Beg Algebra: Solving Absolute Value Inequalities

This lesson has not been reviewed.

Please purchase the lesson to review.

Page 1 of 2 www.thinkwell.com
SOLVING ABSOLUTE-VALUE INEQUALITIES
I want to start thinking about inequalities that have absolute values in them. This actually requires a little bit of discussion and really to think back to what absolute value means. Remember that absolute value is a measure of distance, how far you are from something. So, what does it mean if I say to you that the A, which is some number, is less than B. What does that mean exactly? Well, it means that A in absolute value is pretty small. It’s less than B. So, graphically, if here is 0 and if here is B, then therefore, -B would be right over here. If A < B, then A would have to live somewhere inside here. See that? Because all the values in here, positive or negative, have absolute values that are less than B.
But once you escape, once you go outside, then the absolute value, no matter where you are, would exceed B. So, if someone gives you an inequality with absolute values of this form, what you have to realize is that you can actually restate that inequality without any absolute values at all. What would it be? Well, it would look like this. It would just be all the points that are in here. So, A would be between B and -B. In particular, I’d write -B < A < B. So, the cool thing here is that this kind of statement is identical to this one. You just make a conversion. This thing we know how to solve. We talked about that earlier. So, if you see something like this you can go to here.
What if you see something that looks like the opposite? Suppose you see something that looks like this, A > B. What would that mean? That means that the size of A is actually large. It’s even beyond B. So, what would a picture look like? Let’s draw a picture for this. If here’s zero, if here’s B and then therefore, here would be -B, if A > B, that means that A is far away from zero. It’s going to be way out here or maybe, since we’re taking absolute values, way out here. So, now A would live in this region here. So, in this case we would say that A would live way out here or way out here.
So, if you see an inequality like this you can actually convert that to two different inequalities. One inequality that looks like this, which would be what? This would be that A > B. That’s what this is saying. A is somewhere bigger than B or, A < -B. So, when you see an inequality that has an absolute value bigger than something that actually can be converted into two different inequalities, something or something else. So, then you have to solve these two inequalities together to find the solution to this. So, the moral so far is that if you see an inequality, which has absolute value less than something, you convert it to an inequality that looks like this. That thing is tracked between the something and negative the something.
If the absolute value of something is bigger than something else then you have two inequalities to solve. That thing is greater than the something else or that thing is smaller than negative the something else. Don’t memorize this. Think of the little chart. Think what it means for an absolute value to be small. It means that it’s near zero. Think what it means for an absolute value to be big. It means that it’s far away from zero. Then write down the inequalities that would correspond to these things. So, this is the idea. Let’s take a look at some examples.
Let’s solve the following, 3x + 1 7. Okay, well how would this look? Well, I see an absolute value is bigger than or equal to something. So, how do I think? Do I memorize that? No, I’ll tell you exactly how I do it. This is just me now. You don’t have to do it this way. I actually draw that little picture I just showed you. I say, “Okay, let’s think about that.” If the absolute value is bigger than or equal to 7, where does that put me? If here is negative 7 and here is 0 and here is 7, I’m living way out here or way over here. I’m far from 0. So, this means there’s two inequalities. There’s 3x + 1 7, that’s this wing right here, and there’s this wing, which says 3x + 1 is actually ??? -7. So, actually, if I were any good I would have had these wings written the other way.
So, this wing right here corresponds to this and this wing right here corresponds to this. So, this one inequality with absolute value turns into two inequalities without absolute values. Just like with equalities, remember? One equality with absolute values gave rise to two equals to solve. The same thing here. Well, now we can solve these things. I’ll bring the plus one over to this side as a negative 1. So, I see 3x 6. Dividing through by 3, I don’t have to switch the signs, since 3 is positive. I see x is greater than or equal to 6 over 3, which equals 2. So, x 2, we have that, or now we have to solve this one. So, I bring the 1 over to this side. So, I see 3x ??? -8. So, I see x ? - 8/3. I don’t have to switch the signs since I’m dividing by a positive 3. So, there’s the other wing. © Thinkwell Corp.
Page 2 of 2 www.thinkwell.com
So, these would be the answer. So, it’d be x is either greater than or equal to 2 or x is less than or equal to minus eight-thirds. Any one of those would actually give an answer to this that would be true. Namely, if you pick any number that’s bigger than or equal to 2, that will make this thing bigger than ore equal to 7. If you pick any value of x that’s less than or equal to minus 8/3, that will produce a value here that’s bigger than or equal to 7. So, two answers, and both of them are inequalities.
Let’s try one last one together. Suppose I tell you that 64x+ is strictly less than 1. What’s my thinking here? Well, now I’m saying the absolute value is small. So, that means I must be near 0 somehow. So, I draw a little thing with 0. I put in -1 and then 1, these are the two values, the negative and positive. Since I’m smaller than 1, I must be living in here. So, that means I’m setting up an inequality that says that -1 is less than this thing, which in turn is less than 1. Notice how that little sandwich inequality actually captures this little sandwich interval here. Those are all the values that satisfy this. Well, now I just solve this compound multiplex inequality by, just remember, doing whatever I do on one side, I do to the two other sides.
I want to get rid of the four so I subtract four everywhere, -1 - 4 is -5. I keep the inequality exactly as it is, 6x, this is 0. Then 1 - 4 is -3. I want to now divide through by 6. It’s positive, so I don’t have to worry about changing the inequality symbols. Then I see -5/6 > x > -½. So, that’s the answer. What does that mean? It means that if you pick any value for x that’s bigger than -5/6, yet smaller than -½, so any number in between that little interval, any such number will actually satisfy this and make this true. Whereas if you pick any number outside of that region, this will be false. So, the solution to this inequality is actually this inequality. It’s this collection of x’s. Any x in this region will actually satisfy that. Okay, great. So, that is some discussion of inequalities with absolute values and I’ll say some more about this up next. I’ll see you there.
© Thinkwell Corp.