PRACTICE WITH $\,x\,$ and $\,-x\,$

by Dr. Carol JVF Burns (website creator)
Follow along with the highlighted text while you listen!

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 $\,x\,$’.

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:   $\,x = -4\,$
Master the ideas from this section
by practicing the exercise at the bottom of this page.

When you're done practicing, move on to:
Practice with Products of Signed Variables

Choose the correct answer to this question:
(an even number, please)