PC 30 9 3 of 2 Solving Rational Equations

okay okay so once you get this once you've sold for these values you can check by putting each one of them in separately and make sure the equation works okay so that was how to use one function i'm going to show you again the very same question here it is and instead of using one graph and solving for intercepts we're going to use two graphs okay and we're going to find intersection points make sure that they're the same all right so here is Method number to solving graphically so solving graphically method number two two functions alright so what you want to do here and this might already be given to you in question but you want to have to kind of separate functions that you can graph and by throwing this negative 5 to the other side now have to kind of functions that can graph one function is this equals plus 6 over plus 2 that's 1 at this half of the equation the other one is this half of the equation equals 5 okay and if we go back to our steps from previous for two functions we find the intersection point or points and so this is these are our two functions right here so let's give that go hit equals alright so from the previous example there already had this in here I'll just get rid of that negative 5 okay so so there it is there you can do that as well if you still have that in your calculator and we'll put the equals 5 as second function so again graph this half of the equation has function and then graph this half of the equation as second function and find the intersection points so if graph that now this is what we have so this graph right here do you remember this part when we just grab this as one function right there you see we didn't see this bottom half down there on the screen and this is your asymptote so here is this graph over here and here's the equals 5 so we're going to find the intersection points which one is there and one is there editions how you do that second function calculates and you go down to option 5 now and it's going to say first curve so the line of questioning is little bit different first curve yes second curve yes now when you tell the calculator to guess here's the thing because there's two intercepts it's going to automatically just choose one you should kind of move your cursor closer to the one that you're actually looking for and then the calculator will say okay that's the one wants me to fine so the intersection point is negative 15 so equals negative 1 there is your answer one solution and then we would do the same for the second intersection point and so if you do the same process you find the other intersection point as equals negative 4 and equals 5 so here's your other valued equals 4 and they're not restricted so there you go should be the same as when we graphed single equation so this example 3 asks us to solve both algebraically and graphically okay so again what we're going to want to do is algebraically we need to get rid of the denominators okay need to get rid of the denominators it might be good idea to factor so that you can kind of see what you're dealing with right so if we have like this over here can be factored okay 2x minus 52 and guess the top not so much but bottom certainly we can factor and we can take lot two-out right 2x sorry just move that to the bottom obviously you got to rewrite this part 2 times 2x plus 5 okay so if we if we notice if we factor this we might save ourselves just little bit of time okay instead of multiplying by this right here and by this whole thing then we can just pick the two factors yeah this one's plus 5 over here from the tech thank you copied that donal ok alright lot that makes even simpler thank you right very good so here we only have two two factors in the denominator and not like we have this one and we have to that's it instead of this and this big ugly thing so we're going to multiply everything by 2 and 2x plus 5 okay so what happens when we multiply these two together this times that what happens well on this side of the equation of sign then the two here is going to cancel with the two there the 2x plus 5 with the 2x plus 5 so we just have 8x plus 15 over here that's what's going to happen when multiply these two together what's going to happen when multiply this with this side over here ok so we do that separately each term gets multiplied so let's do first of all let's do this one times this and we see that the 2x plus 5 and the 2x plus 5 are going to count slow and we're just left with 2 times so it's going to 2x4 there right because again this goes with that so it's just 2 times so let's do in pink now let's do this one times this term and there's there's nothing to cancel out or divide out here right no denominators so you just have to kind of multiply this just straight away here so 2x times two is four what's four times 2x plus 5 well let's let's separate that so it's going to be four times 2x plus 5 because you have really have three terms here one two three so you have to multiply them all together there's no distribution the first time through just this one times this and now we have to just distribute okay and so let's do that that last step here that's 2x plus now this is 8x squared plus 20 equals 8x plus 15 48 squared plus 22 28 plus Dean and at this point I'm going to come back to this line because when we do this when we solve this graphically right you could go right back to here and type those two in or you could come to here and type these two as as the two equations and you should get the same answer matter which one so we'll come back to that but for algebraically again we need to get everything over to one side right so 8x squared if subtract 8x over here so 22-8 is plus 14x and then subtract 15 both sides gives us this and if we factor this now if we can pull common factor from the terms we should we can't and so this one we have coefficient other than one here so again if you don't know how to factor this need to come see me at lunch okay we'll go through this but by intuition you can do some trial and error here and that means that we know that the first terms here have to multiply to this one so you might choose 8 and you might choose what else could you choose for + 2 right it's usually good idea usually you have better chances if you pick the factors that are closer together so I'm going to do for + 2 just as trial again and then I'm going to flip over to this term and say what two numbers multiply to negative 15 well we have you know 15 and one with one of them being negative we have five and three with one of them being negative so if we kind of stick with the factors that are closer together again that's usually those are usually the factors that are it so then you go okay well let's try negative 5 here and positive 3 and if you were doing this by intuition all this can go pretty quick in your head but you do quick foil and that's negative 20x plus 6x negative 20x plus 6x does that give us positive 14 it gives us negative 14 so that means we're close and again this is all going going through your head real quick and so you just say hey you know what if these signs were changed then would be okay again click foil from the beginning that's 8 squared yeah that's 20 yeah that's minus six and minus 15 that equals that so you factor this correctly so that's by intuition the decomposition method is something that we can go over at the time anyways factoring this that's how you factor that you could also use the quadratic formula yes which is equals negative plus or minus root squared minus 4ac all over 2a yes that's the annoying little thing that taught you is stuck in your head know is it sad sad you have math formula stuck in your head you were officially nerd like me okay that's all right okay to be nerd hey have wonderful life I'm just very okay mm-hmm do you can if you're if you're nerd like me you probably shouldn't call people nerds life said have wonderful life have wonderful wife too have nerd loving wife it's true that is true love you know it's real when your wife is smoking hot and you're nerd that's how you know it's for sure okay can we get back to the lesson apples thank you alright so yes this is the quadratic form thank you yes you can use that to this sometimes is quicker and easier sometimes not well what is the value here that we're looking for positive 3/4 easy this one is negative 5/2 just again you just skipped few just kept writing out step here instead of doing 2x plus 5 equals 0 and then solving step by step you can actually get right to the answer from here going hey know that this has got to go to the other side it's going to become negative then have to divide by 2 so change the sign this number divided by that number that always works change the sign this number divided by that number done if you plugged 8 14 negative 15 into this puppy over here you should get the exact same answers well okay okay good no it's try this tried to pause it and then end up hitting stop but didn't haven't stopping it because pick control stop yeah I'm gonna go back and edit this now thank you no don't have to add it either way if hit pause right don't have edited edit it takes to be lot of time you'll help ease the items teacher anyone want to guess no three we have an answer is three don't hope if you are free three out of what two yes three out of two that's exactly how busy am boy okay so now you have to watch this video to see if actually edited that out okay wonder if you headed that out let's watch the whole math video again see this is part of my plan no don't fast for you can't fast forward that's just that's illegal youtube doesn't you do that alright moving on moving on for all the future generations that are just fast forwarded through this part okay actually have nothing else to teach you so that's pretty much the end what why would show my class these videos next year why why would do that yeah what might do what might do just sit over there with my feet up and just say hey listen to this video I'm gonna sleep and drink off your rear hopefully not the same time and they might aspirate and die okay look at all this that we've done this is amazing okay so here's the review the review the key ideas review to the section yes you are gonna have test soon now don't know let's see okay read the key ideas on the board okay here is your small assignment isn't it cute you didn't want to make song you want small one there it is