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The absolute value operation can be defined piecewise as \(\left| x \right| = \left\{ \begin{align}
&x{\rm \thinspace{for}} \thinspace x \ge 0 \\
-&x{\rm \thinspace {for}} \thinspace x < 0 \end{align} \right.\). This means the inverse operation varies, depending on the value of the expression inside the absolute value symbol.
&x{\rm \thinspace{for}} \thinspace x \ge 0 \\
-&x{\rm \thinspace {for}} \thinspace x < 0 \end{align} \right.\). This means the inverse operation varies, depending on the value of the expression inside the absolute value symbol.
- When the expression is zero or positive, the absolute value symbol does not change the expression. That is, if the contents are positive, you can just drop the absolute value symbol.
For example, if \(2x + 1\) is positive, then \(\left| 2x + 1\right| = 2x + 1\). - When the expression is negative, the absolute value symbol requires the expression be multiplied by \(–1\) to make it positive (a negative times a negative gives a positive). The inverse operation, in this case, is to multiply by \(–1\). That is, you can drop the absolute value symbol by making the expression negative.
For example, if \(2x + 1\) is negative, then \(\left| 2x + 1\right| = -(2x + 1)\).