Lesson 7 6 Solving Rational Equations

Okay guys, so today we are doing our very last lesson of the year. It is lesson 7 6 solving rational equations and it is short lesson and hopefully you will enjoy it as much as do. All right, so let's take look at example one. Here we're going to have two rational expressions and we're setting them up to equal each other. You have seen this before in your previous math life. know you have. How do you solve these? Who thinks they know how to solve these? Cross multiplication. Yes, cross multiplication. my markers are not going to make it to the end. my god. All right, cross multiplication. So what you do, and it's important you multiply the right ones with the right ones. You take this guy, you multiply it with that, and then you take this guy and you multiply it with that. So, we're going to do 3 * 4x + 5 = 9 * + 1. Yeah. And then we just distribute. This is 12x + 15 = 9x + 9. And then just solve for Easy peasy. I'm going to subtract 9x from both sides. So I'm going to get 3x and am going to subtract 15 from both sides and I'm going to get6. So is2. That was fun, right? You have to admit that was that was really cool. All right, let's try example two and I'm going to let you guys do it. 3 over 5x = 2 over - 7. Go ahead and give that shot. All right. Did anybody get an answer? X= -3. Excellent. So, you guys already know this. We're going to take this guy, multiply it with this. That'll give you 3x - 21. This guy times that. That's going to be 10 we're going to subtract 3x from both sides. So, is -3. Easy peasy. All right. now we're going to take look at least common denominators for adding and subtracting these guys. And we did that already last lesson. believe it was adding and subtracting, but we're just extending it now to equations. So there the sum of two rational expressions equals something and you need to find it. So let's go ahead and do those example three. So for part we have 5x + 7 over 4 = -9 overx. How do we do this? We need the least common denominator. So you've got to look at all your denominators here. What we're really looking for is common denominator. Here you have Here you have four. Here you have So what denominator can we use? What is the 4x? Perfect. Nothing crazy. So, this one's got to be multiplied by 4 over 4. This one's got to be multiplied by overx. And this one's got to be multiplied by 4 over 4. All right. And this will give me over 4x cuz that's our common denominator, also known as the least common denominator. + 7 over 4x = -36 over 4x. At this point, our denominators are all the same. So, basically, if we're looking for the value of we can ignore the denominators and just focus on the numerators, right? So 20 + 7 must be - 36. And then we just solve for So subtract 20, we're going to get 7 = -56, which means is8. So really easy, really, really straightforward. I'm going give one for you to do. Go ahead and think about this one. It's 1 - 8 /x - 5 = 3x. You're still going to have to find the least common denominator and do what you need to do. Go ahead and give you few minutes to try to figure This is actually the last problem of the lesson. So, go ahead and try to do this one. First of all, find the least common denominator. And if you're at home, stop the video and try to do this on your own. And then start it up again when you think you have an answer. Let's put our little thinking caps on and think, what could the least common denominator be? Here we have an Here we have an - 5. Here we have whole number. Yeah. But one we can make it whatever we want. So no, my least common denominator is * - 5. don't have to worry about it with this guy because you're going to see what happens in just second. So, really only have to worry about these two guys. So, this is my least common denominator. That means this guy over here has to be multiplied by - 5 over - 5. This guy over here is going to be multiplied by overx. And this guy is going to be multiplied by - 5 over - 5. Yeah. Times 1. Yes. because the same thing over the same thing is just one. All right. So now we have - 5. Let's do the common denominator here. Minus 8x, right? equals distribute the 3. That's going to give me 3x - 15. So once again, our denominators are the same. So we can ignore those and just set the numerators to be equal. So * - 5 - 8 has to equal 3x - 15. This is x^2 - 5x - 8 = 3x - 15. Gather all of our like terms. That'll give me x^2. We have -5x - 8x is -3x - 3x that'll be -16x. So all these guys are gone, right? And then + 15 = 0. What is that? That is what kind of an equation? Quadratic. All right. So let's try and factor it. This one's actually easy to factor. factors of 15 that give me 16 15 15 and one. Mhm. And it'll both be negatives. So my answers are = 1 or = 15. Yeah. And don't think this is the shortest lesson of the entire year. We're under 10 minutes. I'm looking at the timer. So, that's it. The big ideas is already out there and, the next video that you'll be getting will be one for the study guide. And then, just wrap it up. Put little bow on it. We're done. Bye.