In this exercise, you will
determine the sign (positive or negative) of some simple expressions.
Most expressions—but not all—involve absolute values.
Keep in mind that if a number is positive or negative, then it is guaranteed
to be nonzero (not zero).
When a number is nonzero, then its absolute value—its distance from zero—is
positive.
EXAMPLES:
Question:
Suppose that
$\,x\,$ is negative. Then, $\,-x\,$ is:
Solution:
POSITIVE
Note:
Remember that
$\,-x\,$ denotes the opposite of $\,x\,$.
If $\,x\,$ is positive, then its opposite is negative.
If $\,x\,$ is negative, then its opposite is positive.
The presence of a minus sign in front of a variable does not necessarily mean that you have a negative number!
Question:
Suppose that
$\,x\,$ is positive. Then, $\,-|2x|\,$ is:
Solution:
NEGATIVE
Note:
In this example, it doesn't matter if
$\,x\,$ is positive or negative.
The expression $\,|2x|\,$ will always be positive, since it reports a distance from zero.
Then, it is being multiplied by $\,-1\,$, which gives a negative number.
Remember that a minus sign outside the absolute value indicates multiplication by $\,-1\,$.
Question:
Suppose that
$\,x\,$ is negative. Then, $\,|-x|\,$ is:
Solution:
POSITIVE
Note:
In this example, it doesn't matter if
$\,x\,$ is positive or negative.
The expression $\,|-x|\,$ will always be positive, since it reports a distance from zero.
Remember that the absolute value of ANY nonzero number is positive.