College Algebra: Solving Compound Inequalities

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So solving one inequality where you just have x's and some constants in it are not too bad. You just treat it like a regular linear equality and just be careful of the bookkeeping with that sign, the inequality symbol flipping back and forth. But what if you have more than one inequality that you want to consider at the same time? These are sometimes called compound inequalities. So you're given a couple of inequalities that you want to look at together. And there are two basic flavors of them. There's the kind where I say solve this inequality and this inequality and the other type is solve this inequality or this inequality. So what will distinguish those two types is the and or the or and I'll show you exactly what those differences are. Let's take a look at some very simple examples just to get in the spirit of this.
Suppose I say to you that x 2 and x is > than -1. So what does that mean? What's the solution to both those things together? Now what does the and here mean? Well, and means that these both must be satisfied at the exact same time. So and means they both must hold simultaneously. In some sense you can think of it as like the intersection. Where does this answer intersect with this answer? That intersection is where they both happen at the same time. So a lot of times you should think of and as intersection. Or if I were the Cajun mathematician, it would be intersection. I don't know if you've ever seen that. It's a great show. Anyway let me now show you how you would find this intersection. The best way to do it is to graph it. So if we graph this on a number line, all I would do is put the relevant points like -1 is over here, 0 would be over here, but I'm not even going to mark in 0. 1, 2, I'm just going to put in the 2 point because that's the only ones we really need to look at here. What's the graph of the first solution here? X 2. That means all the points to the left of 2, but I'm also allowed to equal 2. So, in fact, I put a right bracket that looks like this which means I'm allowed to equal 2, and then all these points here. Those are all the points that will satisfy this first inequality. Now what about the second inequality? Well, the second inequality says x is going to be greater than -1. So that means all the points to the right of -1, but since it's a strict inequality, I'm not allowed to equal -1. So I'll put a parenthesis there and then put this there. And those are all the solutions to this.
Now where are these both satisfied at the same time? When do we have this sort of cyan color and green color happening at the same time, this and that? Well, it's the intersection. It's the intersection of these two things, which you can see is this thing right in here. So I could say it this way. I could write it as it's all the points that are between -1 and 2. And notice that I include 2 because 2 is part of the intersection. The green actually contains that point, but notice that the green does not contain -1. -1 is the point they don't have in common. It's in the cyan one, but it's not in the green one. So it's not in the intersection. That's what and means. And means intersection. Now the other possibility is that we look at something that is an or kind of problem.
Let me show you an or kind of problem. Let's consider the following collection of inequalities. x < -1 or x is bigger than 5. Well, what's the solution to this look like? Well, what does or mean? Or means that a solution could be a solution to one or the other. It doesn't have to be both. It could be just either one or the other. So in that case, that's much less confining. I'm just going to mark on the number line my points that I need, -1. I'm not going to mark 0, 1, 2, 3 or 4. I'll just mark the 5 here. x < -1. Well, that means that I'll put in something that looks like this anything to the left of -1, but I don't include -1. So I put that in like that. That's this picture. And this picture is anything bigger than 5. So that would be this picture. Notice those two people don't overlap at all. They have nothing in common. But I'm not asking for a common solution. I'm asking for a solution to either one of them which means that the solution set is actually all of this put together. Any of these things will satisfy this system of inequalities because if I pick anything in this region, like suppose I pick 6.
Does 6 satisfy this? Well, this says either 6 < -1 or 6 is bigger than 5. And notice that is true because, in fact, one of these is the case, namely, 6 is bigger than 5. So for an or statement, they don't both have to be true. They only both have to be true if you have an and connective. But if you have an and or connective, all you need is one or the other to be true or maybe both. It doesn't make a difference. So now how would you write this answer? Well, now you could think of or not as intersection, but you could sort of think of it as putting it all together, taking the union. This is sometimes called a union, like uniting all different kind of solutions together, bringing it all together at the very opening.
So how would I write this? Well, this thing here I could write that as all the points from -up to -1 but not including -1. That's why I have this open parenthesis and then I've got all these points here. I'm also going to have to consider all the points bigger than 5 up to infinity. Now how do I show that in fact it's this or that? Well, I say I'm going to take the union which means this thing and then also this thing. Put them both together. I don't care about the intersection. I can take this or that and so sometimes we denote that by putting a u here to stand for union. And what that means is that if something is in the union, it's either here or it's here. So this thing, this symbol means the union of this interval with that interval which means all the points here with all the points here union together. So,any point where my fingers are all in the union. These points here are not in the union. They're not in the union. So or means you put intervals together. And means you look for where they overlap.
Let me try one little last example. Suppose I say to you x 3 and x < 0. And means these both have to be satisfied at the same time so I'm looking for an x that will do both of these at the same time. Well, let's try to graph this and see what it looks like. So here's 0, 1, 2 and I'll mark 3. x 3 so that means I'm allowed to actually include 3 since I'm allowed to equal 3, but then I'm all the points bigger than 3. So there's that. And at the same time, I have to be less than 0. So this is the less than 0 one and this is the 3 one. And the and means I have to look for the overlap. Where do these overlap? Well, they don't overlap at all. There is not one value for x that will satisfy this at the same as satisfying this because these don't overlap. And if you think about it, that's right. You can't have a number that's negative but also be bigger than or equal to 3. It's impossible. So what's the answer to this?
The answer to this is there are none so no x. Sometimes we call this the empty set. Nothing is in it. So sometimes it's called the empty set and sometimes we denote the empty set as, well, you want it to say 0, right? But 0 is actually a number. So to denote an empty set, we take 0 and put a line through it. That means nothing. It doesn't mean 0 because 0 is a number. This means nothing at all. So in fact, sometimes there may be no solution because we're putting so many restrictions on a system of inequalities, that in fact, no one can solve it. You can't both be negative and bigger than 3 at the same time. You're just demanding too much. Why are you being so demanding? Well, you're being less demanding than you can actually find solutions and we'll take a look at some of those coming up next.
Equations and Inequalities
Solving Inequalities
Solving Compound Inequalities Page [1 of 2]