Why is $|-x|=-x$ if $x<0$?

Question

Here is the question:

If $x < 0$, then what should $|-x|$ be equal to?

From my understanding, everything within the brackets should equate to a non-negative value. Which means no matter what number $x$ is, $|-x|$ should always be equal to a positive number but the book says it's equal to $-x$.

Let's say $x = -1$, then it would be $|-(-1)| = 1$, which is still a positive number.

So what am I getting wrong? I am very confused! Thank you in advance.

Answer 1

If $x$ is negative, then $-x$ is positive and the brackets $||$ do not influence the result, so you definitely get $-x$ as a result.

Answer 2

Short observation that precedes thinking about absolute values: a numerical expression that starts with a $-$ sign does not necessarily represent a negative number.

That's exactly what your example shows. The expression $-z$ starts with a $-$ sign. It may or may not evaluate to a negative number. That depends on whether $z$ is negative to begin with.