In mathematics, an inequality is a relation which makes a non-equal comparison between two numbers or other mathematical expressions. It is used most often to compare two numbers on the number line by their size. There are several different notations used to represent different kinds of inequalities:
The notation a < b means that a is less than b.
The notation a > b means that a is greater than b.
In either case, a is not equal to b. These relations are known as strict inequalities, meaning that a is strictly less than or strictly greater than b. Equivalence is excluded.
In contrast to strict inequalities, there are two types of inequality relations that are not strict:
The notation a ≤ b or a ⩽ b means that a is less than or equal to b (or, equivalently, at most b, or not greater than b).
The notation a ≥ b or a ⩾ b means that a is greater than or equal to b (or, equivalently, at least b, or not less than b).
The relation "not greater than" can also be represented by a ≯ b, the symbol for "greater than" bisected by a slash, "not". The same is true for "not less than" and a ≮ b.
If the values in question are elements of an ordered set, such as that of the integers or the real numbers, then they can be compared in size, that is, described using the above inequality relations. On the other hand, the notation a ≠ b means that a is not equal to b, and is sometimes considered a form of strict inequality. It does not say that one is greater than the other, or even that they can be compared in size.[clarification needed]
In engineering sciences, less formal use of the notation is to state that one quantity is "much greater" than another, normally by several orders of magnitude. This implies that the lesser value can be neglected with little effect on the accuracy of an approximation (such as the case of ultrarelativistic limit in physics).
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 strictlymonotonic functions.
In other words, the inequality relation is preserved under addition (or subtraction) and the real numbers are an ordered group under addition.
Multiplication and division
If x < y and a > 0, then ax < ay.
If x < y and a < 0, then ax > ay.
The properties that deal with multiplication and division state that for any real numbers, a, b and non-zero c:
If a ≤ b and c > 0, then ac ≤ bc and a/c ≤ b/c.
If a ≤ b and c < 0, then ac ≥ bc and a/c ≥ b/c.
In other words, the inequality relation is preserved under multiplication and division with positive constant, but is reversed when a negative constant is involved. More generally, this applies for an ordered field. For more information, see § Ordered fields.
If both numbers are positive, then the inequality relation between the multiplicative inverses is opposite of that between the original numbers. More specifically, for any non-zero real numbers a and b that are both positive (or both negative):
If a ≤ b, then 1/a ≥ 1/b.
All of the cases for the signs of a and b can also be written in chained notation, as follows:
If 0 < a ≤ b, then 1/a ≥ 1/b > 0.
If a ≤ b < 0, then 0 > 1/a ≥ 1/b.
If a < 0 < b, then 1/a < 0 < 1/b.
Applying a function to both sides
The graph of y = ln x
Any monotonically increasing function, by its definition, may be applied to both sides of an inequality without breaking the inequality relation (provided that both expressions are in the domain of that function). However, applying a monotonically decreasing function to both sides of an inequality means the inequality relation would be reversed. The rules for the additive inverse, and the multiplicative inverse for positive numbers, are both examples of applying a monotonically decreasing function.
If the inequality is strict (a < b, a > b) and the function is strictly monotonic, then the inequality remains strict. If only one of these conditions is strict, then the resultant inequality is non-strict. In fact, the rules for additive and multiplicative inverses are both examples of applying a strictly monotonically decreasing function.
A few examples of this rule are:
Raising both sides of an inequality to a power n > 0 (equiv., −n < 0), when a and b are positive real numbers:
0 ≤ a ≤ b ⇔ 0 ≤ an ≤ bn.
0 ≤ a ≤ b ⇔ a−n ≥ b−n ≥ 0.
Taking the natural logarithm on both sides of an inequality, when a and b are positive real numbers:
0 < a ≤ b ⇔ ln(a) ≤ ln(b).
0 < a < b ⇔ ln(a) < ln(b).
(this is true because the natural logarithm is a strictly increasing function.)
A set with a partial order is called a partially ordered set. Those are the very basic axioms that every kind of order has to satisfy. Other axioms that exist for other definitions of orders on a set P include:
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.
Besides from being an ordered field, R also has the Least-upper-bound property. In fact, R can be defined as the only ordered field with that quality.
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. By the above laws, one can add or subtract the same number to all three terms, or multiply or divide all three terms by same nonzero number and reverse all inequalities if that number is negative. Hence, for example, a < b + e < c is equivalent to a − e < b < c − e.
This notation can be generalized to any number of terms: for instance, a1 ≤ a2 ≤ ... ≤ an means that ai ≤ ai+1 for i = 1, 2, ..., n − 1. By transitivity, this condition is equivalent to ai ≤ aj 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, it is not possible to isolate x in any one part of the inequality through addition or subtraction. Instead, the inequalities must be solved 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. This notation exists in a few programming languages such as Python.
An inequality is said to be sharp, if it cannot be relaxed and still be valid in general. Formally, a universally quantified inequality φ is called sharp if, for every valid universally quantified inequality ψ, if ψ⇒φ holds, then ψ⇔φ also holds. For instance, the inequality ∀a ∈ ℝ. a2 ≥ 0 is sharp, whereas the inequality ∀a ∈ ℝ. a2 ≥ −1 is not sharp.
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 ≤ a2; this means that i2 > 0 and 12 > 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"). Sometimes the lexicographical order definition is used:
a ≤ b, if
Re(a) < Re(b), or
Re(a) = Re(b) and Im(a) ≤ Im(b)
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 vectors. If we let the vectors (meaning that and , where and are real numbers for ), we can define the following relationships:
, if for .
, if for .
, if for and .
, if for .
Similarly, we can define relationships for , , and . This notation is consistent with that used by Matthias Ehrgott in Multicriteria Optimization (see References).
The trichotomy property (as stated above) is not valid for vector relationships. For example, when 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.
The cylindrical algebraic decomposition is an algorithm that allows testing whether a system of polynomial equations and inequalities has solutions, and, if solutions exist, describing them. The complexity of this algorithm is doubly exponential in the number of variables. It is an active research domain to design algorithms that are more efficient in specific cases.