Editing Inequality (mathematics) (section)

== Properties on the number line ==
Inequalities are governed by the following [[Property (philosophy)|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 ''strictly'' [[monotonic function]]s.

=== Converse ===
The relations ≤ and ≥ are each other's [[Converse relation|converse]], meaning that for any [[real number]]s ''a'' and ''b'':
{{block indent|text=''a'' ≤ ''b'' and ''b'' ≥ ''a'' are equivalent.}}

===Transitivity===

The transitive property of inequality states that for any [[real number]]s ''a'', ''b'', ''c'':<ref>{{cite book |last1=Drachman |first1=Bryon C. |last2=Cloud |first2=Michael J. |title=Inequalities: With Applications to Engineering |date=2006 |publisher=Springer Science & Business Media |isbn=0-3872-2626-5 |pages=2–3 |url=https://books.google.com/books?id=sIbfBwAAQBAJ}}</ref>
{{block indent|text=If ''a'' ≤ ''b'' and ''b'' ≤ ''c'', then ''a'' ≤ ''c''.}}
If ''either'' of the premises is a strict inequality, then the conclusion is a strict inequality:
{{block indent|text=If ''a'' ≤ ''b'' and ''b'' < ''c'', then ''a'' < ''c''.}}
{{block indent|text=If ''a'' < ''b'' and ''b'' ≤ ''c'', then ''a'' < ''c''.}}

===Addition and subtraction===
[[File:Translation invariance of less-than-relation.svg|thumb|300px|If ''x'' < ''y'', then ''x'' + ''a'' < ''y'' + ''a''.]]
A common constant ''c'' may be [[addition|added]] to or [[subtraction|subtracted]] from both sides of an inequality.<ref name=":1" /> So, for any [[real number]]s ''a'', ''b'', ''c'':
{{block indent|text=If ''a'' ≤ ''b'', then ''a'' + ''c'' ≤ ''b'' + ''c'' and ''a'' − ''c'' ≤ ''b'' − ''c''.}}

In other words, the inequality relation is preserved under addition (or subtraction) and the real numbers are an [[Partially ordered group|ordered group]] under addition.

===Multiplication and division===
[[File:Invariance of less-than-relation by multiplication with positive number.svg|thumb|If ''x'' < ''y'' and ''a'' > 0, then ''ax'' < ''ay''.]]
[[File:Inversion of less-than-relation by multiplication with negative number.svg|thumb|If ''x'' < ''y'' and ''a'' < 0, then ''ax'' > ''ay''.]]
The properties that deal with [[multiplication]] and [[division (mathematics)|division]] state that for any real numbers, ''a'', ''b'' and non-zero ''c'':
{{block indent|text=If ''a'' ≤ ''b'' and ''c'' > 0, then ''ac'' ≤ ''bc'' and ''a''/''c'' ≤ ''b''/''c''.}}
{{block indent|text=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|§ Ordered fields]]''.

===Additive inverse===

The property for the [[additive inverse]] states that for any real numbers ''a'' and ''b'':
{{block indent|text=If ''a'' ≤ ''b'', then −''a'' ≥ −''b''.}}

===Multiplicative inverse===

If both numbers are positive, then the inequality relation between the [[multiplicative inverse]]s is opposite of that between the original numbers. More specifically, for any non-zero real numbers ''a'' and ''b'' that are both [[Positive number|positive]] (or both [[Negative number|negative]]):
{{block indent|text=If ''a'' ≤ ''b'', then {{sfrac|1|''a''}} ≥ {{sfrac|1|''b''}}.}}

All of the cases for the signs of ''a'' and ''b'' can also be written in [[#Chained notation|chained notation]], as follows:

{{block indent|text=If 0 < ''a'' ≤ ''b'', then {{sfrac|1|''a''}} ≥ {{sfrac|1|''b''}} > 0.}}
{{block indent|text=If ''a'' ≤ ''b'' < 0, then 0 > {{sfrac|1|''a''}} ≥ {{sfrac|1|''b''}}.}}
{{block indent|text=If ''a'' < 0 < ''b'', then {{sfrac|1|''a''}} < 0 < {{sfrac|1|''b''}}.}}

===Applying a function to both sides===
[[File:Log.svg|right|thumb|The graph of ''y'' = ln ''x'']]
Any [[Monotonic function|monotonic]]ally increasing [[function (mathematics)|function]], by its definition,<ref>{{Cite web | url=http://www.cs.yale.edu/homes/aspnes/pinewiki/ProvingInequalities.html|title=ProvingInequalities | website=www.cs.yale.edu | access-date=2019-12-03}}</ref> may be applied to both sides of an inequality without breaking the inequality relation (provided that both expressions are in the [[Domain of a function|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:<!--
--> {{block indent|text=0 ≤ ''a'' ≤ ''b'' ⇔ 0 ≤ ''a<sup>n</sup>'' ≤ ''b<sup>n</sup>''.}}<!--
--> {{block indent|text=0 ≤ ''a'' ≤ ''b'' ⇔ ''a''<sup>−''n''</sup> ≥ ''b''<sup>−''n''</sup> ≥ 0.}}
* Taking the [[natural logarithm]] on both sides of an inequality, when ''a'' and ''b'' are positive real numbers: <!--
--> {{block indent|0 < ''a'' ≤ ''b'' ⇔ ln(''a'') ≤ ln(''b'').}} <!--
--> {{block indent|0 < ''a'' < ''b'' ⇔ ln(''a'') < ln(''b'').}} <!--
--> (this is true because the natural logarithm is a strictly increasing function.)