 Let's do a problem, an equation, where we have the x in the denominator, just a little bit more complicated to get with the little twists. I'm just going to use the same numbers, I'm just going to use the same numbers as the last equation. All I'm going to do is just flip the axis. So we've got another equation and all I've done with this equation is flip the x terms. So we've got instead of having 2x over 3 to the 3 over 2x as compared to the last question as we did. So what we're going to do is let's solve this equation again using both methods as multiplying by a common denominator, the whole thing by the common denominator, or doing cross multiplication. Let's do the first method, which is multiplying by the common denominator and getting rid of the fractions right away. So the common denominator between all these terms, and those are just over 1, right, 2 over 1, 1 over 1, so we don't care, right, we don't care. Between x to 2x and 3, the common denominator is 3x, and again, if you're having a hard time with this, go to series 1, because, and look at fractions, and how to add fractions, multiply fractions. If you don't know that, it's useless to listen to these lectures right now. You have to get that stuff down, Pat, before you continue on with what's coming. You've got to learn your fractions. So you multiply this whole thing by 6x, right? Now, x eliminates the x term there, so all that multiplies this thing, it's just 6. Again, let's do this one more time on the side, but the last time I'm going to do this on the side, because it's just every single term, that's what's going to happen, right? So if you have 1 over x times 6x, right, the x is going to kill the x, so this is just going to be equal to 6, right? And you do this thing with every single term, okay? And this 6x doesn't change, the x doesn't disappear when you go to the next term, right? It's still there, it's just each term you deal with individually. So over here, this just becomes 6 minus 2 reduces 6x down to 3x. 3x times 5 is 15x, minus 6x times 2 is going to be minus 12x is equal to 2x reduces this down to 3, so it becomes 9, minus 3 reduces 6 down to 2, so it's 2x multiplied by minus 6x, right? And again, what you do is you combine your life terms on both sides, and you combine up your useful side. So you've got 6 minus, negative 15x minus 12x is negative 27x. 6 minus 27x, and negative 2x minus 6x is not 8x, so 9 minus 8x. Bring all your x's to one side, take your numbers to the other side. So I'm just going to grab this here, let's take this and move it on this side. So it becomes plus 27, over here becomes minus 9, so 6 minus 9 is negative 3. 27x minus 8x is 19x, right? And then divide by 19, divide by 19, so x is equal to negative 3 over 19, right? That's your answer. Now, is that a valid solution? What you would have to do is plug it in to see if it works out, the left side of the equation and the right side of the equation. And you would have to check your restrictions. Now, we didn't write down our restrictions, we should have done them, right? So our restrictions, before we start doing anything, our restrictions here is x cannot equal 0, right? So x cannot equal 0, right? So when we write down our final answer, we're going to have to leave a little room here and say x is equal to negative 3 over 19. And, well, x cannot equal. Now, this is really unnecessary right now, but it's a very good habit to get into because the more complicated questions you get into, the more complicated equations you get, the more complicated functions you get, what's going to happen is you have to keep track of your restrictions. You're going to have to keep track of what happens to your function as you move along on an axis, as your variable changes. So the restrictions come in superhand. So right now, make a mental note of it, you don't have to do this every time, but it's really important to keep track of your restrictions. It's a very good habit to get into. So we just solved this using the method which is in general taught in most schools, which is multiply your equation by the common denominator, so it gets rid of the fractions right away. And again, the problem with this is what happens if the common denominator is some gigantic number that you really want to deal with. Personally, I prefer cross multiplication, so we're going to do that again. One more time. Same equation again. Again, your restriction is going to be x cannot equal to zero, because that's what you get if you set the denominator to zero. Wherever you have a variable, the numbers, you don't have to worry about it. So x cannot equal to zero, that's straight down. 2x cannot equal to zero, which means x cannot equal to zero. That's your restriction. So what we're going to do is use cross multiplication. You know, break it down to one fraction equal to another fraction. Cross multiply and solve the equation. So common denominator on this side is going to be 2x. Over here, multiply this by 2. So whatever you multiply the bottom by, you multiply the top by, you multiply this by 2, you multiply the top by. Over here, multiply by x, so 5x minus 4x. 2x and 3 is going to be 6x. So you multiply this thing by 3, multiply that by 3, 9, minus, multiply that by 2x, so 2x minus 6x. So what you're going to do is combine your life terms, right? So line up your equal sign. So 2 minus 9x over 2x. 9 minus 8x over... What you're going to do now is cross multiply. This guy comes up, multiplies that. That guy comes up, multiplies that. You're cross multiplying anything that's similar on this side, which is identical on both sides, they can kill each other. So when you cross multiply this up, this x eliminates this x. So that's gone. And 2 goes into 6, 3 times. So the only thing that happens is just that 3 is multiplying up here, because that's just 1 coming up here. So keep this in mind, whatever you cross multiply, anything that's similar between the terms, when you cross multiply this way, they kill each other in the middle. So over here is 3. So it becomes 3 times 2 is 6 minus 27x is equal to... It's just 9 minus 8x. So you break your terms over. So this guy comes over, becomes plus 27x. That guy goes over, it becomes minus 9, right? So 6 minus 9 is going to be negative 3. Negative 8x plus 27x is going to be 19x. And divide by 19, divide by 19. So x is going to be equal to negative 19 over 3. So x is equal to... Oh, sorry, negative 3 over 19, right? Negative 3 over... And that's our solution using cross multiplication.