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PRACTICE WITH x and -x
Jump right to the exercises!
The concepts for this exercise are summarized below.
For a complete discussion, read the text.
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 = (-1)x | comment |
| 2 | (-1) · 2 = -2 | x is positive, so -x is negative |
| -3 | (-1) · (-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 < 0
if x = -5 , then -x = 5
if -x = 4 , then x = -4