.e., the real numbers are an ordered group.
The properties which 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:
If either a or b is negative then
We consider two cases of functions: monotonic and strictly monotonic.
Any strictly monotonically increasing function may be applied to both sides of an inequality and it will still hold. Applying a strictly monotonically decreasing function to both sides of an inequality means the opposite inequality now holds. The rules for additive and multiplicative inverses are both examples of applying a monotonically decreasing function.
If you have a nonstrict inequality (a ≤ b, a ≥ b) then:
It will never become strictly unequal, since, for example, 3 ≤ 3 does not imply that 3 < 3 however 103x<4 means that the answers can be 1, 2, or 3 but not 4
If (F, +, ×) is a field and ≤ is a total order on F, then (F, +, ×, ≤) is called an ordered field if and only if:
Note that both (Q, +, ×, ≤) and (R, +, ×, ≤) are ordered fields, but ≤ cannot be defined in order to make (C, +, ×, ≤) an ordered field, because −1 is the square of i and would therefore be positive.
The nonstrict inequalities ≤ and ≥ on real numbers are total orders. The strict inequalities < and > on real numbers are strict total orders.
The notation a < b < c stands for "a < b and b < c", from which, by the transitivity property above, it also follows that a < c. Obviously, by the above laws, one can add/subtract the same number to all three terms, or multiply/divide all three terms by same nonzero number and reverse all inequalities according to sign. But care must be taken so that you really use the same number in all cases, e.g. a < b + e < c is equivalent to a − e < b < c − e.
This notation can be generalized to any number of terms: for instance, a_{1} ≤ a_{2} ≤ ... ≤ a_{n} means that a_{i} ≤ a_{i+1} for i = 1, 2, ..., n − 1. By transitivity, this condition is equivalent to a_{i} ≤ a_{j} for any 1 ≤ i ≤ j ≤ n.
When solving inequalities using chained notation, it is possible and sometimes necessary to evaluate the terms independently. For instance to solve the inequality 4x < 2x + 1 ≤ 3x + 2, you won't be able to isolate x in any one part of the inequality through addition or subtraction. Instead, you can solve 4x < 2x + 1 and 2x + 1 ≤ 3x + 2 independently, yielding x < 1/2 and x ≥ 1 respectively, which can be combined into the final solution 1 ≤ x < 1/2.
Occasionally, chained notation is used with inequalities in different directions, in which case the meaning is the logical conjunction of the inequalities between adjacent terms. For instance, a < b > c ≤ d means that a < b, b > c, and c ≤ d. In addition to rare use in mathematics, this notation exists in a few programming languages such as Python.
Every inequality (except those which involve imaginary numbers) can be represented on the real number line showing darkened regions on the line. A < or > is graphed by an open circle on the number. A ≤ or ≥ is graphed with a closed or black circle.
There are many inequalities between means. For example, for any positive numbers a_{1}, a_{2}, …, a_{n} we have H ≤ G ≤ A ≤ Q, where
(harmonic mean),  
(geometric mean),  
(arithmetic mean),  
(quadratic mean). 
When manipulating inequalities it is sometimes useful to take the logarithm of both sides of the inequality. To do this the following result is needed.
Sometimes with notation "power inequality" understand inequalities which contain a^{b} type expressions where a and b are real positive numbers or expressions of some variables. They can appear in exercises of mathematical olympiads and some calculations.
See also list of inequalities.
Mathematicians often use inequalities to bound quantities for which exact formulas cannot be computed easily. Some inequalities are used so often that they have names:
The set of complex numbers with its operations of addition and multiplication is a field, but it is impossible to define any relation ≤ so that ,+,*,≤ becomes an ordered field. To make (,+,*,≤) an ordered field, it would have to satisfy the following two properties:
Because ≤ is a total order, for any number a, either 0 ≤ a or a ≤ 0 (in which case the first property above implies that 0 ≤ − a). In either case 0 ≤ a^{2}; this means that i^{2} > 0 and 1^{2} > 0; so − 1 > 0 and 1 > 0, which means ( − 1 + 1) > 0; contradiction.
However, an operation ≤ can be defined so as to satisfy only the first property (namely, "if a ≤ b then a + c ≤ b + c"). A definition which is sometimes used is the lexicographical order:
It can easily be proven that for this definition a ≤ b implies a + c ≤ b + c.
Inequality relationships similar to those defined above can also be defined for column vector. If we let the vectors (meaning that and where x_{i} and y_{i} are real numbers for ), we can define the following relationships.
Similarly, we can define relationships for x > y, , and . We note that this notation is consistent with that used by Matthias Ehrgott in Multicriteria Optimization (see References).
We observe that the property of Trichotomy (as stated above) is not valid for vector relationships. We consider the case where and . There exists no valid inequality relationship between these two vectors. Also, a multiplicative inverse would need to be defined on a vector before this property could be considered. However, for the rest of the aforementioned properties, a parallel property for vector inequalities exists.
Inequality is when one object is:
Inequality is sometimes used to name a statement that one expression is smaller, greater, not smaller or not greater than the other.
