# 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 $\,x\,$’.

## Examples

If $\,x\,$ is positive, then:  $\,-x\,$ lies to the left of zero
If $\,-x\,$ is greater than zero, then:  $\,x \lt 0\,$
If $\,x = -5\,,$ then:  $\,-x = 5\,$
If $\,-x = 4\,,$ then:  $\,x = -4\,$