Int Algebra: Solving Quadratic Inequalities

There is a reason I was not a math major.

12/31/2008

~ brittanie

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Quadratic Equations and Inequalities
Nonlinear Inequalities
Solving Quadratic Inequalities Page [1 of 2]
So solving inequalities where we just have x’s, you know, sort of linear inequalities. It’s not that awful of a thing. You're just really careful with the switching signs when you're dividing or multiplying by a negative, but outside of that pretty much everything else is straightforward. What’s a little bit more a challenge is if you want to solve an inequality that is a quadratic so there are x squared’s appearing. And this really is a challenge until you sort of feel comfortable with the idea. So let’s walk through one really carefully and see how you would solve a quadratic inequality. Suppose I want to find all the solutions to 2x2 + 5x - 12 ? 0. Well, how would I do that?
The first thing that I would try to do as I would if this were an equal sign is factor this thing. So let’s factor and let’s hope, in fact, that it can be factored. And let’s see. So I'm going to try a 2x here and an x here. This tells me that both signs will be opposite but I can’t just cavalierly put in plus minus or minus plus because these aren’t the same. So I'll have to do a little bit thinking in seeing how I am going to actually get the combination to work out just right. I'm going to want two numbers that multiply to give 12, but combined with this 2 to actually give me a +5.
So let’s see. How about if I think about 4 x 3? What if I put the 4 here? Let’s think about this together and the 3 here. Here I'd get a 3x; here I'd get an 8x. If I subtract that just right, I would get the 5. So I think what I want to do is put a plus here and a minus here and that seems to work. And you can check that and see if this really does foil out to give me this.
Well, now I've got, you know, sort of this going on here and what I have to do now is figure out when is this positive or, in fact, greater than or equal to 0. Well, here’s what I'm thinking this. Look, I've got two numbers and they’re product is bigger than or equal to 0. What are the possible signs for these numbers? Well, there’s two possibilities. Either this number is positive and this number is positive or the other possibility is this number is negative and at the same time this number is negative. So actually there’s two different cases we have to consider. The first case is positive, positive and the second case is negative, negative. Well, now how should we proceed here?
Well, an easy way of actually putting together the signs of these things is actually to make what’s called a sign chart. And a sign chart basically means just putting pluses and minus wherever these things are positive or negative. So the first thing I would do is find out when these things actually equal 0 and that’s easy to do because for this thing to be 0, I just would solve this and find out when that’s 0. Let’s do that really fast. So I would have—oops! I'm going to have a 3. I'm going to have a 3 later in life, but not now.
Let me start out here. I would have a 2x - 3 = 0, so where does that equal 0? Well, 2x would equal 3 so x would equal 3 over 2. So this is going to be 0 at 3 over 2. When is this 0? Well, that’s easy to see. That’s going to be 0 when x = -4. So if I make a sign chart. I'll just mark down these points here. I have -4, that’s where this is 0 and then I've got 3 over 2 way over here and let’s see. If I put little lines here like this, I know that if I plug in -4 into here, that’s going to give me 0 and why? Because this term is 0 and 0 times anything would be 0. So I know that this thing is 0 here. And what about if I plug in 3 over 2 into this? Well, I know that’s going to be 0, too, because now this term will be 0 and that term multiplied by anything will still be 0. So, in fact, I know this thing is 0. But in these intervals here I don’t know what’s going on. So how would I find this out?
Well, all you have to actually do is pick just a random point in here and see what the sign of this is going to be. It’s either going to be positive or it will be negative. So let’s do this one here in this interval right here. So all I'm going to do in this region, do you see what happened? I cut the world up into these three pieces. This piece, this piece and that piece. I cut up by seeing where this thing equals 0. So that’s sort of the balance point and now I'm going to look at each of these regions and determine what the sign of this thing is. I'm searching for where that thing is positive and I'm going to see where that is. So all you have to do is literally pick a random point in between. So what’s a good point to pick? Any number will work between -4 and 3 over 2. Well, a good point between -4 and 3 over 2 is 0. That’s in here. Suppose I take 0. If I plug in 0, which would maybe be like over here, if I plug in 0 for x, what would I see? 0 + 0 - 12. So this thing here would equal -12. So that means that 0, this thing is negative. That means it’s going to be negative no matter what point you pick in here. So all this is negative region. Negative, negative, negative, negative, negative region. And I found out it was a negative region because I just picked one point, an easy point, plugged it in and saw what the sign was.
Quadratic Equations and Inequalities
Nonlinear Inequalities
Solving Quadratic Inequalities Page [2 of 2]
Now what would I do over here in this region? Well, again, I'd pick a point. I could pick any point at all. It doesn’t matter. Let’s see. Three halves is 1 1/2, I need something bigger than that. How about if I just picked 2? So I'll pick 2 as my point and I'm going to plug 2 into here. Now you could plug 2 into one of two places. You could either plug it into the original thing, 2 x 22 + 5 x 2 - 12 and compute what that number is. All you care about is if it is a positive or a negative. But I'll tell you what I'd do. I'm lazy and remember, the good mathematician is the lazy mathematician. I just take 2 and I'm going to plug it into these things and multiply it and It’ll be easier. Watch what happens. If I plug 2 into here, that gives me 4 - 3. Now I know that’s 1, but all I care about is is it positive or negative. So I'm going to say, well, it’s positive and I'm going to forget that it’s actually 1. I just keep in my head positive.
What happens if I put in 2 here? When I put in 2 here, I see that 2 + 4 is 6. All I care about is positive. This is positive and that’s positive, what’s their product? It’s positive. So at 2 this thing is positive which means that everywhere from this point onward, this is positive land. This land is positive land, this land . . . now what about here in this region? Well, I have to pick a point, you know, way over here and you may think we’ll only pick like -5. We can pick -5 if you want, but let me actually show you that you can be quite dramatic. Let me pick -100 just to show you that it doesn’t make a difference. All I'm concerned with is just the sign. So if I plug in -100 here, will this be positive or negative. Let’s see. -100 x 2 is -200. -200 - 3 is actually -203 but all that matters is minus. We don’t care about the actual value, just the sign. This is negative here.
Now what happens when I plug in -100 in here, I have -100 + 4. Well, that +4 helps a little bit, but that thing is mighty negative, right? In fact -96. So this is negative. Negative times a negative, what is that? That’s a positive something. Well, if that’s positive that means this whole land is positive. And so now I see exactly the sign chart for this particular function. So all I want to do is know where it’s positive. I want to know when it’s bigger than or equal to 0. Well, I could read it off the chart. It’s positive on this part of the wing. See these two wings here? Am I allowed to actually use these end points? Well, these end points are where the thing actually equals 0. Am I allowed to equal 0? Yes, in this case I am allowed to equal 0, so in fact, my solution would be, I can actually show you the solution set. It’s going to be equaling -4 and then all this positive land or equaling three halves and all this, I'm sorry. All this negative land, which makes it positive or this land here. So that’s the graph of the solution. You could actually use that as the graph of the solution. Some people actually would accept the answer or want the answer in this form. So, in fact, the green actually represents the answer. If you wanted to write it out in interval notation, how would that look? I just would copy this down -out to -4 including the -4,but never including the -?? and then I pick up the action at three halves with a bracket because I'm including that out to +? and then since I'm taking one or the other, I take their union.
So this is how to write the answer in interval notation and this is how to write the answer sort of graphically, but notice all that’s happening here is I'm finding all the values for x, which make that positive. Great. I'll show you another one up next and you can see sort of how this thing all plays out in general using a sign chart. So try to get this into long term memory because it is really tricky. The thing to remember is always factor, make a sign chart putting down where the function actually equals 0. That which will separate the positive land from the negative land, because that’s where it switches and then just pick a point in each region and find out what the sign is and then presto, you can report your answer. I'll see you up next.