There are two crucial viewpoints that you should have when you see an expression like
‘$\,-x\,$’ ;
i.e., a variable, with a minus sign in front of it.
For the moment, read
‘$\,-x\,$’ aloud as ‘the opposite of $\,x\,$’.
Firstly, the symbol
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$\,-x\,$ denotes the opposite of $\,x\,$.
If
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$\,x\,$ is positive, then $\,-x\,$ is negative.
If
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$\,x\,$ is negative, then $\,-x\,$ is positive.
Study the chart below:
[beautiful math coming... please be patient] $\,x\,$ | [beautiful math coming... please be patient] $\,-x\,$ | comment |
$2$ | $-2$ | [beautiful math coming... please be patient] $x\,$ is positive, so $\,-x\,$ is negative |
$-2$ | $2$ | [beautiful math coming... please be patient] $x\,$ is negative, so $\,-x\,$ is positive |
Secondly, the expression
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$\,-x\,$ is equal to
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$\,(-1)x\,$.
That is, the minus sign can be thought of as multiplication by
$\,-1\,$.
Notice how this interpretation is used in the chart below:
[beautiful math coming... please be patient] $\,x\,$ | [beautiful math coming... please be patient] $\,-x\,$ | comment |
$2$ | $(-1)\cdot 2 = -2$ | [beautiful math coming... please be patient] $x\,$ is positive, so $\,-x\,$ is negative |
$-3$ | $(-1)\cdot (-3) = 3$ | [beautiful math coming... please be patient] $x\,$ is negative, so $\,-x\,$ is positive |
The symbol
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$\,-x\,$ can be read as
‘the opposite of $\,x\,$’ or
‘negative $\,x\,$’.
Both are correct, and both are commonplace.
Although the phrase
‘the opposite of
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$\,x\,$’ is a bit longer,
it's also safer for beginning students of algebra.
The reason is this: when you say
‘negative
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$\,x\,$’ aloud,
there is a temptation to think that you're dealing with a negative number
(i.e., one that lies to the left of zero on the number line).
Not necessarily true!
If
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$\,x\,$ is negative, then $\,-x\,$ is positive.
If you can say
‘negative
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$\,x\,$’ with full knowledge that it's not necessarily a negative number,
then go ahead and use this phrase.
Otherwise, say
‘the opposite of
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$\,x\,$’.