**Absolute Value**

The absolute value of a number is its distance on the number line from 0. Since distance is a positive number, absolute value of a number is positive. Two vertical bars denote the absolute value of a number: |*x*|. For example, |3| = 3 and |-3| = 3. This can be illustrated on the number line:

Students rarely struggle with the absolute value of numbers: if the number is negative, simply make it positive; and if it is already positive, leave it as is. For example, since â€“2.4 is negative, |-2.4| = 2.4 and since 5.01 is positive |5.01| = 5.01.

Further, students rarely struggle with the absolute value of positive variables: if the variable is positive, simply drop the absolute value symbol. For example, if *x* > 0, then $\left|x\right|$= *x*.

However, negative variables can cause students much consternation. If x is negative, then |*x|*= -*x*. This often confuses students because the absolute value is positive but the â€“*x* appears to be negative. It is actually positiveâ€”it is the negative of a negative number, which is positive. To see this more clearly let *x* = â€“ *k*, where k is a __positive__ number. Then x is a negative number. So |*x*| = - *x* = - (- *k*) = *k*. Since k is positive so is â€“ *x*. Another way to view this is |*x*| = - *x*= (-1) â€¢ x = (â€“1)(a negative number) = a positive number.

**Example:**

If *x* = â€“ |*x*|, then which one of the following statements could be true? (Select One or More Answer Choices)

A. *x* = 0

B. *x*< 0

C. *x* > 0

Choice A could be true because â€“ |0| = -(+0) = - (0) = 0. Choice B could be true because the right side of the equation is always negative [ â€“ |*x*| = â€“(a positive number) = a negative number]. Now, if one side of an equation is always negative, then the other side must always be negative, otherwise the opposite sides of the equation would not be equal. Since Choice C is the opposite of Choice B, it must be false. But letâ€™s show this explicitly: Suppose *x* were positive. Then |*x*| = *x*, and the equation *x* = â€“ |*x*| becomes *x* = â€“ *x*. Dividing both sides of this equation by *x* yields 1 = â€“1. This is contradiction. Hence, x cannot be positive. Thus, the answer consists of choices A and B.