10.3 Add and Subtract Rational Expressions

Holy cow. Is that Mister Bean made the cover of thrift shop? I can't believe it. Never would have guessed it. Check this out, you know, sometimes I got a Photoshop some of the other algebras, you know, kind of make them look silly. Didn't have to do a thing here. That is all Mister Bean in this coat that he found here. Looks like he got it from baby gap or something, squeezing into that. You know, almost looks like he's sagging his jeans and everything. That's pretty awesome. What is gonna make fun of him, but actually that's the coolest I've ever seen Mister Bean look. I could call him a young Beasley Whitney's wearing that one very, very impressive Mister Bean. So 10.3 add subtractive rational expression. So this is a really awesome, really awesome section here. Let's go ahead and hammer this out. All right, here we go. So we're gonna start off with a quick review of how to do fractions because this is our goal down here in the corner. That's our thing. This is adding two rational expressions together. To get there though, let's bring it back old school style, make sure we can add fractions. If you have one third plus 5 thirds, what do you have? You have 6 thirds. Remember as long as they have that common denominator, you're good to go. That's how many thirds that you have. And then you can reduce it to two over one, which is just plain old two. So really we're looking at fractions, what are we going to do? We're going to throw some variables in them though. So now all of a sudden we've got a rational expression because we've got this very variable on top. So can I still subtract I'm sure they still have this common denominator of three, no worries, what's going to happen on top is just X -5 all over three. Boom, done and done pretty cool. So when they commented armor is already there, it's nice when we have a light denominator. The denominator is three Y squared. It's going to stay three Y squared. So in this case, I have a variable on the bottom. I'm adding on top 7 plus 5 is 12. Can I reduce here? Sure. You can cancel out three goes into that once. It goes into that four times. So this case, we're really looking at four over Y squared. Awesome. How about this one down here? We've got some traction. The same denominator, so I'm going to leave it like this. I'm not going to simplify it. I like to leave it in that factored form. And then make sure you subtract on top you've got 9 minus three. I'm not going to write that out. 9 minus three is 6. So let's just put a 6 in there. And again, sometimes some things cancel here, you know, because it is multiplication on the bottom, I can say three goes in that once. It goes into that twice. So technically this is this one right here. Fantastic. All right, moving on. So these are not like denominators. Are we are they, oh, they are, if I factor it, check this out. This factor is Y plus three Y minus three. So all of a sudden I can add these if I write it as Y plus three on bottom, Y minus three on bottom. So they have the same denominator. Then I'm adding on top so I'm two Y plus 5. Excellent. So everything is great when the denominators are the same. Things are going to get a little bit trickier though when we got to change that denominator. So in the first case, I can't add these. They're not the same as 7th of something as not like a third of something. They're different sizes. So I can't just add them together. Unless I make them the same denominator. So what am I going to do? I'm going to make this 21. What number can both of these become? This is like that least common multiple. 7.3 can both be 21. So what is the 7 need to get there? Well, you're going to have times it by three. But if you do that to the bottom, you're going to have to do it to the top to keep the fraction equal. Awesome, what is the 3D will it needs the time to buy 7? Whatever you do the bottom, you've got to do it to the top. So we're really looking at three 21st plus 35 21st. Now they have that same denominator they're the same size so I can add them together in say 38 21st. Awesome. So we're going to do the same thing. Is it going to freak you out if we throw some variables in there? Hopefully not. So again, I use the same common denominator here. I'm going to times it by three. This time I'm going to times by 7. What's the only difference here? I'm going to say this is going to be three X over 21. -35 over 21. So I got that common denominator. And then that's on top is going to be three X -35. All over 21. Boom, love it. There it is. How about if the variables on the bottom here can I still get a common denominator? Sure, it's like, the way I look at it with variables is what is it missing? So this X in this three, if I put them together, that he needs a three to be to kind of match up here. So I'm going to multiply by three. Whatever you do the bottom, you've got to do it to the top though. The three doesn't have an X, so I'm going to times this side by X, whatever you do the bottom, do it to the top. So I'm really looking at three times four is 12. On bottom is that three X -5 times X is 5 X on bottom is at three X so now I have that common denominator. Now it's okay to subtract them. I'm looking at 12 -5 X all over three X and you're done. Don't try to cancel or do anything weird with that. You are good to go. Excellent. How about this one? They look like they're kind of close here, but they're not exactly the same. So let's take a look at the three Y squared. What is it missing from this? Well, this is a 9. This is only three, so I need to get it to be a 9. So what I'm going to do, I'm going to multiply it by three. So I'm going to multiply whatever I do the bottom. I'm going to do the top. How about this 9 Y? So the three and the three make it 9, but this is Y squared. It's missing a Y to be Y squared. So they've got to be exactly the same. So it's missing that Y so whatever you do the bottom, do it to the top. So I'm really looking at this is 21 over three Y squared. -5 Y over, oh, did I say three? That should be 9 Y squared in my bed. It should be 9 Y squared. So now I know they're exactly the same when I multiply it all out. They're both 9 Y squareds. That's the whole point is that common denominator. So now I can subtract them 21 -5 Y all over 9 Y squared, and I am good to go. Pretty cool, huh? Yeah, this is really awesome. How about this right here? So again, I'm going to try to make it the same thing. So I like to think about these as one quantity. One group thing, X minus four. So if I look over here, what is this missing that this has? Well, this is three X. It's missing all of it. So I'm going to multiply this all by three X, whatever I do to the bottom. I'm going to do it to the top. What is he missing? Well, he's missing the X minus four. So that whole term, so whatever I do one side, I'm going to do to the other. We have room here. I'll go to I'll go down on this one. So three X times 9 is 27 X on bottom I'm good. I'm going to leave it as that factored form. I've got this on bottom, plus distribute this three, we're looking at three X -12. And then the bottom is the three X X minus four like this. Fantastic. So now I can finally add them together. So now we've got some kind of an interesting here. 27 X plus three X actually gives me 30 X and my 12. So you actually combining like terms now. And this will be three X minus four. Leave the bottom. Kind of factored here. So you could, if you really want to factor the top and try to make things cancel, I'm not going to worry about that too much if you're really excited to do that, go for it. I'm going to leave it alone, though, but you could. If you were trying to match up if you were using a textbook or something, they'd probably take this out, but I'm okay with leaving like that for now. The key is can you add subtract these? All right, let's bring the pain a little bit here. So now we got some tricky ones. So what I do is I kind of factor it, you know, to see what I'm actually starting with. So this is a difference of squares. We got Y plus three, Y minus three. So I've got this difference of squares, four Y minus three. So I look at this and say, okay, what is it missing? Well, over here's why my story I've got the Y minus three in common, but it's missing the four. So he actually needs the times this side by four. And then what is this side missing? Well, it's missing a Y plus three. So once you have it all factor, then you can look through it and say, okay, who's missing what here? So on the left side over here, I'm going to have four times two Y is 8 Y the whole bottom I know I'm getting this big thing here. So I'm going to do it in one step. This is the whole bottom. That's the common denominator. And I wrote it wrong, sorry. It's Y minus three here. So that's the whole goal, so I know they both have it because that's what I'm multiplying by. Now, be careful with this negative. This is where it gets really tricky when you're subtracting. You're subtracting the whole thing, and what is the whole thing? Well, the whole thing is distribute this 5 Y plus 15. So I distribute the 5, you know, I multiply by Y plus three. So distribute that 5, it gives you 5 Y plus 15, but then you're subtracting the whole thing. So really, you have to now distribute the negative. You're subtracting every part of it. So it's going to be 8 Y -5 Y -15. So that is a monster killer mistake right there. Is that negative? Got to distribute your subtract and the whole quantity. So a lot of people have issues with that one. Now I'm here and then finally I can combine like terms here. So when I combine like terms, I'm looking at 8 Y -5 Y is three Y -15. All over and then I'm going to leave that in my factored out form like that. Awesome. There we go. So again, if you really wanted to factor the top and see if something cancels, I don't think it will in this case, but you could factor it out. But I'm cool with that. Leave it like that. We're going to count that a match checks and whatnot. So you'll be good to go. All right, moving on. Let's scroll down to the bottom one. So again, this one looks tricky. I'm going to factor the bottom. That's how I like to start. So if I undistributed to, I've got this two times X minus one. Then over here, does this factor? Yes, the only thing that multiplies by three is one and three. So the sign's got to be the same. So I'm looking at this. And again, now it's just a matter of trying to match them up. What is he missing? He's two times X minus one. Well, he's missing the X minus three. So whatever you do to the bottom, bring it up here, do it to the top. And then what is he missing? Well, he's only missing the two, right? Out of this. So you're going to multiply the whole top, remember that's all in parentheses. And in fact, this is all in parentheses. Oh, this is going to be a good one. So when I go to write this out, what's going to happen here? Well, I'm actually going to have to double distribute this. You're actually going to have to multiply all of this out. So on top, I'm going to end up with X times X is X squared. Plus two X, then I'm going to say minus three X and I'm going to say -6. So I got to double distribute some people call that foil, but we're just double distributing. Dang, this is going to be a long one here. Then I'm subtracting. So again, there's that key that subtraction. So I'm subtracting. I usually do subtracting and then I put like a bracket or a princess, let me know. I'm subtracting the whole quantity. Luckily I'm just distributing a two here. So this turns into negative four X, then two times negative one is negative two, something like this. And the bottom, you know, I usually put that number first, but the order doesn't really matter. It looks something like this. So I know the bottom is the same. So I've got that common denominator. So I'm good to go there. The key is, does this simplify at all? So hopefully, it's going to simplify a little bit here. There's just way too much stuff in there. What I always like to start with is get rid of that negative. So you've got to distribute it. So go ahead and distribute that out. We're looking at and then we can combine some terms here. I'm looking at X squared, two X minus three X is minus one X -6. Then we're going to say negative times a negative is plus four X we're going to say negative times negative is plus two. Again, all of that is over that common denominator. Holy cow. These are going to take some space. I hope I left you a lot of room to work. Oh my goodness, we got that going on here. The final step, combine like terms again, so I've only got that one X squared. How many XS do I got? I got negative one X here plus four X so I'm looking at a positive three X and I'm looking at negative 6 plus two is negative four. Throw that bad boy over this and we are good to go. In credible. Get a top looks like it does factor, but we're not going to mess with that. We're pretty happy with this answer right here. And we are good to go. Wow, that's amazing. Amazing. All right, so we can add subtract them. Let's get back to this complex fraction. So we did some last section. We're going to kind of up the end here. We have two methods. You get this, we've got more of an expression on top now. We've got this edition going on in the fraction of subtraction on the bottom. You can get common denominators. You can say the common denominator of two and three is 6. And you could do something like this, get a common denominator and say, oh yeah, really two thirds is four 6. Plus three 6. And again, you know, we're not going to use method one. I don't know if you looked ahead where I say method two is really awesome. We're going to recommend method two. So if you just want to watch this, make sure you got it. That's cool. And then on bottom, what do you do? You've got to say the common denominator is 15. So I'm going to do this to both sides. So I'm really looking at bottom. I've got 20 fifteenths minus was this one over here times it by three. So 6 fifteenths. So you common denominator in the top and the bottom. Now you can finally add them to say, yeah, four 6 plus three 6 is 7.6. That's all over. Let me change colors here because that's going to look crazy right here. So I'm going to say 7 6 all over subtract those 14 fifteenths. Now I can finally do what I did last chapter, flip the bottom fraction and multiply so this is going to equal 7.6 times 15 14th and hopefully something cancels here. I think it will here that goes into that twice. It goes in that once. Three goes in there twice, three goes into there 5 times. So I'm looking at 5, four. So that's not terrible. It's not the end of the world you can do it. It's just kind of like a couple of problems blended into one. Here's a cool little trick though if you want to do it. Same problem. If you notice method two is the same problem. What if you just take the common denominator for everybody? So three, two, and 5. What can they all become? Like 5 can be 5, ten, 15, 20, 25, 30. What number can they all be? It looks like I'm thinking 30, right? Maybe there's one smaller, but they can all be 30. So I'm going to use 30. So multiply the whole thing by 30. So check this out multiply the top by 30, multiply the bottom by 30. This is just one. You're just multiplying everything by one. It's just a special one. So what's going to happen here? You're going to distribute this 30 to everybody. Well, what's cool about that? Well, what is two thirds of 30? So you can say 30 times two is 60 divided by that three, though, what happens? That turns into 20, what is half of 30? That's 15. Boom, just like that. Now do the same thing here. I'm going to distribute the 30 to the bottom. So if you want to do 30 divided by three is ten times four is 40. 35 O three is 6 times two is 12, something like this. That's cool if you want to do it like that. Then go ahead and add them. I'm looking at what 35 over what is that 28. And does that reduce? Hopefully that reduces sure. 7 divides both of these 7 goes in there 5 times four times. Look at that. So to me, that is a nice way to do it. I don't have to get all those common denominators and cancel and blah, blah, blah. It gets me right down to it. Pretty nicely. So I'm going to go with method two. If you want to do method one, you can. It's going to get your right answer. It just may need a little more space. So how do I apply this when I've got some variables in here? So I'm looking for if there's nothing written here, it's over one. So that's like three over one. And I'm saying, okay, I have an X, a one, a three, and X and a 5. So they can all be what? Three and 5 are 15. I've got that one X though, so really 15 X is what I'm going to multiply everything the top and the bottom by. So when I do that, what's going to happen, distribute that. So I'm looking at 15 X, the X's will cancel, and I'm left with what, 15 times two is 30, the X cancels out a 30, 15 X times three is 45 X, then I distribute it to the bottom, so 15 X divided by three X is just what, 5, it's a plain old 5 times four is 20. So if you need to write that out, write that out. No worries, you know? I need to multiply this together. Well, what happens when I do that? Well, I've got a lot of canceling. That's the whole point of this. One, and 5. So that's why I get 20. So feel free. I know I'm going quickly here, but if you want to write that out, no worries. So I'm going to subtract on bottom. Let's put it down here on the bottom. So I'm looking at 15 X times 7 fifths. So again, what happens here? It's like over one. So that goes 5 cancels that to the three. I'm looking at 21 X so really saying 21 X and does this break down or simplify? I'm not too worried about that. If you want to try to factor down, you can, but again, I'm pretty happy with that answer. It looks like we got it. Boom, there it is. That is a great little trick. Does the trick work here, holy cow, look at this monster. Fraction in a fraction. So really, this bottom one factor. So before we get going too far, we should factor the bottom, something like this. And now what is the common denominator for everybody? X plus two, X minus two X plus two, X minus two. So I'm going to do X plus two X minus two. All over X plus two X minus two. So again, just multiply by a special little one there. What's going to happen when I multiply it by this? Well, it's cool. The X plus twos will cancel, yeah. So I'm left with on top, I got to go two times that'll be two X minus four. When you distribute that to. All right, this is going to be crazy here. So I'm looking at this whole fraction. I'm going to multiply this by this. Well, what's going to happen? It's the whole thing, isn't it? They all cancel this will cancel out. So I'm left with plain old four. I'm going to write it down here. Hopefully, I hope I gave you a lot of space. Minus what happens here? Well, the X minus two is going to cancel that X minus two. So we're looking at 7 times X plus two. So that's going to make the bottom of this fraction. So really so on bottom we're looking at four minus distribute to 7, we've got -7 X -14. So a lot of things can go on here. Make sure you're careful with your signs. So that would be the bottom of our fraction. If I combine like terms, I'm looking at what a negative 7 X combine that minus ten, that is pretty awesome right there. Excellent. Moving on. So give these a shot. Pause them. You know, I'm going to show the answers up there and you can check them out and see how you did. Good luck. All right, if you want to see these actually worked out, I think I'm going to think I'm going to include an optional video beneath the application walkthrough. If you want me to work them out step by step, go ahead and switch videos now because this is the end unless you want to watch the funny clip at the end. There's a pretty great video to end it. So I'm going to pause it right now, show the answers if you want more of it. Go underneath the application walk through I'll have a couple of this worked out step by step. All right, there are the answers to the try it. I hope those match what you got. I get a box up there for your system. Interesting things, especially the last one I want to make sure you got that if there isn't a denominator in there, it's always over one. Then you can build your common denominator. This fraction down here is crazy. It's not going to get any tougher than that. If you've got that, you're a boss. All right, so I've got to give a shout out to these guys who make these math videos. Probably the funniest math video I've ever seen make this function awesome along the same lines is the macklemore to start this off. So good luck on the magic. Peace out. What what? What, what, what, what, what, what, what, what what what? What what what? What? What? What about it? I'm gonna up my grade only got a calculator in my pocket. I'm hunting looking for a value to make this function awesome. Now I walk into the room like, what up? I got this big book. I'm just pumped from the homework that I took unknown values. 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Looking for a value to make this function a song I'm gonna ask my brain when we got a calculator in my pocket. I'm hunting. Looking for a guy to make things function also what you know about rocking these bars on your paper. What you ever know about the actor Christian Slater. I'm writing. I'm writing trying to attack this problem. Gotta clear the boss before my teacher comes to come and solve a navy is equation just like I'm supposed to do and I finally get this right then I'm gonna need two. I'm graphing values with the bars that make a B what I'm a I'm a fussing in my big white you skipper your sagging your sleeper you tweeted I take those suckers money out the hands 'cause they can't rock this kicking later. I'm seeing exponential growth on the speculator. I'm fitting graphs to scatter plots. I'm just 'cause you later. They be like, oh, Mr. T that's hella tight. I'm like, yo, I've been working on this all night. Number two edition, let's do some simple addition. I'll begin with parent function than I add to them. Call that transformation with the move. I call that taking care of my business. That graph's hella though and not seeing functions like the other people in this class is a hell of a peak game go through my algebra telescope. Try to get the count of this knowledge man you hella want. Man, you hella won't. I've been great yeah I'm gonna up my grave only got a calculator in my pocket I'm hunting looking for a baby to make this fucking awesome my man games getting tight. The future is looking bright. My mom won't get her call from those two guys down the hall. Getting tight so tight the future is looking bright. My mom will get a call from those two guys down the hall I'm gonna up my grade only got a calculator in my pocket I'm hunting looking for a value to make this function a song.