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.