Up until now, we've always known exactly what's been inside of the absolute value bars. We happen to think we might be a little psychic. Pretty cool, right? We promise to use this ability only for good. And to impress people at parties.
Now though, whatever psychic power let us peer inside those bars has failed. We have some unknown sitting in there. Like this one here: |x| = 5.
Dealing with absolute values is like dealing with matter and anti-matter, Superman and Bizarro, Spock and Goatee Spock: they're basically the same, but the opposite. But the same. What values of x would give us an absolute value of 5?
If we think about it illogically, we would just stare at x until our latent psychic powers kick back in, letting us see the answer. However, if we think about it logically, we can rewrite the expression in two different ways:
|-x| = 5 and |x| = 5
These are actually identical statements, because the absolute value bars make everything positive in the end. We can now get rid of the absolute value bars and solve for x in these two different equations.
|-x| = 5 becomes:
-x = 5
x = -5
We also have:
|x| = 5
x = 5
Those are exactly the answers we needed for x. We plug them both into the original equation, and yep, they both give us a 5. Not a high-5, though.
Sample Problem
Solve |x – 4| = 9.
Settle down—no need to get all worked up over the extra stuff in the middle. To deal with it, we treat the "x – 4" expression as one whole entity. As before, we can rewrite the expression in two different ways:
|-(x – 4)| = 9
|(x – 4)| = 9
We drop those bars like they're hot and solve for x.
|-(x – 4)| = 9
-(x – 4) = 9
x – 4 = -9
x = -5
Oh, you again. Can we get that high-5 now? No? Okay, moving on.
|(x – 4)| = 9
(x – 4) = 9
x = 13
If we plug x = -5 into our original equation, we get |-9|, and if we plug 13 in, we get |9|. If we try plugging our phone charger, we won't get a thing.
We've successfully found the unknowns inside of the absolute value bars. Maybe we weren't psychic this whole time. Guess we won't be getting in the X-Men after all.