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Which college will you attend? Who knows. Will that cutie in your history class ask you to prom? No clue. Will it rain the night of homecoming? Hopefully not, but it could. Will you beat every level in Peggle? Possibly, but probably not.
The point is that few things in life are absolute—except absolute values, that is.
The absolute value of any number is how far away the number is from 0. If you walk 5 blocks west from your house, you will be 5 blocks away from your house. If you walk 5 blocks east from your house, you will again be 5 blocks away from your house.
This sort of thinking can also be applied to the absolute values of real numbers. The number 5 is 5 units away from 0. Similarly, -5, which is on the opposite side of 0 on a number line, is also 5 units away from 0.
To show that we're looking for an absolute value, we put two little bars on the outside of the number or expression, like this: |5| or |-5|. Those bars make everything inside turn positive, so |5| and |-5| both have the exact same value.
|5| = 5
|-5| = 5
Absolute value. Is it cool? Absolutely. You probably saw that one coming from |±20 miles| away.
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.
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.
Absolute value expressions don't need to be absolutely equal; they can have inequations, too. Er, inequalities. Things get a bit weird when we have absolute values and inequalities in the same room together, though.
Solve |x| < 1.
If this were an equation, we would have this in the bag: x = 1 or x = -1. But think about what an absolute value actually means. It's a number's distance away from 0. If we have |x| < 1, we want all of the numbers whose distance from 0 is less than 1.
Yep, that's them. On the number line, we can see x = -1 and x = 1 are the boundaries of this compound inequality, -1 < x < 1. Put another way, that's x > -1 and x < 1. Or a third way, the solutions to our inequality are between ±1. Do we need to put it a fourth way? Hopefully not, because we're tapped out.
This is a pattern that repeats itself for a lot of absolute value inequalities. If we have |(some stuff)| < some number n, then our answer is -n < (some stuff) < n. Then, we solve for our variable. Not n, the other one, inside the stuff. We didn't show it here, but you know it's there.
Solve |2x + 3| ≤ 5.
The partial equality of the sign doesn't change anything. A lot like that cheap motel we stayed at on our road trip last summer. We get the willies just remembering the sheets at that place.
We have a "less than" sign of some kind, so any solutions we can find will be between -5 and 5.
-5 ≤ 2x + 3 ≤ 5
Yeah, like that. Now we can solve for our variable. Let's start by subtracting 3 from each section.
-8 ≤ 2x ≤ 2
It's time to divide out the coefficient from the variable. Our palms have started to sweat; how about yours? They should, since we're dividing with inequality signs around. A 2 is positive, though, so we're all good. Wiping our hands on a towel and dividing gets us our solutions:
-4 ≤ x ≤ 1
This is all well and good, but what do we do when the absolute value is "greater than" a number? The answer won't look the same as the "less than" case. It can't be simple, can it?
Solve |x| ≥ 6.
Let's think about this in terms of distance again: we want all of the numbers that have a distance greater than or equal to 6 units away from 0. Once we have all those numbers, we'll put them in dresses and have a tea party.
Looks like our solutions can be less than or equal to -6 or greater than or equal to 6. This is another compound inequality, but it's joined by "or" this time:
x ≤ -6 or x ≥ 6
It may seem strange that numbers smaller than -6 would be solutions to a problem that starts with a "greater than" sign, but plug in a number and check it: is |-7| ≥ 6? Yeah, |-7| is 7. And 7 is definitely bigger and scarier than 6, because 7 8 9.
Solve |-2x – 1| > 7.
The first thing we do is stand up and stretch. We've been working on these inequalities a long time, and it's killing our back. Once we've done that, we check the inequality sign. It's a "greater than" sign, so we know we'll have a compound "or" inequality. One of the inequalities will be our original expression, just without the absolute value bars.
-2x – 1 > 7
-2x > 8
x < -4
Did you see that? Negative division—we have to change all of the signs. All of them, including the inequality sign.
The other half of the solution will be the opposite of what's inside the bars.
-(-2x – 1) > 7
-2x – 1 < -7
We've got more negative multiplication. Flip that inequality, flip it good.
-2x < -6
x > 3
And then we have to change it back again. Sheesh, make up your mind already. Do you want to point left or right?
Our full solution is x < -4 or x > 3. We're a little nervous about our answer; there was a lot of sign flippage going on there, and we've heard some unsettling noises while he's been in the other room. So let's double check.
Plugging in -5, which is less than -4:
|-2(-5) – 1| > 7
|10 – 1| > 7
|9| > 7
Plugging in 4, which is bigger than 3:
|-2(4) – 1| > 7
|-8 – 1| > 7
|-9| > 7
Okay, everything's cool. And it turns out that those noises were just his out-of-tune violin. We thought he was trying to summon a bunch of spiders. We were this close to calling in an exterminator.
Faced with an absolute value inequality and don't know what to do?
Quick question: what happens if we try to find |x| < -1? Or |x| > -1? What happens is, "It's a trap!" An absolute value will always be positive, so there's no solution. Any value of x will be greater than any negative number, and it can never be less than a negative number. Don't fall for it, or squids in space suits will keep yelling at you.