# Practice with $\,x\,$ and $\,-x$

## A Signed Variable: $\,-x\,$

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*
$\,-x\,$
*denotes the opposite of*
$\,x\,$:

- If $\,x\,$ is positive, then $\,-x\,$ is negative.
- If $\,x\,$ is negative, then $\,-x\,$ is positive.

Study the chart below:

$\,x\,$ | $\,-x\,$ | comment |

$2$ | $-2$ | $x\,$ is positive, so $\,-x\,$ is negative |

$-2$ | $2$ | $x\,$ is negative, so $\,-x\,$ is positive |

*Secondly, the expression*
$\,-x\,$
*is equal to*
$\,(-1)x\,$.

That is, the minus sign can be thought of as multiplication by $\,-1\,.$

Notice how this interpretation is used in the chart below:

$\,x\,$ | $\,-x\,$ | comment |

$2$ | $(-1)\cdot 2 = -2$ | $x\,$ is positive, so $\,-x\,$ is negative |

$-3$ | $(-1)\cdot (-3) = 3$ | $x\,$ is negative, so $\,-x\,$ is positive |

## Reading ‘$\,-x\,$’ Aloud

The symbol $\,-x\,$ can be read as ‘the opposite of $\,x\,$’ or ‘negative $\,x\,$’. Both are correct, and both are commonplace.

Although the phrase ‘the opposite of $\,x\,$’ is a bit longer, it's also safer for beginning students of algebra. The reason is this: when you say ‘negative $\,x\,$’ 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 $\,x\,$ is negative, then $\,-x\,$ is positive.

If you can say ‘negative $\,x\,$’ with full knowledge that it's not necessarily a negative number, then go ahead and use this phrase. Otherwise, say ‘the opposite of $\,x\,$’.