Algebra/Chapter 3/Absolute Value Equations

Introduction
You might have learned about absolute value before. Absolute value is defined such that:
 * $$|a| = \begin{cases} a, & \mbox{if } a \ge 0  \\ -a,  & \mbox{if } a < 0. \end{cases} $$

More simply, it can be defined as the distance between a number and zero on the real number line. Absolute value is written as |a|. So, |-3| is the absolute value of -3, which is 3. Variables, however, are unknown, so absolute value equations should have two possible solutions. In an equation such as |x + 4| = 25, the equation would hold if x + 4 was positive, negative or zero, so to solve that equation, two should be written down first; x + 4 = 25 or x + 4 = -25, with one equation turning the side of the equation without the absolute value into its opposite. From there, you should solve the two equations as you would any equation with variables. You only need to subtract 4 from both equations to get the solution: x = 21 or x = -29. In absolute value equations, the absolute value must be isolated, like you would a variable. In an equation such as 4|x - 7| + 9 = 21, first subtract 9 and divide by 4.