![]() The resulting piecewise function will have the exact same graph as the original function. We can break this equation up into two equations that are part of a single piecewise function, neither of which involves an absolute value sign. Each “piece” will start and end at a point where an expression inside absolute value bars changes sign. Solving Equations Involving Absolute ValuesĮquations that involve one or more absolute value expressions can be solved by breaking them into piecewise functions. ![]() This same process can be also be used to solve absolute value equations. Once we do know a value for “x”, if the expression within the bars is not negative, we can drop the bars if it is negative, we must change its sign (negating it, making it positive) before dropping the absolute value signs. And we won’t know that until we know the value of “x”. In such a case, we don’t know whether the absolute value will have any effect on the quantity until we know whether “x-4” evaluates to a negative value or not. While this is fairly straightforward when working with constant values, as shown above, what happens when a pair of absolute value signs contains a variable? Just as with parentheses, absolute value symbols serve as grouping symbols: the expression inside the bars must be evaluated and expressed as either zero or a positive quantity before the bars may be dropped. Only the final result, after evaluating the entire expression inside the absolute value signs, should be made positive.Ībsolute Value expressions that contain variables Note that absolute value signs do not instruct you to make “all” quantities inside them positive. These bars mean: evaluate what is inside and, if the final result ( once the entire expression inside the absolute value signs has been evaluated) is negative, change its sign to make it positive and drop the bars if the final result inside the bars is zero or positive, you may drop the bars without making any changes: The notation used to indicate absolute value is a pair of vertical bars surrounding the quantity, sort of like a straight set of parentheses. In other words, its distance from zero expressed as a positive number. ![]() The term “Absolute Value” refers to the magnitude of a quantity without regard to sign.
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