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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.

calculatormathematical
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Liang enhao
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  • "Let's say $x = -1$, then it would be $|-(-1)| = 1$, which is still a positive number." Yes. $-x$ is positive. Not $x$. And so $|-x|=-x$. Just because there's a minus before $x$ it doesn't mean it is negative, it still depends on $x$. – freakish Nov 09 '23 at 08:59
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    If the symbol $\color{red}{«-»}$ confuses you, then immediately focus on simplifying it. You know that, by definition $|x|=|-x|$. Therefore, you have $$|-x|=|x|=-x$$ where $x<0$ . – lone student Nov 09 '23 at 09:01
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    When $x$ is negative, $-x$ is positive. – Peter Phipps Nov 09 '23 at 09:05
  • "Let's say x=−1, then it would be |−(−1)|=1, which is still a positive number." Yes but $1$ is exactly $-x$ when $x=-1$. – Mauro ALLEGRANZA Nov 09 '23 at 09:22

2 Answers2

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

calculatormathematical
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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.

Ethan Bolker
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