# Absolute Value Equations and Inequalities and More on Solving Systems

Absolute value and absolute value equations are two types of equations in mathematics. Absolute value functions are functions whose values are constant for all values of the independent variable. If the absolute value of a number is positive, then its absolute value is positive. If the absolute value of a number is negative, then its absolute value is negative. Absolute value equations are used to specify that a variable's absolute value should always be positive. Absolute value inequalities are used to specify that a variable's absolute value must be greater than or less than some value. In the field of mathematics, a function is a relationship between a set of ordered pairs of real numbers. The set of ordered pairs is a set of ordered pairs of real numbers, written in the form , each of which is known as the graph of the function. The set of ordered pairs is called the domain of the function, and the set of real numbers that are paired in the function is called the codomain of the function. The set of ordered pairs of real numbers is called the range of the function. A function is a rule which associates each ordered pair with a unique real number. For example, the function which associates each ordered pair (1, 4) with the real number 4 is defined by the rule , where is the set of ordered pairs, and is the set of real numbers. The value of a function is the value of the ordered pair associated with the value of the independent variable. For example, the value of the function which associates with the ordered pair (1, 4) is 4. In this example, the value of the function is equal to the value of the ordered pair (1, 4). The graph of a function is a set of ordered pairs of real numbers; the graph of a function is the set of ordered pairs of real numbers such that the function assigns to each ordered pair a unique real number. The graph of a function is a subset of the Cartesian plane, called the graph of the function. The graph of a function may have more than one set of ordered pairs of real numbers as the graph. For example, the graph of the function is a subset of the Cartesian plane, with the ordered pairs (1, 6), (2, 7), (3, 5), and (4, 8) in the graph. Thus, the graph of the function consists of the ordered pairs (1, 6), (2, 7), (3, 5), and (4, 8). The graph of a function may not be defined, in which case the graph of the function is the empty set. A function is denoted by a symbol, which is a string of letters and/or numbers. The string of letters and numbers is called the name of the function. The symbol of the function is called the function name or symbol. The string of letters and numbers may be written in any order, such as , , , , etc. The symbol of a function is written between parentheses, such as or . The domain of a function is the set of all ordered pairs of real numbers for which the function is defined. The codomain is the set of all real numbers for which the function is defined. The domain of a function is the set of real numbers that the function associates with ordered pairs. The codomain of a function is the set of ordered pairs of real numbers for which the function is defined. The range of a function is the set of all real numbers for which the function is defined. The graph of a function is the set of all ordered pairs of real numbers for which the function is defined. The graph of a function is a subset of the Cartesian plane, called the graph of the function. The graph of a function may have more than one set of ordered pairs of real numbers as the graph. The set of all ordered pairs of real numbers that are in the graph of a function is called the support of the function. Thus, the graph of the function contains the ordered pairs (1, 4), (2, 5), (3, 6), and (4, 7). The sum of the values of the ordered pairs in the graph is equal to the value of the independent variable. The range of a function is the complement of the support of the function. Absolute value refers to the distance of a number from zero on a number line. In an absolute value equation, the absolute value of a variable is always positive. Thus, in absolute value equations, the absolute value of the variable of the left side is always positive. The absolute value of the variable on the right side of an absolute value equation is always positive. In an absolute value inequality, the absolute value of a variable is always greater than or equal to some value. Thus, in absolute value inequalities, the absolute value of a variable is always greater than or equal to the value of the absolute