A2 10.4 Solve Rational Equations

All right, here we go. We're gonna solve some rational equations, 10.4. Hopefully gotta look at, I don't know if you guys know Moby Moby is kind of old school made me for you guys. DJ famous DJ turns out he looks just like DJ Sully. Check out DJ Sully with the specs. It's pretty awesome. I would love to see him spin it. If you didn't notice the theme of this one was, you know, I mashed up some of the algebras with the, you know, the artist I thought they represented. So there's, you know, Mister Bean with macklemore. Me and Drake and then of course in the beginning, the fantastic mister Kelly and Miley Cyrus. So let's get this thing rolling this rock out here at the last one here. Solving some equations. Let's put it all together here. All right, so let's solve some of these rational equations here. So it says solve check for extraneous solutions. Remember, when we're looking for extraneous solutions, we're looking for an answer that actually got canceled out, so it won't count as one. And the first one here we don't have to worry about it, because we're never dividing by a variable. X is on the bottom. So it's not going to be impossible to divide by zero. So in this case, we're safe. This actually is kind of old school. We're just looking forward to get rid of these fractions here. So to do that, we're saying, okay, three and four, this is a fraction if you want it's over one. What can they all become? Well, three times four is 12. So the common denominator, what they all can be is 12. And so what's going to happen is here, when we multiply everything by 12, we're going to distribute this on out. So everybody is going to get 12 times bigger here. So why do we do this? Well, it works out great when we do it. We'll come over here and say 12 times one is just 12. 12 times X is 12 X and again, if this is over the three, this is over the four, and then 12 times two is 24. And you could say over one. So when we do this, then things are going to start canceling here. So 12 over three is what that goes in there, four times. So we're looking at four. And then four goes in there three times. So we're looking at three X and then 24 over one is 24. So if you've done it correctly, your fractions will be gone here. So we're just getting rid of those fractions, so we can solve it. And now we're just solving for X, you know, subtract four from both sides. We're looking at three X equals 20, divide both sides by three. We're looking at 20 over three. So there's our solution for this one. And that'll get us rolling here. That's like general ideas getting rid of that fraction. All right, how about the next one over here? Let's slide over here. Excluded value. So remember, we can't divide by zero. So in this case, X is not allowed to equal zero. So if we solve it, we get X equals zero, we just got to get rid of us in extreme solution. We can't have it because we can't divide by zero. So in this case, what is the common denominator? Well, really, you can just take them and multiply them all together. Three times X times 5 is what is 15 X so if you want to write this out, we've got 15 X and what am I going to do with that? I'm going to distribute that to everybody over here. So we're distributing that 15 X so 15 X times one is going to give us what 15 X still over that three. Then I'm going to say 15 X times four is going to be what? That should be 60 X and that's over X and then I'm going to say 15 X times two is 30 X awesome and all of that is over 5. So when you distribute that, I'm just distributing to the top of it. Leave the bottom alone. And then that's why I like to cancel in this next step. Now start canceling three goes into 15. What, 5 times? So we're left with 5 X, the X's cancel here, so I've got plus 60. And then 5 goes into that 6 times so it equals 6 X awesome. So once we're there, we're good to go. The fractions gone, you've done it correctly. There's no fraction. What are you going to do? Get all your X's on one side. So I'm going to subtract 5 X from both sides. I think so that cancels I'm looking at 60 X subtract that equals one X so X equals 60 is my solution. It's okay because it's not saying that X can't be zero. It's allowed to be 60. I'm good to go. And you can always plug it back in and check it. Does it work? Plug 60 in right here. The double check that makes sure these fractions add up and yes, it does work out there, so that's pretty good one. Awesome. Great, moving on. What happens is make this a little bit trickier. So we've got three X 6 and three X. So again, you can multiply this out and say 18 X, that's great if you want to do that. If you want to leave it in parts, you're more than welcome to say, well, one of them is three X, another one of them is just 6. Three, there's the three X, there's a 6, three extra piece, so I don't need to do it again. So you can do all the parts. And then you can multiply it like this. Or if you prefer the 18 X to go ahead and multiply it out, no worries. I think it will show up better in the next one. So if I did leave it all in the parts, this would be, I still have my two here, so it would be two times three X times 6, all of that is over three X and then I would do my every I've got one here, so this will just be three X times 6. All over 6. And then do the last one here, the three X times 6, oops, and then times that four, don't forget the four, all over three X so if you want to write out in each individual part, you can, what happens here well, the three XS are going to straight up cancel. The 6s are going to straight up cancel, and the three X will have canceled. So this way shows it just the piece canceling out, which is nice. So what do we have? We have two times 6 is 12. And then we're left with what on this one. It's just the three X and then 6 times four is 24. So again, all my fractions are gone. I'm good to go. Now just solve this, so to solve this, subtract your 12 from both sides. And we're looking at three X equals 12. So what's X got to be X has got to be four? Is four cool answer? Am I okay with that? Go back to the problem. Check it out. What is X not allowed to be? Well, X the bottom of fraction can ever equal zero. So three X can never equal zero. So if I solve that, just divide by three, X is not allowed to equal zero. Is that true? Yes, I'm happy. That's good to go. That's the worst happy face I've ever drawn. That's a little rough. Excellent. Moving on here. So let's go check this next guy out. So again, this may be one where I want to do it by the parts here. So you could multiply this out, remember, this is a fraction over one. And sometimes like right off the bat, I like to look at, okay, what is my excluded value? So in this case, X is not allowed to be what? Well, put in the corner somewhere down here at the bottom, X is not allowed to be negative three. If it was, then I get zero. If I write before I feel like I'm more likely to check it at the end to make sure I don't forget it. All right. So what am I going to multiply this one by? Well, I'm going to multiply this one by. I'm going to give myself a little room here. I'm going to multiply it by two. This is one quantity, X plus three, and then one. So in this case, it's nicer, definitely to write out. You could distribute it, but you're going to do extra work if you distribute that out. So I would leave it as parts. And now you're going to multiply each term in the equation. Boom. Just like that. So when we multiply this out, what do we get? We're going to get two times X plus three. What do we multiply that by? We're multiplying that by X and all of that is all over two. Then we're going to minus, this is just one times it. So this will be two times X plus three. All over what X plus three. And I feel like my hand is a little rough in this one. I apologize for that. Let's see if I can make the last one nice here. Then we've got X over one. So I'm going to say the two X plus three to this X out here is going to be two times X plus three times X and all of that is over just that one. So now let's get our canceling things to start canceling here. So we're going to say, boom, boom, the twos cancel out in this. And what's left? Well, I'm going to have to distribute it. So in this case, you do have to distribute that out and you end up getting what X times X is X squared. Plus three X, the twos cancel out, so we're good. Now be careful, we've got a minus here. So this gets tricky. So luckily, this cancels, so I'm okay. But if it would be the whole quantity of something happened, but look at that, cancel, so it's just minus two. So be very careful with subtraction. And then nothing cancels on this one. So does the order you do this matter? No, it doesn't matter. So if you want to distribute this two first, you're more than welcome to say it. Just be careful that's going to be two X plus 6, but it's the whole quantity still times X so it's like you got to distribute again. So you can distribute the two, but now you have to distribute the X so you should end up with two X times X is what two X squared. And then we've got plus 6 X so I'm feeling pretty good about that. Fractions are gone. Now it's just a matter of getting everything on the same side of the equation. So I like to have my X squared's positive. So I'm going to move X squared over to this side. So I'm going to subtract X squared from both sides, so they cancel. Some left with three X minus two equals X squared plus 6 X and then it's a quadratic, so what do I got to do? I got to set it all equal to zero. So I'm going to go ahead and subtract three X from both sides. And I'm looking at what here, I'm looking at negative two equals X squared. Plus three X and I think I'm going to need a little more room here. I'm going to slide that baby down a little bit. And then to set equal to zero, what do I got to do? I've got to add two to both sides. So I'm going to add the two. Finally, I'm at zero here. So I'm setting it equal to zero, because of that square. And once I get it down to this, now I can solve it. So you can use quadratic formula. Graph it look for the roots. Or does it factor? Yes, it looks like this one's going to factor. That's nice, so it's going to turn into what multiplies the two as the three should be X plus two times X plus one. Equals zero. So now what makes that zero? That would be if X equals negative two or negative one. Is that cool? Come back over here and check it. It's just said you can't be negative three. So we're happy with both of these answers. Sometimes one of them gets ruled out or both of them could even get ruled out. But that one works for us. We're good to go. Awesome. So we've solved some rational equations. Let's bring the pain here. Now we're going to up the ante. Look at this monster. Well, sometimes they look pretty rough, but usually they work out kind of friendly. So I always look at this. Does this factor to get this going? Does it factor? Yes, I can say, you know, what multiplies the 12 adds respect to negative one would be X minus four, X plus three. Usually good hit, you know, these problems are usually made up for a reason. They're pretty set standard problems. If you see X plus three and X minus four over here, you may want to be thinking or hoping that it's going to factor. Because that makes your life much easier. So what am I going to do in this case, I'm going to multiply everybody by each thing. So X plus three, X minus four, this is really exposed to three X minus four. So my two quantities, I'm going to multiply by our X minus four and X plus three. So I'm going to take that whole thing into distributed to each term in this. Like that. And when I do what's going to happen, so I'm going to have my two X times this, so it's going to be two X times X minus four. X plus three, all of that is over what? I'm going to put the factored form in here. So I like that factored form, and we'll see why here. It should make things work out nicer for us. And then what do I have to 5? So it's this times 5, so it'll be 5 times, whoo, I hope you got plenty of room to write. You gotta use some small handwriting here or grab a second sheet of paper, and on bottom is what the X plus the three. And this is going to equal the multiply it all by this. You've got the two, so it'll be two times your X minus four, X plus three. Then that is all over just X minus four. Fantastic. So now hopefully things will start canceling. Look at this first one. This is a great X minus four X minus four gone. X plus three gone. So I'm left with just two X that's great. Plus what happens here? X plus three X plus three. And then I'm going to go ahead and distribute the remainder part. So luckily it's a dish in here for subtraction. I'd have to distribute a negative 5 to all that, but it's addition. So I'm cool with saying 5 X -20. And that's going to equal what happens these cancel and this one. So I'm going to distribute the two into both of this. So this should be two X plus 6. So luckily I don't have any quadratics in this one. I'm going to combine some terms on the left. So I've really got what, 7 X is -20 equals two X plus 6. And now it's just a matter of solving this. Hopefully that's not too bad. At this point, subtract two X from both sides. I'm going to do it in one step. That's going to give me 5 X, then you're going to add the 20 because you're trying to get the numbers on the other side. We're looking at 26. And yes, this is going to be 26 fifths. So I like fractions. I like improper fractions like that. That's great. If you really feel like making it 5.2, that's cool too. But I prefer this right here. And then double check it, is that okay? What was my excluded values? Remember back here, X is not allowed to be four, because it would cause this to be zero. And X is not allowed to be negative three. So just double check that. Those are my excluded values. Yeah, I'm cool. I'm good to go. No worries there. Awesome. So it takes up a lot of space, but they're cool. You know, this is the type of thing you can show your parents, hang it on the fridge, but look at this. Awesome. All right, let's do one more of these together. Then I'm going to think I'm going to have you try a couple. See how that goes for you? If I look at this, I see an X minus three and X plus three does this factor yes, this is our difference of squares. So we've got X plus three X minus three, which is great, isn't it? Because now I'm going to multiply everybody by the same thing, so I'm going to say, sure, we've got that X plus three. And that X minus three, I love it. Love it. What's going to happen here? So that's all my terms. I'm going to start distributing so distribute that bad boy. Boom. Boom. Oh, I see a negative sign. I'm getting pretty excited about that. All right, so what happens when I distribute? So this is going to be 6 times that whole quantity. So let's start writing and feel free to work ahead of me if I'm right too slow here. See what you come up with. So I've got X plus three X minus three. That's all over just the X minus three on bottom. Now it equals what it equals. I've got 8 X squared, so kind of a big number here. Times my X plus three, X minus three, and all of that is over what I like the factored form, the X plus three, the X minus three. I should have picked different number three, my threes are not flowing today. And the last one we've got the four X so we're going to say minus this four X times X plus three X minus three, all of that is over X plus three. So I'm doing this and hopefully things are going to cancel these things out. Hopefully a lot is going to cancel for us. So boom, X, minus three, X minus three, gone. And then what do we do with the leftover? We're going to distribute that out. So we're looking at 6 X plus 18. And then on the right side, hopefully some things cancel, we got X plus three X plus three, X minus three X minus three. We're looking at 8 X squared. And then O be careful of this negative. Remember, the whole thing is negative. So let's cancel parts of it. Gone and gone. You can think of this as distributing a negative four X to that. But I mean, really what's happening in your subtracting the entire mess over there, the entire quantity. So we've got four X times X minus three. So I'm going to be very extra super careful with that because I hate it. If you make this big long problem, you make one little sign problem. But if you're good with say negative four X, go ahead and distribute, you'll get the same thing I get. I'm just going to make sure I show all my work here. So now I've got this going on and I'm still subtracting let's simplify inside of this print. So I'm going to distribute. So I'm looking at four X times X is four X squared. Four X times that. Times negative three is negative 12 X and then there it is. There's that negative I want to be so careful about. I've still got this going on a little left side. I've got my 8 X squared, and then what happens here, I'm changing all the signs distribute this negative to both of them. So you need to end up with negative four X squared. Here's the key plus 12 X so if that's a sign gets messed up, we're in trouble. It's going to throw everything off. All right, so can we combine some terms? Left side is cool. It's good to go. The right side, I can say 8 X squared minus four squared is that. And then now what I'd like to do here is set it equal to zero. I get quadratic. I got this square. So I'm going to set equal to zero. So I'm going to do one step. I'm going to go ahead and subtract both of these in one line if that's Coolio. Awesome, awesome. So I'm looking at what zero equals we've got four X squared plus 6 X -18 excellent. So in this point, you could go to quadratic formula if you wanted to. I'm going to check it out. Does it factor? Let's see. Well, first I got to be careful because you can divide it to outright to begin with. So I can say two divides all of these. So we're really looking at two X squared plus three X -9. I'm going to extend the page. I'm just going to take some room up here a dang. A great problem, mister breast. They won't get any hard. This is as hard as we can make it right here. So now I'm looking at the inside. Does that factor? Well, I've got to remember, say, the 9 times the two, so does anything multiply to negative 18. Well, adding or subtracting to three, I think 6 and three will do the trick, won't they? So if I do this, I'm going to look at who's going to be negative. He's going to be negative. So the way we were factoring, this nice little shortcut is I'm going to bring the two down into both of them, so it's two X plus 6. Two X minus three. So we know this doesn't work. That can't be the answer because I made this 9 times two, so I've brought in this extra two times. So I got to kind of reduce it here a little bit, so when I do that, I got to take the two back out so that two I put in, I got to take it out. So it comes right out of here. Two divides both V so this is really going to be X plus three. Nothing comes out of that. So now I got rid of that too. It's kind of shady kind of magic math, but it works. I got the answer, so I'm good to go here. Excellent. So I factored it. So what are my possible solutions? Nothing out here because there's no variable, but X is either going to be negative three. Or if you have a hard time with this, you can feel free to go out here and say, when does two X minus three equal zero add three to both sides divide by two. You get three halves. So it's going to be negative three or three halves. Something I got to be careful of did I look at my extraneous solutions though. Right off the bat, I should have done this at X is not allowed to be what. Can't be three because that's going to be a zero. Can't be negative three. So be careful of these if that happen. Oh, it did. Holy cow. So I don't want that in there. Negative three can't happen. So this is not a solution. This is an extraneous. It can't happen. Solution. So really, this is my only solution X is three halves. Holy cow, if you get that and understand that your boss, you're going to blow it up this section. I like it. That's the full page as a screen. That's amazing. Fantastic. Old school stuff before I have you tried a couple. Maybe you remember doing this in the past. This is really a rational equation here. Because I get this fraction. So what happens when it looks just like this, this one's nice, because what do I do to get rid of it? I'm just going to multiply both sides by two X minus one. So when it's just one thing on the bottom, you know, whatever you do once I do it to the other, I'm going to think of it more like this where I'm straight canceling out. I like doing that. And then all I have to do is distribute this side. So I can say 12 times two X 24 X -12 equals that canceled out and you're left with a 5 X plus 6. Then it's just a matter of again your variables on one side. So again, I'll do it, let's just finish it 'cause I have some closure here. Subtract 5 X and then you want to move this guy over here. When I do it, it looks like I'm ending up with 19 X equals 18. So what do I do? Divide both sides by 19. So in this case, I'm good. Was that am I okay on extraneous solutions? So again, you have to go back here and say X could not have been one half. That would have made that zero. I'm cool. I got 18, 19. So I am good to go. Awesome. Everything's happy. Yay. It works out. Getting better, unfortunately, that's my better smiley face. How about this? This is definitely old school like algebra. This is really a proportion, but technically what are you doing? Technically, just like our rules, we're saying you're going to multiply by four X minus one and two X plus one. And then you're going to distribute to both of these. That's really what we're doing to cancel it out. Do we want to do it that way? You can. Does anybody remember the shortcut that you can it works? Do that and you're good to go. If you want though, you can think about this shortcut that works is remember. These are proportions. You just cross multiply. So if you want to straight up just cross multiply, we can say 6 times this is 6 times four X minus. That and in this case, 12 times two X plus one. It's the same thing that's going to happen if you multiply the other way. But it's our rules of proportion. It's nice to remember those rules because a lot of things are proportional. So let's solve it out just so we have it distribute the 6. We're looking at 24 X -6. And 24 X plus. 12 and oh, this one's kind of interesting. What happens here? When I go to get my X's on one side, what's going to happen? They're gone. They're gone. Boom. Boom. So one of my left with, negative 6 equals 12, is that possible? No, this one actually has no solution. It's actually impossible to solve this one, so that can happen. I didn't even check for my extraneous solutions, but there is no solution. So it doesn't even matter. I'm good to go there. Awesome. Very nice. So I just threw a lot of equations at you. Why don't you take a break, pause it, try these. I'm going to go ahead and work them out. But I'm just going to fly the answer. If you want to see how I did it, I'm going to throw the tribe video down below the application walk through. So if you really enjoy it, you can see that awesome. Excellent. So here are all the answers, the first one, remember double check it, you can't be zero, but we're okay because we got X equals four one. The second one I get X equals zero, but my experience solution was one for some safe. The last one's kind of crazy. Remember, don't forget, it can't be zero or three because of this one right here. You can't divide by zero. Well, I actually ended up when I solved this. It was a lot of work going through this. I got X equals three, which is extraneous. There's not another answer to go to. So this is actually no solution. So there it is. They're solving rational equations, and we are done with the chapter. And this is my last real video chapter of the year. Enjoy the rest of it. I hope the master check goes well for you. I've got a little DJ fail here. This is kind of imagine how if there was a real DJ Sully, how it go for him. It all goes well until he zooms in on a camera. Good luck and peace out. You know you need to be moving