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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: .
For example,  and . 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 –2.4 is negative,  and since 5.01 is positive .

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 ξ > 0, then .

However, negative variables can cause students much consternation. If x is negative, then . This often confuses students because the absolute value is positive but the –x appears to be negative. It is actually positive—it is the negative of a negative number, which is positive. To see this more clearly let x = –k, where k is a positive number. Then x is a negative number. So . Since k is positive so is –x. Another way to view this is  (–1)(a negative number) = a positive number.
If , then which of the following statements could be true?
I.    x = 0
II.   x < 0
III.  x > 0
  1. None
  2. I only
  3. III only
  4. I and II
  5. II and III
Statement I could be true because .
Statement II could be true because the right side of the equation is always negative [ –(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 Statement III is the opposite of Statement II, it must be false.
But let’s show this explicitly: Suppose x were positive.
Then , and the equation  becomes x = –x.
Dividing both sides of this equation by x yields 1 = –1.
This is contradiction. Hence, x cannot be positive.
The answer is (D).

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