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.
Sample Problem
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.
Sample Problem
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?
Sample Problem
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.
Sample Problem
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.
Summary
Faced with an absolute value inequality and don't know what to do?
- If the inequality has a "less than" sign (<) with the absolute value expression on the left, then the solution is a compound "and" inequality between ± a constant.
- -n < (something) < n
- If the inequality has a "greater than" sign (>) with the absolute value expression on the left, then the solution is a compound "or" inequality.
- The first half is the original expression with the absolute value signs removed.
- (something) > n
- The second half flips the inequality sign, and changes everything outside the absolute value bars to the opposite sign.
- (something) < -n
- The first half is the original expression with the absolute value signs removed.
- If you can't remember these rules, or you're just not sure what to do, draw out a little number line. Looking at it will keep you from messing up.
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.