{"history":[],"article":{"id":"1c0dc865-e48e-416e-8384-3d465fe0ee96","title":"Inequality (mathematics)","slug":"inequality-mathematics","content":":::summary\n:::\n\n# \nIn [mathematics](https://bliptext.com/articles/mathematics), an **inequality** is a relation which makes a non-equal comparison between two numbers or other mathematical expressions.\nIt is used most often to compare two numbers on the [number line](https://bliptext.com/articles/number-line) by their size.\nThe main types of inequality are **less than** (<) and **greater than** (>).\n# [The feasible regions of linear programming are defined by a set of inequalities. | upload.wikimedia.org/wikipedia/commons/0/0c/Linear///Programming///Feasible///Region.svg]\n\n# Notation\nThere are several different notations used to represent different kinds of inequalities:\nIn either case, a is not equal to b.\nThese relations are known as **strict inequalities**, meaning that a is strictly less than or strictly greater than b.\nEquality is excluded.\nIn contrast to strict inequalities, there are two types of inequality relations that are not strict:\nIn the 17th and 18th centuries, personal notations or typewriting signs were used to signal inequalities.\nFor example, In 1670, [John Wallis](https://bliptext.com/articles/john-wallis) used a single horizontal bar above rather than below the.\nLater in 1734, ≦ and ≧, known as \"less than (greater-than) over equal to\" or \"less than (greater than) or equal to with double horizontal bars\", first appeared in [Pierre Bouguer](https://bliptext.com/articles/pierre-bouguer)'s work.\nAfter that, mathematicians simplified Bouguer's symbol to \"less than (greater than) or equal to with one horizontal bar\" (≤), or \"less than (greater than) or slanted equal to\" (⩽).\nThe relation **not greater than** can also be represented by a \\ngtr b, the symbol for \"greater than\" bisected by a slash, \"not\".\nThe same is true for **not less than**, a \\nless b.\nThe notation a ≠ b means that a is not equal to b; this [inequation](https://bliptext.com/articles/inequation) sometimes is considered a form of strict inequality.\nIt does not say that one is greater than the other; it does not even require a and b to be member of an [ordered set](https://bliptext.com/articles/ordered-set).\nIn 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](https://bliptext.com/articles/order-of-magnitude).\nThis implies that the lesser value can be neglected with little effect on the accuracy of an [approximation](https://bliptext.com/articles/approximation) (such as the case of [ultrarelativistic limit](https://bliptext.com/articles/ultrarelativistic-limit) in physics).\nIn all of the cases above, any two symbols mirroring each other are symmetrical; a < b and b > a are equivalent, etc.\n\n# Properties on the number line\nInequalities are governed by the following [properties](https://bliptext.com/articles/property-philosophy).\nAll of these properties also hold if all of the non-strict inequalities (≤ and ≥) are replaced by their corresponding strict inequalities and — in the case of applying a function — [monotonic functions](https://bliptext.com/articles/monotonic-function) are limited to strictly [monotonic functions](https://bliptext.com/articles/monotonic-function).\n\n## Converse\nThe relations ≤ and ≥ are each other's [converse](https://bliptext.com/articles/converse-relation), meaning that for any [real numbers](https://bliptext.com/articles/real-number) a and b:\n\n## Transitivity\nThe transitive property of inequality states that for any [real numbers](https://bliptext.com/articles/real-number) a, b, c:\nIf either of the premises is a strict inequality, then the conclusion is a strict inequality:\n\n## Addition and subtraction\nA common constant c may be [added](https://bliptext.com/articles/addition) to or [subtracted](https://bliptext.com/articles/subtraction) from both sides of an inequality.\nSo, for any [real numbers](https://bliptext.com/articles/real-number) a, b, c:\nIn other words, the inequality relation is preserved under addition (or subtraction) and the real numbers are an [ordered group](https://bliptext.com/articles/partially-ordered-group) under addition.\n# [If x < y, then x + a < y + a. | upload.wikimedia.org/wikipedia/commons/1/1b/Translation///invariance///of///less-than-relation.svg]\n\n## Multiplication and division\nThe properties that deal with [multiplication](https://bliptext.com/articles/multiplication) and [division](https://bliptext.com/articles/division-mathematics) state that for any real numbers, a, b and non-zero c:\nIn other words, the inequality relation is preserved under multiplication and division with positive constant, but is reversed when a negative constant is involved.\nMore generally, this applies for an [ordered field](https://bliptext.com/articles/ordered-field).\nFor more information, see [§ Ordered fields](https://bliptext.com/articles/).\n# [If x < y and a > 0, then ax < ay. | upload.wikimedia.org/wikipedia/commons/0/03/Invariance///of///less-than-relation///by///multiplication///with///positive///number.svg]\n# [If x < y and a < 0, then ax > ay. | upload.wikimedia.org/wikipedia/commons/0/0f/Inversion///of///less-than-relation///by///multiplication///with///negative///number.svg]\n\n## Additive inverse\nThe property for the [additive inverse](https://bliptext.com/articles/additive-inverse) states that for any real numbers a and b:\n\n## Multiplicative inverse\nIf both numbers are positive, then the inequality relation between the [multiplicative inverses](https://bliptext.com/articles/multiplicative-inverse) is opposite of that between the original numbers.\nMore specifically, for any non-zero real numbers a and b that are both [positive](https://bliptext.com/articles/positive-number) (or both [negative](https://bliptext.com/articles/negative-number)):\nAll of the cases for the signs of a and b can also be written in [chained notation](https://bliptext.com/articles/), as follows:\n\n## Applying a function to both sides\nAny [monotonically](https://bliptext.com/articles/monotonic-[function](https://bliptext.com/articles/function-mathematics)) increasing [function](https://bliptext.com/articles/function-mathematics), 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](https://bliptext.com/articles/domain-of-a-function) of that [function](https://bliptext.com/articles/function-mathematics)).\nHowever, applying a monotonically decreasing function to both sides of an inequality means the inequality relation would be reversed.\nThe rules for the additive inverse, and the multiplicative inverse for positive numbers, are both examples of applying a monotonically decreasing function.\nIf the inequality is strict (a < b, a > b) and the function is strictly monotonic, then the inequality remains strict.\nIf only one of these conditions is strict, then the resultant inequality is non-strict.\nIn fact, the rules for additive and multiplicative inverses are both examples of applying a strictly monotonically decreasing function.\nA few examples of this rule are:\n# [inequality | upload.wikimedia.org/wikipedia/commons/e/ea/Log.svg]\n\n# Formal definitions and generalizations\nA (non-strict) **partial order** is a [binary relation](https://bliptext.com/articles/binary-relation) ≤ over a [set](https://bliptext.com/articles/set-mathematics) P which is [reflexive](https://bliptext.com/articles/reflexive-relation), [antisymmetric](https://bliptext.com/articles/antisymmetric-relation), and [transitive](https://bliptext.com/articles/transitive-relation).\nThat is, for all a, b, and c in P, it must satisfy the three following clauses:\nA set with a partial order is called a **[partially ordered set](https://bliptext.com/articles/partially-ordered-set)**.\nThose are the very basic axioms that every kind of order has to satisfy.\nA strict partial order is a relation < that satisfies:\nSome types of partial orders are specified by adding further axioms, such as:\n\n## Ordered fields\nIf (F, +, ×) is a [field](https://bliptext.com/articles/field-mathematics) and ≤ is a [total order](https://bliptext.com/articles/total-order) on F, then (F, +, ×, ≤) is called an **ordered [field](https://bliptext.com/articles/field-mathematics)** if and only if:\nBoth (\\mathbb Q, +, \\times, \\leq) and (\\mathbb R, +, \\times, \\leq) are [ordered fields](https://bliptext.com/articles/ordered-field), but\n≤\ncannot be defined in order to make (\\mathbb C, +, \\times, \\leq) an [ordered field](https://bliptext.com/articles/ordered-field), because −1 is the square of i and would therefore be positive.\nBesides being an ordered field, **R** also has the [Least-upper-bound property](https://bliptext.com/articles/least-upper-bound-property).\nIn fact, **R** can be defined as the only ordered field with that quality.\n\n# Chained notation\nThe notation a < b < c stands for \"a < b and b < c\", from which, by the transitivity property above, it also follows that a < c.\nBy 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.\nHence, for example, a < b + e < c is equivalent to a − e < b < c − e.\nThis 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.\nBy transitivity, this condition is equivalent to ai ≤ aj for any 1 ≤ i ≤ j ≤ n.\nWhen solving inequalities using chained notation, it is possible and sometimes necessary to evaluate the terms independently.\nFor 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.\nInstead, 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.\nOccasionally, chained notation is used with inequalities in different directions, in which case the meaning is the [logical conjunction](https://bliptext.com/articles/logical-conjunction) of the inequalities between adjacent terms.\nFor example, the defining condition of a [zigzag poset](https://bliptext.com/articles/zigzag-poset) is written as a1 < a2 > a3 < a4 > a5 < a6 > ... . Mixed chained notation is used more often with compatible relations, like <, =, ≤.\nFor instance, a < b = c ≤ d means that a < b, b = c, and c ≤ d.\nThis notation exists in a few [programming languages](https://bliptext.com/articles/programming-language) such as [Python](https://bliptext.com/articles/python-programming-language).\nIn contrast, in programming languages that provide an ordering on the type of comparison results, such as [C](https://bliptext.com/articles/c-programming-language), even homogeneous chains may have a completely different meaning.\n\n# Sharp inequalities\nAn inequality is said to be sharp if it cannot be relaxed and still be valid in general.\nFormally, a [universally quantified](https://bliptext.com/articles/universally-quantified) inequality φ is called sharp if, for every valid [universally quantified](https://bliptext.com/articles/universally-quantified) inequality ψ, if ψ ⇒ φ holds, then ψ ⇔ φ also holds.\nFor instance, the inequality ∀a ∈ R. a2 ≥ 0 is sharp, whereas the inequality ∀a ∈ R. a2 ≥ −1 is not sharp.\n\n# Inequalities between means\nThere are many inequalities between means.\nFor example, for any positive numbers a1, a2, ..., an we have\nwhere they represent the following means of the sequence:\n\n# Cauchy–Schwarz inequality\nThe Cauchy–Schwarz inequality states that for all vectors u and v of an [inner product space](https://bliptext.com/articles/inner-product-space) it is true that\nwhere is the [inner product](https://bliptext.com/articles/inner-product).\nExamples of inner products include the real and complex [dot product](https://bliptext.com/articles/dot-product); In [Euclidean space](https://bliptext.com/articles/euclidean-space) Rn with the standard inner product, the Cauchy–Schwarz inequality is\n\n# Power inequalities\nA **power inequality** is an inequality containing terms of the form ab, where a and b are real positive numbers or variable expressions.\nThey often appear in [mathematical olympiads](https://bliptext.com/articles/mathematical-olympiads) exercises.\nExamples:\n\n# Well-known inequalities\n[Mathematicians](https://bliptext.com/articles/mathematician) often use inequalities to bound quantities for which exact formulas cannot be computed easily.\nSome inequalities are used so often that they have names:\n\n# Complex numbers and inequalities\nThe set of [complex numbers](https://bliptext.com/articles/complex-number) \\mathbb{C} with its operations of [addition](https://bliptext.com/articles/addition) and [multiplication](https://bliptext.com/articles/multiplication) is a [field](https://bliptext.com/articles/field-mathematics), but it is impossible to define any relation\n≤\nso that becomes an [ordered field](https://bliptext.com/articles/ordered-field).\nTo make an [ordered field](https://bliptext.com/articles/ordered-field), it would have to satisfy the following two properties:\nBecause ≤ is a [total order](https://bliptext.com/articles/total-order), for any number a, either 0 ≤ a or a ≤ 0 (in which case the first property above implies that 0 ≤ −a).\nIn either case 0 ≤ a2; this means that i2 > 0 and 12 > 0; so −1 > 0 and 1 > 0, which means (−1 + 1) > 0; contradiction.\nHowever, an operation ≤ can be defined so as to satisfy only the first property (namely, \"if a ≤ b, then a + c ≤ b + c\").\nSometimes the [lexicographical order](https://bliptext.com/articles/lexicographical-order) definition is used:\nIt can easily be proven that for this definition a ≤ b implies a + c ≤ b + c.\n\n# Systems of inequalities\nSystems of [linear inequalities](https://bliptext.com/articles/linear-inequalities) can be simplified by [Fourier–Motzkin elimination](https://bliptext.com/articles/fourier-motzkin-elimination).\nThe [cylindrical algebraic decomposition](https://bliptext.com/articles/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.\nThe complexity of this algorithm is [doubly exponential](https://bliptext.com/articles/double-exponential-function) in the number of variables.\nIt is an active research domain to design algorithms that are more efficient in specific cases.\n\n# Sources","current_revision":null,"created_by":"kACnuQmH5k9Nr5VNVEpVeg7XpvUmMctD","created_at":"2025-01-25T17:53:08.788Z","updated_at":"2025-01-25T17:53:08.788Z","search_vector":"'inequ':1 'mathemat':2","search_content":" \n \n\n# \nIn [mathematics](https://bliptext.com/articles/mathematics), an **inequality** is a relation which makes a non-equal comparison between two numbers or other mathematical expressions.\nIt is used most often to compare two numbers on the [number line](https://bliptext.com/articles/number-line) by their size.\nThe main types of inequality are **less than** ( ).\n# [The feasible regions of linear programming are defined by a set of inequalities. | upload.wikimedia.org/wikipedia/commons/0/0c/Linear///Programming///Feasible///Region.svg]\n\n# Notation\nThere are several different notations used to represent different kinds of inequalities:\nIn either case, a is not equal to b.\nThese relations are known as **strict inequalities**, meaning that a is strictly less than or strictly greater than b.\nEquality is excluded.\nIn contrast to strict inequalities, there are two types of inequality relations that are not strict:\nIn the 17th and 18th centuries, personal notations or typewriting signs were used to signal inequalities.\nFor example, In 1670, [John Wallis](https://bliptext.com/articles/john-wallis) used a single horizontal bar above rather than below the.\nLater in 1734, ≦ and ≧, known as \"less than (greater-than) over equal to\" or \"less than (greater than) or equal to with double horizontal bars\", first appeared in [Pierre Bouguer](https://bliptext.com/articles/pierre-bouguer)'s work.\nAfter that, mathematicians simplified Bouguer's symbol to \"less than (greater than) or equal to with one horizontal bar\" (≤), or \"less than (greater than) or slanted equal to\" (⩽).\nThe relation **not greater than** can also be represented by a \\ngtr b, the symbol for \"greater than\" bisected by a slash, \"not\".\nThe same is true for **not less than**, a \\nless b.\nThe notation a ≠ b means that a is not equal to b; this [inequation](https://bliptext.com/articles/inequation) sometimes is considered a form of strict inequality.\nIt does not say that one is greater than the other; it does not even require a and b to be member of an [ordered set](https://bliptext.com/articles/ordered-set).\nIn 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](https://bliptext.com/articles/order-of-magnitude).\nThis implies that the lesser value can be neglected with little effect on the accuracy of an [approximation](https://bliptext.com/articles/approximation) (such as the case of [ultrarelativistic limit](https://bliptext.com/articles/ultrarelativistic-limit) in physics).\nIn all of the cases above, any two symbols mirroring each other are symmetrical; a a are equivalent, etc.\n\n# Properties on the number line\nInequalities are governed by the following [properties](https://bliptext.com/articles/property-philosophy).\nAll of these properties also hold if all of the non-strict inequalities (≤ and ≥) are replaced by their corresponding strict inequalities and — in the case of applying a function — [monotonic functions](https://bliptext.com/articles/monotonic-function) are limited to strictly [monotonic functions](https://bliptext.com/articles/monotonic-function).\n\n## Converse\nThe relations ≤ and ≥ are each other's [converse](https://bliptext.com/articles/converse-relation), meaning that for any [real numbers](https://bliptext.com/articles/real-number) a and b:\n\n## Transitivity\nThe transitive property of inequality states that for any [real numbers](https://bliptext.com/articles/real-number) a, b, c:\nIf either of the premises is a strict inequality, then the conclusion is a strict inequality:\n\n## Addition and subtraction\nA common constant c may be [added](https://bliptext.com/articles/addition) to or [subtracted](https://bliptext.com/articles/subtraction) from both sides of an inequality.\nSo, for any [real numbers](https://bliptext.com/articles/real-number) a, b, c:\nIn other words, the inequality relation is preserved under addition (or subtraction) and the real numbers are an [ordered group](https://bliptext.com/articles/partially-ordered-group) under addition.\n# [If x 0, then ax ay. | upload.wikimedia.org/wikipedia/commons/0/0f/Inversion///of///less-than-relation///by///multiplication///with///negative///number.svg]\n\n## Additive inverse\nThe property for the [additive inverse](https://bliptext.com/articles/additive-inverse) states that for any real numbers a and b:\n\n## Multiplicative inverse\nIf both numbers are positive, then the inequality relation between the [multiplicative inverses](https://bliptext.com/articles/multiplicative-inverse) is opposite of that between the original numbers.\nMore specifically, for any non-zero real numbers a and b that are both [positive](https://bliptext.com/articles/positive-number) (or both [negative](https://bliptext.com/articles/negative-number)):\nAll of the cases for the signs of a and b can also be written in [chained notation](https://bliptext.com/articles/), as follows:\n\n## Applying a function to both sides\nAny [monotonically](https://bliptext.com/articles/monotonic-[function](https://bliptext.com/articles/function-mathematics)) increasing [function](https://bliptext.com/articles/function-mathematics), 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](https://bliptext.com/articles/domain-of-a-function) of that [function](https://bliptext.com/articles/function-mathematics)).\nHowever, applying a monotonically decreasing function to both sides of an inequality means the inequality relation would be reversed.\nThe rules for the additive inverse, and the multiplicative inverse for positive numbers, are both examples of applying a monotonically decreasing function.\nIf the inequality is strict (a b) and the function is strictly monotonic, then the inequality remains strict.\nIf only one of these conditions is strict, then the resultant inequality is non-strict.\nIn fact, the rules for additive and multiplicative inverses are both examples of applying a strictly monotonically decreasing function.\nA few examples of this rule are:\n# [inequality | upload.wikimedia.org/wikipedia/commons/e/ea/Log.svg]\n\n# Formal definitions and generalizations\nA (non-strict) **partial order** is a [binary relation](https://bliptext.com/articles/binary-relation) ≤ over a [set](https://bliptext.com/articles/set-mathematics) P which is [reflexive](https://bliptext.com/articles/reflexive-relation), [antisymmetric](https://bliptext.com/articles/antisymmetric-relation), and [transitive](https://bliptext.com/articles/transitive-relation).\nThat is, for all a, b, and c in P, it must satisfy the three following clauses:\nA set with a partial order is called a **[partially ordered set](https://bliptext.com/articles/partially-ordered-set)**.\nThose are the very basic axioms that every kind of order has to satisfy.\nA strict partial order is a relation a3 a5 ... . Mixed chained notation is used more often with compatible relations, like 0 and 12 > 0; so −1 > 0 and 1 > 0, which means (−1 + 1) > 0; contradiction.\nHowever, an operation ≤ can be defined so as to satisfy only the first property (namely, \"if a ≤ b, then a + c ≤ b + c\").\nSometimes the [lexicographical order](https://bliptext.com/articles/lexicographical-order) definition is used:\nIt can easily be proven that for this definition a ≤ b implies a + c ≤ b + c.\n\n# Systems of inequalities\nSystems of [linear inequalities](https://bliptext.com/articles/linear-inequalities) can be simplified by [Fourier–Motzkin elimination](https://bliptext.com/articles/fourier-motzkin-elimination).\nThe [cylindrical algebraic decomposition](https://bliptext.com/articles/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.\nThe complexity of this algorithm is [doubly exponential](https://bliptext.com/articles/double-exponential-function) in the number of variables.\nIt is an active research domain to design algorithms that are more efficient in specific cases.\n\n# Sources","revisionCount":0},"hasMore":false,"nextCursor":null}

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