Absolute Value and Solving Absolute Value Equations

Absolute value, designated by surrounding a number or algebraic expression by parallel lines, is a commonly used process in mathematics. Many students translate absolute value as: throwing away the positive or negative sign associated with a number and making everything positive. For example:  |2| = 2, or |-7| = 7.

Absolute value, therefore, appears to be a way in which one can look at a number and report its actual physical amount ... disregarding any negatives. The formal definition of absolute value is given by the following 2-part definition:

The definition indicates that one of two different things can happen whenever you work with absolute value. If the original number or algebraic expression "N" is positive (technically greater than or equal to zero), then the number/expression goes unchanged (and is reported "as is").  If ... on the other hand ... the original number or expression "N" is negative, then the OPPOSITE of the number/expression is to be reported.

The consequence of this 2-part definition create an interesting situation when solving absolute value equations. Basically, absolute value equations must allow for both portions of the definition and, consequently, cause the equation worker to solve two separate equations. For example, consider the following equation:

Solving this equation requires setting up and solving two sub-equations. The first sub-equation corresponds to the top part of the absolute value definition ... that the absolute value of a number/expression is itself:

The second sub-equation is produced by taking the exact opposite of the expression inside the absolute value symbols. This corresponds to the bottom part of the absolute value definition ... that the absolute value of a number/expression is the opposite of itself.

If you check both answers ... 9 or -6 ... both satisfy the original absolute value equation. In general, it is important (imperative) to check all absolute value equation solutions ... in the event that an extraneous root (solution) is introduced.

The graph of an absolute value equation often produces some interesting results. Since absolute value has the ability to "so-to-say" remove negatives, a graph of an equation with absolute value frequently produces a "sharp" reversal or turn, and often creates symmetric image. In the equation above, "y = |2x - 4|" ... a symmetric V-shape curve is generated.

The key critical values that should be located in an absolute value equation graph are usually those at and near those x-values at which the algebraic expression in absolute value change from positive to negative (when the expression in absolute value takes on the value of 0).  In this example, when |2x - 4| = 0 ... which occurs when x = 2. Building an "x-y" table around  x=2 (at and near x=2) generates the following:

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