 With all these tools that we've just acquired to solve equations, there is, you know, certain restrictions, certain limitations that we have. Now in mathematics, limitations or what we can't do is refer to as restrictions. So what we're going to do is take a look at the restriction that we need. One, the main restriction actually, that we have in solving equations. And it's really related, and it is related to the real number set. If you remember, you know, one of the first videos that we ever did was talking about the real number set and the boundary between natural numbers and whole numbers. And that boundary is basically us introducing the number zero in our vocabulary for the language of mathematics, right? You know, initially we had the natural numbers which was one, two, three all the way up to infinity. And then, you know, we were able to define the number zero. And for this new discovery, we created a new subset, the whole number set, which included the natural numbers, and in addition it included zero, right? And when we're solving equations, this zero comes up. Zero comes up basically throughout mathematics. It's something that we haven't been able to deal with completely. So, you know, it's going to create problems in a lot of things that we do. What we have to do is understand what those problems are, what those limitations are, what those restrictions are, right? And, you know, that way we can understand how far we can go with things, right? You know, what we're capable of doing, what the problem when it comes to zero. As far as we know, as far as we're concerned, you know, you can't divide by the number zero, right? Because when you divide by the number zero, our equations explode. Basically, you get infinity. You know, we get it not unknown, okay? So what we have to do when we're solving equations is make note of where we have restrictions, where we have limitations. We have to know our limitations when we're solving the equations, okay? And in mathematics, a lot of courses initially, when you're solving equations, they're going to ask you to give the restrictions to your solution. Let's say we have something like, we're going to use their chalks, the little chalks. By the way, I've called them near the classroom at UBC, okay? So hopefully there won't be anybody coming in and we can work here a little bit. So we're going to use a little chalk and an actual blackboard in a classroom to do these videos. So let's say we have something like this, five over three. Let's say we had this equation and they asked us to solve this equation, okay? Before, this is a general rule, you should always try to do this. Before you start crunching numbers, moving numbers around and trying to solve an equation, you should take a look at an equation and list your restrictions, okay? The reason you want to do this is because at a certain point when you're crunching, you know, when you try to solve this equation, you might start losing some of the solutions, okay? So you have to keep note of your restrictions. As we stated in mathematics, our restriction, one of our restrictions that we're going to have is we cannot divide by zero because the equation explodes, right? So what we do with this equation, whenever we get an equation, if there's a fraction involved, all we do, we grab the denominator, right? And we say we can't divide, we can't divide by zero, zero. So what we do is we solve for this, not equaling zero, and that will be our restriction, okay? So what we do, we take the denominator and we say 3x cannot equal zero. Because if 3x is equal to zero, that means the denominator is equal to zero, and we're dividing by zero, we cannot divide by zero, okay? So if you divide, you divide by three, you divide by three, three equals three, so x cannot equal zero, okay? And this is our restriction when we're solving for this equation. Now this side also has an equal side, and this side also has a fraction, so you also have to take a look at this side and say, what's the restriction here? Now the restriction here is exactly the same as the restriction here, because if you go 6x cannot equal zero, you divide by six, so x cannot equal zero. So you get the same restriction for this side of the equation, okay? So before we start crunching this equation, before we start solving this equation, we always try to list our restrictions. And what I personally do, and I think this is the general standard of the way things are done, is I always list my restrictions on the side, and what happens when you get more complicated equations, this is going to, this is a straightforward equation, and there's only one restriction to this. When you get large equations, restrictions might pop up half way through your solution, okay? So what I end up doing is listing my restrictions on the side, on the right side of, you know, the piece of paper where I'm doing the work. And at the end, I look up the right side, and for my solution, I always list the restrictions, I just add them all up at the bottom, okay? Not add them all up, or list them all at the bottom. So over here, I would say our restriction is x cannot equal zero. As we stated, our restriction is, you know, comes out with the number zero, right? We cannot divide by zero. So before you would go ahead and start solving this equation, what you need to do is take the denominators from both sides, and if you had about a term here, let's say you have two over x here, right? You have to take the denominators in every single term and say that cannot equals zero. So for example, for this one, it would be, let's do this in blue. For over here, you would go five x minus one cannot equals zero. So five x cannot equal one, you just grab the one, bring it over, right? Divide by five, divide by five. So x cannot equal one over five, okay? That's one restriction. That's for this term only. You have to deal with this one. For this one, you take each term and you say cannot equals zero. You set the whole thing cannot equal to zero, so you're going to go x plus one times two x minus three cannot equal zero. You've got two things multiplied together to give you zero. That means each one you solve for not equal to zero. So x plus one cannot equal zero. Two x minus three cannot equal zero. Grab the one, bring it over. So x cannot equal negative one. And two x minus three cannot equal three. So two x cannot equal three. Divide by two. So x cannot equal three over two. So those are three restrictions so far we have for this equation. And we also have over here. Well over here, it's just straightforward, x cannot equal zero. So before you even start looking at solving this equation, what you would do is write down all the restrictions on the right-hand side here. And all your restrictions are x cannot equal zero, x cannot equal three over two, x cannot equal negative one, and x cannot equal one over five. And I'm just going to jump ahead a little bit for anyone who's gone far enough. These are going to be our vertical asymptotes if we're graphing the function. So these are unknowns because if we set x equal to any single one of these in the above equation, we're going to have to divide by zero because we don't know how to divide by zero.