Variables embody a key concept in problem-solving. A variable is simply a representative member of a population called its domain. As a representative of its domain, a variable has the flexibility and capacity to take-on the value of any of the members of its domain. Often, variables represent members of their domains whose values are unknown. It is in this capacity that they are most useful.
Mathematicians generally use letters of the alphabet to symbolize unknown variables. For example, the letter "x" is a commonly used symbol for variables. But before "x" can be a mathematical variable, it must be established by a mathematical statement that indicates the domain it represents. For example, the statement "x such that x is a real number" establishes the symbol "x" as a variable that represents any real number. In mathematical terms, x : x belongs to R.
Variables allow mathematicians to establish mathematical relationships. A mathematical relationship involving only one variable is univariate while a mathematical relationship involving more than one variable is multivariate. Determining the values of variables that satisfy the mathematical relationships they belong to, is a primary focus in mathematics. Equations and Inequalities are two important types of mathematical relationships.
Some domains of the real number line using the variable "x" as a representative are as follows:
All positive numbers (x > 0)
All positive numbers including 0 (x ≥ 0)
All negative numbers (x < 0)
All negative numbers including 0 (x ≤ 0)
All positive fractions (0 < x < 1)
All negative fractions (-1 < x < 0)
Numbers within the open interval bounded by the positive numbers "a" and "b" (a < x < b)
Numbers within the closed interval bounded by "a" and "b" [a ≤ x ≤ b]
Numbers within the open interval bounded by the negative numbers "-a" and "-b" (-a < x < -b)
Numbers within the closed interval bounded by "-a" and "-b" [-a ≤ x ≤ -b]
Numbers within the interval lowerly bounded by "a" and upperly open at "b" [a ≤ x < b)