Editing Elementary algebra (section)

=== Equations ===
[[File:Pythagorean theorem - Ani.gif|thumb|Animation illustrating [[Pythagorean theorem|Pythagoras' rule]] for a right-angle triangle, which shows the algebraic relationship between the triangle's hypotenuse, and the other two sides.]]
{{Main|Equation}}
An equation states that two expressions are equal using the symbol for equality, {{=}} (the [[equals sign]]).<ref>Mark Clark, Cynthia Anfinson, ''Beginning Algebra: Connecting Concepts Through Applications'', Publisher Cengage Learning, 2011, {{ISBN|0534419380}}, 9780534419387, 793 pages, [https://books.google.com/books?id=wCzuRMC5048C&q=equation&pg=PA134 page 134]</ref> One of the best-known equations describes Pythagoras' law relating the length of the sides of a [[right angle]] triangle:<ref>Alan S. Tussy, R. David Gustafson, ''Elementary and Intermediate Algebra'', Publisher Cengage Learning, 2012, {{ISBN|1111567689}}, 9781111567682, 1163 pages, [https://books.google.com/books?id=xqio_Xn4t7oC&dq=algebra+Pythagoras+hypotenuse&pg=PA493 page 493]</ref>

:<math>c^2 = a^2 + b^2</math>

This equation states that <math>c^2</math>, representing the square of the length of the side that is the hypotenuse, the side opposite the right angle, is equal to the sum (addition) of the squares of the other two sides whose lengths are represented by {{mvar|a}} and {{mvar|b}}.

An equation is the claim that two expressions have the same value and are equal. Some equations are true for all values of the involved variables (such as <math>a + b = b + a</math>); such equations are called [[identity (mathematics)|identities]]. Conditional equations are true for only some values of the involved variables, e.g. <math>x^2 - 1 = 8</math> is true only for <math>x = 3</math> and <math>x = -3</math>. The values of the variables which make the equation true are the solutions of the equation and can be found through [[equation solving]].

Another type of equation is inequality. Inequalities are used to show that one side of the equation is greater, or less, than the other. The symbols used for this are: <math> a > b </math> where <math> > </math> represents 'greater than', and <math> a < b </math> where <math> < </math> represents 'less than'. Just like standard equality equations, numbers can be added, subtracted, multiplied or divided. The only exception is that when multiplying or dividing by a negative number, the inequality symbol must be flipped.

==== Properties of equality ====

By definition, equality is an [[equivalence relation]], meaning it is [[reflexive relation|reflexive]] (i.e. <math>b = b</math>), [[symmetric relation|symmetric]] (i.e. if <math>a = b</math> then <math>b = a</math>), and [[transitive relation|transitive]] (i.e. if <math>a = b</math> and <math>b = c</math> then <math>a = c</math>).<ref>Douglas Downing, ''Algebra the Easy Way'', Publisher Barron's Educational Series, 2003, {{ISBN|0764119729}}, 9780764119729, 392 pages, [https://books.google.com/books?id=RiX-TJLiQv0C&dq=algebra+equality+++reflexive++symmetric++transitive&pg=PA20 page 20]</ref> It also satisfies the important property that if two symbols are used for equal things, then one symbol can be substituted for the other in any true statement about the first and the statement will remain true. This implies the following properties:

* if <math>a = b</math> and <math>c = d</math> then <math>a + c = b + d</math> and <math>ac = bd</math>;
* if <math>a = b</math> then <math>a + c = b + c</math> and <math>ac = bc</math>;
* more generally, for any function {{mvar|f}}, if <math>a=b</math> then <math>f(a) = f(b)</math>.

==== Properties of inequality ====

The relations ''less than'' <math> < </math> and greater than <math> > </math> have the property of transitivity:<ref>Ron Larson, Robert Hostetler, ''Intermediate Algebra'', Publisher Cengage Learning, 2008, {{ISBN|0618753524}}, 9780618753529, 857 pages, [https://books.google.com/books?id=b3vqad8tbiAC&dq=algebra+inequality+properties&pg=PA96 page 96]</ref>
* If &nbsp; <math>a < b</math> &nbsp; and &nbsp; <math>b < c</math> &nbsp; then &nbsp; <math>a < c</math>;
* If &nbsp; <math>a < b</math> &nbsp; and &nbsp; <math>c < d</math> &nbsp; then &nbsp; <math>a + c < b + d</math>;<ref>{{cite web|url=https://math.stackexchange.com/q/1043755 |title=What is the following property of inequality called? |date=November 29, 2014 |work=[[Stack Exchange]] |access-date=4 May 2018}}</ref>
* If &nbsp; <math>a < b</math> &nbsp; and &nbsp; <math>c > 0</math> &nbsp; then &nbsp; <math>ac < bc</math>;
* If &nbsp; <math>a < b</math> &nbsp; and &nbsp; <math>c < 0</math> &nbsp; then &nbsp; <math>bc < ac</math>.
By reversing the inequation, <math> < </math> and <math> > </math> can be swapped,<ref>Chris Carter, ''Physics: Facts and Practice for A Level'', Publisher Oxford University Press, 2001, {{ISBN|019914768X}}, 9780199147687, 144 pages, [https://books.google.com/books?id=Ff9gxZPYafcC&q=turned+around&pg=PA50 page 50]</ref> for example:
* <math>a < b</math> is equivalent to <math>b > a</math>