 While we're on the topic of restrictions, let's do a couple more harder problems so you guys get a pretty good idea of where this goes, which is basically our restriction is you can't divide by zero, so whatever is in the bottom of a fraction, you take that and set it equal to zero or not equal to zero, and that's going to be your restriction, right? All you do if they ask you what the restrictions are for this problem, or even if they ask you to solve this problem, what you should do is initially the first step you should do is find your restrictions and make a little note of it on the side of the paper so you know what your restrictions are. So after you solve this equation, when you get to the bottom, you can cross-reference your restrictions with your solutions, that way any of the restrictions appearing in your solutions can knock them off, right? So right now let's just assume they ask you what would the restrictions for this equation be. So let's find your restrictions. All you do is take the denominator and say that can't equal zero. So all you do, you go x plus two can't equal zero, square root of x minus one can't equal zero, two x squared plus one can't equal zero, okay? Those are your restrictions, but obviously you're going to have to solve these, you know, not equations, but not equal to equations I guess. So all you do for this one is just grab this, x cannot equal negative two, that is a restriction, right? Over here you grab the one over, so you get square root of x is equal to one, and the way you get x by itself here, this is square root, you do the opposite which is square, right? So you square both sides, so x is equal to one squared is just one, right? Over here, so you got your restriction is going to be, oops, this should be a non-equal two by the way, so this is x can't equal two, negative two, x can't equal one, and over here we're going to do the same thing, move the one over and solve for the x, right? So you got two x squared can't equal, bring the one over, it's negative one, divide by two, divide by two, you got x squared can't equal negative square root of, negative one over two, okay? And over here what you do is you square root both sides, right? Now on this side you're going to get square root of x squared, x can't equal square root of negative one over two, is that still in the word, that's still in the word, square root of negative one over two, but again that is another restriction that we have which with the real number set anyway, there is a way to work ourselves around this, and we will get into that later on in series that are going to come up, right? These are called complex number or imaginary numbers, and that's the square root of a negative number, so for us what that means is there are no restrictions, because we can't take the square root of a negative number, so for this there's nothing we could put in here to make this denominator equal to zero, and that should be, after you look at this and sort of consider what's being done here, this is two times x squared plus one, right? The only way we could get the denominator to equal zero, if this part of it was equal to negative one, right, because it would be negative one plus one that's equal to zero. Now what's happening here, this is x squared, we can't, if you put a negative number here, because the only way you can turn this negative, if you have a negative number here, right, because that's positive, two is positive, if you put a negative number here, negative number squared is always going to be positive, so there is no way for us to make this term negative, that means there is no way for us to make this, this denominator equal to zero, so this fraction doesn't have any restrictions, okay? So as far as restrictions goes here, there are no restrictions here, if you were dealing, if you were in higher level mathematics, when you're going beyond, outside of the real number set, when you go into the imaginary number set, there would be a solution to this, and the solution to this would be x is equal to, can I write this here, that's right here, this would be x is equal, cannot equal one over two i, okay? So for anyone that's done imaginary numbers, i represents the square root of negative one, and oops, sorry, let's put a square root here, so it would be one over square root of two i, because yeah, you still have to take the square root of a half to bring it out, right? So this would be basically i on top of i over square root two, and that would be our restriction for this equation, for that fraction, I just thought I'd like to text with you. Oh, okay, yeah. Is that the textbook right there on that? Oh yeah, that's it? Awesome. They're expensive. Yeah, yeah, yeah, too much too. Yeah, well thanks for helping me fight it, man. Oh yeah, no problem. Later. I guess he thought I was a teacher or something, or a TA. Good, good, good, good, good. I think the fear helps. Anyway, so this is a harder problem right now, right? So we got two x minus one over x squared plus five x plus six is equal to x minus two x squared minus five x plus three. Let's take a look at this. Even if you get something like this more complicated, again, one more level, it's not really in order of magnitude, it's one more level more complicated. We've gone into quadratic equations now. Now the way you solve for this, you find the restrictions. Again, all you do, you take the denominator and say this can't be equal to zero. You take this denominator and say this can't be equal to zero. Now these two might look similar, but there's a difference here. That's a negative number and that's a positive number. And there's the solutions to this are going to be different. So this is x squared plus five x plus six can't equal zero. And the way you solve for this is you have to factor this. This is a simple trinomial. For anyone that's done this, you know, you should know the answer to this. What you're looking for is two numbers to multiply to give you six. And those same two numbers have to add up to give you positive five. And the way you do this is you want to break this down into two things multiply together. So all this is going to be is going to be xx plus two plus three. And that's this quadratic equation factor. And what you do, you take each one of these not equal to zero. So all you do is go x plus two, can't equal zero and x plus three. And those two are going to be our restrictions for this side of this equation. And for the other side, you're going to do the same thing. Your answer is just going to be different. Over here, you're going to take this same thing again and say this cannot equal zero. And the way you solve for this is you factor in exactly the way you did here. But this time you're looking for two numbers that multiply to give you positive six. Remember the sign in front of the number goes to the number? So you're looking for two numbers that multiply to give you positive six and have to give you negative five. Over here, you're positive five. Well, it's the same two numbers. It's just their signs are different. So it's going to be can't equal. So it's going to be x times x minus two minus three. Over here, you're positive plus three. If you go negative two times negative three, you're going to get positive six. Negative two plus negative three equals negative five. Each one is not equal to zero. So this equation has four restrictions. Simple as this. And it grows from there. You could throw anything in the bottom. You're always going to have to do some kind of analysis on this and find out what your equation can't be solved for. And when you graph these things, you know, they're going to end up later on when we get into functions. These are basically one function equals another function. That's what we have here right now. This equation, really. So one model equals another model. You're comparing them. You know, you can overlay them and change the models. Basically, that's what you're doing with mathematics. The power of it is you take one model and another model and you can incorporate them together and see how those models would work together. With restrictions, the way it works is you end up either getting asymptotes or you get holes, you know, a certain place that an x can't be and you don't have a well, you can get an x but there is no associated y with it. So the function collapses. And these are what our restrictions do is if this side of the equation is a function and that side of the equation is a function is a model, if we're modeling something in the real world or if we're modeling something financial, right? We're comparing two different models. So what happens is when we do this kind of analysis or we, you know, solve these kinds of equations, we find out what our models cannot be, what they don't have a solution for. You know, we can have an x value and but the y value associated with this x, there is none. Our models basically collapse when we approach these restrictions and these restrictions are either going to be asymptotes or they're going to be holes which is basically not known. So an asymptote basically means you can't cross it. You can't, you know, there's no direct link from one side to the other and a whole means nothing exists associated with a certain x value and that's why our restrictions are super important because it gives us our limitations of what we could do with the language of mathematics and using that language, you know, what our restrictions will be for a specific type of model, for a specific type of function. Super important stuff and the higher level mathematics you go is the more you start looking at your restrictions and what your limitations are in the language of math and, you know, you get into more proofs and trying to expand our vocabulary to incorporate some of the unknowns right now that we have.