 just confirm that we're going live we're interrupting an action for a song live stream that we're hosting on our twitch page let's see if it kicks in might have to reboot ah there we go nice good morning good morning everyone good afternoon and happy new year today is january 2nd 2021 and we're doing a math live stream drop in math tutoring session for our first live stream of this year fantastic fantastic fantastic and it's an open discussion basically making myself available for couple hours anywhere between two to four times a month to help people with mathematics if they have any questions if they need any help with mathematics you know make myself available and see if we can help people out if the mathematics is above me there are people that pop in to these live streams that their mathematics is a little bit more powerful they're a lot more powerful and they've been lending a hand with some of the math questions as well and we do have a discord page where people are sharing information and helping each other out and talking about a lot of different things aside from that let me give you a little intro until notifications go out and people start popping in notifications on discord and twitch if you want to follow this work if you want to know what this work is about i am on patreon patreon.com forward slash chico chy cho if you want to support this work if you want to know what this is all about which is basically layered on mathematics that we've been creating work for about 15 years now longer actually i guess and sharing a lot of information everything's sort of layered on mathematics and you can follow the work there for those of you who've been supporting this work on patreon thank you very much for the sport gang i appreciate it and it is in large part because of your sport that we're able to do this and we're able to continue this and i can do the calculus the mathematics and sort of try to figure out where we're going with all the stuff dragons how are you doing what's up chy cho happy new year happy new year here is doing another interesting 2021 right 2020 was pretty interesting let's see what 2021 has in store for us i think it's just going to continue really and i don't put anything beyond paywalls everything's creative comments share or share a like gang and if you do like this work you can follow it and check out everything that we do and if you do have the means if you think this work deserves your support through funds patreon is a great way to make sure that we continue to do what it is that we are doing we are live streaming on twitch twitch.tv forward slash chico live chy cholive if you want to participate in the chat twitch is where it's at okay it's the platform we've chosen to live stream because we can live stream everything that we want to live stream on here for now if the sensor is sort of tightened up on us we might move to another platform but that's not in the works anytime soon and we do love this platform fantastic community we build here and there's a lot of support coming in a lot of discussion a lot of participation and it is linked up with our discord page so loving loving what we are doing here lonely piggy how are you doing chy cho happy new year brother happy new year brother hope you're doing well health health wealth and good blessings to you and yours and to everyone and to everyone and lonely piggy snacks i got pickles out i got had craving for pickles okay so we got homemade pickles with garlic here let's see if you'll focus there just look at that focus focus ability it's garlic cucumber and onions delicious dragons thank you very much for the donation brother very much appreciated for some reason doesn't pop up in our chat i'm glad i caught it on the on the pop-up here uh happy new year brother happy new year if you were here i'd share my pickles with you i'm gonna pop in a pop of garlic stay healthy and eat well i could eat an entire child pickles pickles things so good so good and in the winter it's the way to get your sort of greens and veggies and stuff like this troll possibly possibly but bore bore no fussy really what i risk hang around for a while right listen to a little mathematics see what it's all about your trolling powers will increase tenfold just by sitting back and enjoying a two-hour math session just imagine what your trolling abilities will be if you sit back and learn mathematics where you understand the language of mathematics to deep level right you could become a legend right you could reach legend troll status and there are legends out there right there are legend trolls out there there's a few of them love the legend trolls not you know don't very much like the weak trolls they give trolls bad names right and that's why trolls have bad names because a lot of trolls they're weak they don't know what it really means to do the calculus right hang around for a while you could you could born of Osiris is a pretty cool name oh is that born of Osiris cool name indeed i couldn't make it out born of Osiris indeed it is indeed it is gang i do announce these live streams and by the way thank you for the support gang on twitch thank you for being here thank you for discussions thank you for jumping on to these live streams and months thank you for supporting this work we do announce these live streams on parlor l o mines vk gap and twitter 30 minutes before we go live okay and we do share additional information there and you can come to our discord page anytime you want and go exclamation mark social and all the links here will pop up and we do have a discord page and the link is way at the bottom and there's a lot of people joining our discord participating in conversations helping each other out on the math folder math channel and what not okay so you're definitely welcome to pop into there marco pancia 123 hey chicho happy new year all the best all the best to you as well all the best to you as well corn flakes how are you doing hello all happy new year happy new year oh i hope you guys had a fantastic new year by the way feline juice morning morning how are you doing daily fee fat 20 sub bro not a good day today my team called uh salta sal my team called saltec has lost the league for for you it's soccer but yeah oh so salted uh u k right they do oh it is saturday so they must be playing they lost the game oh no cd death how are you doing for 20 chicho my man happy new year sir happy new year to you as well gang for live streams where we don't have any visuals we will be uploading the audio to sound cloud.com forward slash chicho as podcast and those audio files those podcasts should be available in your favorite podcasting platform including spotify and itunes and we will be uploading this video to both bit shoot and youtube and if you're on those platforms you can support this work by joining subscribing liking sharing and if you're on youtube you can support this work by joining youtube membership and there's a button there and there's a handful of people that have joined youtube membership and are supporting this work through youtube membership thank you very much for the support gang very much appreciate it and it is because of the support we're getting on all these different platforms patreon twitch youtube bit shoot all the other platforms that we're able to do what it is that we are doing hello hello hny chicho chicho chicho are you latin um latin uh no no not latin marco so far so good i've been uh trying to uh conquer 2021 i made the resolutions to pick up boxing again nice and also brush up on guitar nice nice that's good boxing is great exercise right keeps you fit like mad and mad if you've never if you don't appreciate how powerful boxing is you've never done this go in front of a heavy punching bag and try to maintain a punch for just rhythmic punching for two minutes right three minutes you're pouring sweat pouring sweat right my excellent brand nickname was chicho as well aha nice chicho i've heard uh chicho in uh bulgarian means uncle elder god how are you doing oh the elder god's following the footwell soccer he was from uh chalicio mexico mexico yeah my name by the way is a mix of it's a it's a couple of nick's nick names mashed up together and i've been using it for 25 years for a long time long time let me take these guys down so again uh mass stream will deal with your math questions if you have them okay um and it is an open discussion but i do have some stuff i have some students uh that i've been working with focus focus focus focus focus focus focus focus i do have some students that i've been working with i just work with them i we do our own schedule we don't really follow uh school schedule and stuff um but she has some work uh to to hand in to do sort of a take home exam and um these were the later questions that she got done and um i want to check them to make sure she did them right right so if there's any questions that you have regarding mathematics specifically post them in the chat for sure we'll deal with them until any math questions come up right let me go over these there's five of them these are the questions and i wrote down her answers as well so i want to check these to make sure she's doing them right right so if she gets the answer correct then great if she doesn't then i look through the work with my students to see where they might have made a mistake and that's the key if you're doing mathematics if you're able to catch your own mistakes you just amplify your learning process right in order of magnitude better faster right and as a teacher as an instructor the best way to educate someone to help them go through hurdles and help them learn mathematics is to look at their work and see where they might have made a mistake and that is a trigger that is a point that needs to be dealt with and as soon as you deal with that then they can move along right so you just help them along through obstacles right elder god i prefer to kick a heavy bag yeah yeah since it's math i hit 2.7 times harder than my point you're oh indeed their legs are power man if you're if you anybody if you're doing any type of physical training work on the legs but elder god what is it the percentage uh 50 percent of your power comes through your legs i think something like this marco chichol in february i start my new semester and have an advanced functions course is there any type of online tutoring service you offer online tutoring like one-on-one yeah i do have one on one students marco that i work with um you're definitely welcome to pop into these live streams and ask us questions and we'll work work through whatever you're working through right but if you need additional help for sure contact me i do i have my students right now are just in bc and alberta but i've worked with other students from the online with all in the us as well i haven't done any students in europe yet wow you're a teacher i didn't know that i do a private tutor i don't work in an institution a central institution i can't i would be fired within within a week right in boxing the truth isn't is in the legs everything within all types of any type of martial arts or fighting it's the legs that matter it was sure you need upper body and you need flexibility you need this stuff you need the the form but the legs where you get the power right and a punch it's more like 80 percent from the legs 80 percent from legs cool cool cool um let's do the first question these are rational expressions and rational equations so you're simplifying rational expressions and solving rational equations which kicks you up into graphing rational functions okay thanks i'll contact you if need be it's for my site program at your university in toronto okay cool so there's going to be lots of stats in there a g re prep we've we've done some stuff g re is really the i forget what it stands for great equivalency test or whatever it is it's it's basically high school math but sort of takes you to grade 10 and grade 11 mathematics really and gang don't forget free assange free assange free assange julien assange is a publisher and journalist that is being crucified for trying to bring transparency and accountability of capitalist power to humanity for more information please see our julien assange and wiki leaks playlist let's simplify this rational expression to x over x minus five minus x over x plus two x plus two and then we're going to multiply this guy by x minus five x minus five over two x squared plus 10x minus 39 x squared plus 10x minus 39 okay so this is just a rational expression they want to simplify right crunch crunch crunch when you get something like this first over a business factor whatever you can right so this guy can't be factored anymore this guy can't be factored anymore this guy can right this is a quadratic we're looking to for two numbers that multiply to give you negative 39 add to give you positive 10 right so crunch crunch crunch crunch x x and we've done a lot of this i've got a two two playlists right uh in our youtube channel where we did like two summers worth of factoring graphing quadratic functions uh graphing polynomial functions using synthetic division we did a two summers one full year of this basically one full year plus of this and i created a lot of videos so this is a simple trinomial if you want to know how to what the principle behind this is you want to check those things out when you're doing this kind of thing we're under the assumption you already know how to factor polynomials right so two numbers i multiply to give you negative 39 add to give you 10 or positive 13 and negative three right so positive 13 and negative three okay yeah i spent the summer doing it at a military school the factoring stuff or boxing now you can cross that out because this is the one you're dealing with so first order of business when you're doing simplifying rational expressions first order of business factor second write down your restrictions not that you need them for simplifying rational expressions but you do need them for solving rational equations and you need them for graphing rational functions i used to do this in school but i forgot most of it yeah yeah a lot of people do if you don't if you don't there's a saying it says use it or lose it right and that applies to many things right so first thing we're going to do it now is the second thing is restrictions now the only restriction we have in mathematics really is no division by zero you can't divide by zero so what you end up doing is diet tuck thank you very much for the tier one sub so what we do is an daily FIFA 20 redeem 500 points i remember you can save up your points at some point we're going to do auctions again right sort of my appreciation for people being here if we can't divide by zero which is a restriction then the denominators cannot equal zero right we can't have this equal in zero this equal in zero or this equal in zero right so what we can do is just do this on the side here this is the steps and let's find our restrictions here let's do the restrictions here restrictions right restriction is no dividing by zero what does that do just when you spend time watching on twitch or you subscribe or you do different things you gather our points right the 500 points is just you're redeeming points as a as you acquire them for thank you now what you can do i'm not sure how twitch deals with them for creators and i don't know what what they do with it they might be giving me like bonus points or something like this i really don't know but what i ended up doing was creating extra points where we're we're we just we've done one of these where i did auctions i auctioned up some of the comic books that i published some of the jam that i made and honey that we bought and people did auction bidding and the highest bidder for certain something that we're selling not selling but auctioning off i sent all that stuff out out to people so we auctioned up a whole bunch of stuff we did it like a three hour live stream right so you can save up your points okay twitch i remember last year in high school i struggled with limits as x approaches infinity can you explain to me what a limit is is is it as simple as point or line that a function does not pass uh yes but there's different ways of approaching it we'll do one after this because this is related to limits restrictions are related to limits by the way could you prove that your factorial is indefinite the quadrate reverse i guess what i don't know what that means marco of course when you finish this first no rush yeah we'll do we'll do one just related with this actually we can just take this and say that's a function and we'll look at the limit okay hey what's up rambo how you doing my comics are still not in wait a second you you got your comics didn't you but you haven't picked oh you haven't gotten in your hands yet you haven't visited yeah everything's locked down unfortunately i'm getting thank you for the follows two or four has killed my travel plans yeah yeah and i didn't i think this is the first time ever really that i haven't visited family for holidays right so if we're doing restrictions no dividing by zero that means x minus five cannot equal zero and the cannot equal to sign you treat it like an equal to sign so you say x cannot equal five this one is x plus two cannot equal zero so x cannot equal negative two so that's one restriction and then x cannot equal negative two that's another restriction and over here you set both of those equal to zero not equal to zero right i got confused in the factorization i want to multiply it to get the quadratic reverse factorization oh yeah you mean foiling yeah yeah foiling you do this watch this x plus 13 times x minus three right going from here to here is factoring going from here to here is called foiling right so all you do this guy multiplies this and this this guy multiplies this and this x times x is x squared x times negative three is negative three x 13 times x is positive 13 x 13 times negative three is negative 39 combine like terms you get x squared plus 10 x minus 39 which you got that guy back right so it's just foiling right daily FIFA i live all the way in scotland so even if i done a bit i want i wouldn't be able to get it uh you know what i sent uh three packages to the uk in that auction it costs it costs a fair bit of penny man oh my god it costs too much but three people from the uk one and we sent it there right so you can save up your points and uh if you win during the auctions i will send it it might take a while right because it's so far right yeah see graphical stuff is frustrating because solving isn't always the problem it's all not not always the problem so the other restriction we have is this x plus 13 times x minus three cannot equal zero and when you have two things multiplied together to equal zero not equal zero you set each one not equal to zero so x plus 13 cannot equal zero and x minus three cannot equal zero so x cannot equal negative 13 and x cannot equal three so we got four restrictions for this rational expression okay don't want you to spend money that wouldn't be no uh no you know what deli fiva uh it's my way of showing appreciation aside from doing all this right i thought it was a lot of fun uh so that's cool if you win i sent okay now we want to simplify this we got this going on as soon as you factor look into simplifying again so see if you can simplify simplify does anything cancel out no nothing cancels out if this was a x minus five the x minus fives could kill each other but they don't right next thing you do you follow bed mass which is basically follow the rules of mathematics to crunch things together right so for us we're going to do inside the brackets first addition subtraction right so when we're adding two fractions we're looking for a common denominator so this guy we're just going to leave alone right but this guy we're going to say the common denominator is x minus five times x plus two so common denominator is x minus five times x plus two and then you ask yourself what did you multiply x minus five to give you x minus five times x plus two when you multiply x plus two so the top multiplies by x plus two so this becomes two x times x plus two minus x would you multiply this to give you this this x minus five that's just adding a subtracting fractions i hope that's clear and then you just write this guy down x minus five over x plus 13 x minus 13 and then you crunch the top focus focus focus focus focus focus and then you crunch the top right so expand this in expand this in right so this becomes two x squared plus four x negative x times x is negative x squared negative x times negative five is plus five x all over x minus five times x plus two times x minus five over x plus 13 x minus 13 right and then you crunch again right combined like terms so what combines you got two x squared minus x squared these two combined now usually if you can reduce the amount of copying that you're doing right so if i want to crunch this i can crunch this and rewrite everything again but when you're rewriting you might make mistakes right so what i'm going to do is i'm going to crunch this thing here and write it up top so two x squared minus x squared is x squared right so that and that are dealt with four x plus five x is nine x plus nine x right did we do that right i think so right yeah now what you do is you factor what you can over here you could factor out on x now i don't have any more room here so i'm going to rewrite everything right so i'm going to factor out an x up top and you got x plus nine left in the bottom you got x minus five and you got x plus two times x minus five over x plus 13 x oh look at this i made a mistake i made this a 13 and it shouldn't be so three x minus three right check your work okay don't like we went from here to here and because i was looking here and talking i made a mistake so when you're doing problems focus on the problem right don't take your eyes off the page the work that you're doing until you finish and then you can let your eyes wander and stretch or whatever you need to do right so minus three now you got two fractions multiplying each other name it again as simplified before you multiply and this is just more simplification so anything from the top can kill anything from the bottom right pot grimy are you doing so anything from the top can kill anything from the bottom as long as there's no plus and minus between them right so you got x minus five times x minus five this and this kill each other and then that's it nothing else simplifies so the final answer to this is x times x plus nine over x plus two times x plus thirteen times x minus three let's see if she got it right let's check it out this is this was her answer so x x plus nine x plus three x plus thirteen oh she write down x plus three oh i'm going to check this i might have just copied it down wrong right so this should be x minus x plus two because she has x minus three this so that's probably me copying it down wrong okay so now back she got the first one right rock and roll ah yes math and such math and such like ah yes she chose the goat the goat free assange free assange free assange julien assange is a publisher and journalist that is being crucified for trying to bring transparency and accountability of capital as power to humanity for more information police see her julien assange and wikileaks playlist and on monday january fourth okay 2021 there's a decision going to be handed out regarding his extradition to the united states from the uk court okay we're going to do a live stream monday morning eight am okay i don't really get the second step where does the x plus two come from x plus two x plus two this guy here or this guy creeping uh lennie over here let me know and i'll explain it okay the second one yeah so this one okay so check this out right i'm going to erase this i want to leave this up i'm going to erase this part so we can show this work here okay watch this it's called just basic extension of adding and subtracting fractions right so let's say adding fractions you're doing the following you're going to add one over two plus one over three how do you add fractions you add fractions by finding a common denominator right night's all comic how are you doing chicho how can you do this math but not organize your comic books because the problem is night's all comic when i put the stuff in boxes uh i buy more comics coming in right so if i'm buying more comics coming in i would have to have lots of space for for example when i'm buying x-man right x-man continues continues continues so i would have to have a space available empty space available for me to put my comics in as soon as they come in for the x-man lot right now i i don't have that space like a huge warehouse where i can just have empty space waiting for comics to come in so i just put things and then when i try to grab it i'm making excuses they're not organized Marco chicho just out of curiosity have you ever read george or what yeah i have it's a very powerful read very powerful indeed very powerful and so is animal farm animal farm very relevant multiply bottom and cross multiply then simplify well i wouldn't call it cross multiply what you need to do is find a common denominator and the reason you find a common denominator let me show you this right because a lot of people here i'm going to erase these guys and i'm going to give my give ourselves a little bit of space right and the last step was bed mass and then simplify again simplify right and then there's other steps there for more complicated stuff where you find new new restrictions as well when you got division when things flip and stuff we'll go over it right there's a couple of them here now let's go back to adding fractions right let's say you have one over two plus one over three let's do a visual of this right if you're doing a visual of this simplify yes simplify yes if you're doing a visual of this a fraction is part of a whole so one over two means take an object a whole object you have and break it up into two pieces right so we're breaking up this into two pieces and you're taking one of the pieces so one out of two right that's one out of two for this one plus take a whole thing right and break it up into three pieces and you're taking one out of three right here's one out of three now we want to add one over two plus one over three right what is this we will we'll do it simplest form of this right so if you're going to add this plus this we need to like we can't do it right now because they're not broken into compatible parts right so what we need to do is take each one of these holes and break them into equal parts and see what one out of that looks like right so the common denominator or the lowest common multiple for two and three is six so what we can break a whole thing into when we're dealing with one over two and one over three is to break that whole thing into six parts because both two and three go into six right five out of six perfect right but here's the visual so we're going to take this whole thing and break it into six pieces right and one out of two means a half means we're taking three out of six right so this becomes three over six plus this guy we're going to break at the six pieces so we're going to go this this this and we're taking this many which is two so this becomes plus two out of six and three over six plus two out of six because they have the same number of pieces now we can just add them up just add them one two three four five five pieces five over six right good math teacher this guy's good math teacher right five out of six okay now how do you do how do you do this just algebraically without the visuals because you're not going to go through visuals all the time right what you do you find the lowest common multiple you say that's six and you ask yourself what did you multiply two by to give you six you multiply two by three so you multiply the top by three as well so you multiply the bottom by three you multiply top by three and by the way do you know why you can do this because you can multiply any number by one without changing its value and three over three is one right so you're not changing the value of one out of two you're making it look different right you just that's all you're doing first thing i've understood thus far trust me that look mathematics each there's only five axioms everything else is built on that right so what happens is two times three is six which is what we had one times three is three plus what do you multiply three by to give you six you multiply three by two right but you can't just multiply the bottom by two you're going to multiply the top by two because you don't want to change the value the weight of one over three you just want to make it look different so we're going to multiply the top by two and the bottom by two right because two over two is one we're just multiplying it by one so one times two is two three times two is six six cool they have the same number of they've broken the same size pieces now you just add them up you get five over six that's adding fractions right that's adding fractions out for multiplying by one for yassange julien yassange is a publisher and journalist that is being crucified for trying to bring transparency and accountability of capital as power to humanity for more information oh what are the five fundamental axioms it's just a distribution addition addition multiplication distribution they're very basic it's just basically adding subtracting multiplying and dividing and the equal sign right so taking this principle let's add this and this right so again we're just adding fractions right so two x over x minus five minus x over x plus two so we just went from adding one over two plus one over three to adding two x over x minus five minus x over x plus two so you ask yourself what's the common denominator can't wait to see political stream political streams we're going to do we're going to do one of these these sets of streams we got out on monday it's going to be politics right so pi over two is 90 degrees yeah pi over two is 90 degrees yeah marco right because pi is 360 if you converted oh sorry 180 if you convert it 180 divided by two is 90 right trigonometry so common denominator you need this guy and you need this guy so common denominator is x minus five times x plus two okay so again you ask yourself the same thing what do you multiply x minus five by to give you x minus five times x plus two well you multiply by x plus two so you have to multiply the top here by x plus two but you can't just multiply top by x plus two the reason we got x minus five times x plus two is because you multiply the bottom by x plus two and x plus two over x plus two is just one so you're not changing the weight of this fraction you're just making it look different right and i always draw one line when i'm adding fractions to make them because they're over the same thing some people do i like that mathematicians i've said this before are laziest people in the world right they want to reduce the amount of work they do so what you can do is if you want to take that extra stuff to make it separate right you could say oh this is now x minus five times x plus two and the top is two x times x plus two minus well the common denominator is x minus five times x plus two well what do you multiply x plus two by to give you this well that's x minus five so this becomes x minus five times x minus five right so this becomes x times x minus five and i usually don't write them in two separate pieces i just draw a line and put that over both of them because that's their common denominator these are very important when doing Laplace transformations these simplifications if i remember i did Laplace i haven't looked into for a long time right now let's look at limits okay because i forget who it was that wanted us to look at it so let's look at limits anybody at all i'm learning a tensor calculus right now oh my god i love popping in to review everything i've produced they don't nice tensors man i did that a long time ago i don't remember it i don't remember it but let's graph a rational function right marco was you okay done deal take a look at this thing let's graph our rational function here we'll erase this as well just in case we need the space right thanks i get it now awesome creeping creeping Lenny it's important fractions you need to be able to deal with fractions right mathematicians are some of the laziest people in the world ain't that the truth anybody they are i swear an eccentric as crazy right i get tensored anytime someone's so scandalous funny let's graph the following function half of x is equal to let's go x over x plus two do you want to x over x plus so no let's go one let's make it simple we just want to look at the limits right so we're going to go i wrote down x again let's go one over x plus two right on charter days how are you doing that that's use wolfram yeah use wolfram too hey chicho and chat hope you had a happy holidays you too you too as well and happy new year on charter days so let's say we want to graph this right graph this function so the steps are the same right so first of all you factor if you're looking at a function factor well we don't need to factor this it's already in simplest form factored form right then what you do is you find your restrictions restrictions both vertically well let's say you find your restrictions okay we'll talk about asymptotes and stuff so our restriction in mathematics is no dividing by zero really that's the only restriction we have in mathematics on charter days if we have any more let me know but as far as high school mathematics is concerned and they say restriction of no taking even roots of negative numbers but that's not a restriction that's dumbing down society right the indoct like slowing down education education right they used to teach it to us when i was in school complex numbers but they don't anymore right now in my part of world right so the only restriction we have in mathematics is no dividing by zero so what you do whenever you have rational functions or any type of function you look at the denominator and if you can't divide by zero then you say the denominator cannot equal zero right so this part x plus two cannot equal zero so x cannot equal negative two that is your restriction for this function so now that we're starting to get values numbers let's generate our graph so here's our graph right Cartesian coordinate system here's our x here's f of x which is really your y and x cannot equal negative two so here's here's negative two and what does it mean x cannot equal negative two well what you do with that is you say what's your asymptotes right so for next thing you do after you find your restrictions is you find your vertical asymptotes your vertical asymptote if x cannot equal negative two your vertical asymptote is x cannot equal uh x equals negative two so vertical asymptote right it's just terminology right and gang don't forget free assange free assange free assange julien assange is a publisher and journalist that has been crucified for trying to bring transparency and accountability of capitalist power to humanity for more information see our julien assange and wikileaks playlist marco chicho specifically can you inform me on how to use limits to solve for horizontal asymptote i'm going to show you right at right from here right you're going to see what limits means right so x equals to is a vertical asymptote so this is an asymptote is basically a boundary that your function cannot touch or cross right so your function can't equal x is equal to negative two that means it can't this is a no-go zone this is the line right the universe explodes if you try to touch this line okay now that's your vertical asymptote your horizontal asymptote has these three things you have to consider right if you have a rational function f of x if you have a x to the power of n the highest power degree on this thing over b x to the power of n the highest power in the denominator okay what you do is you say if n is greater than m if the highest if the power up top is greater than the power in the bottom of the degree the highest power on the x is greater on top than it is in the bottom then there is no horizontal asymptote no horizontal asymptote if n is equal to m if the power up top is the same as the power in the bottom then the horizontal asymptote is the degree in front divided by this degree right so the horizontal asymptote is going to be y is equal to a over b and if the degree up top is smaller than the degree in the bottom focus focus focus focus focus if n is less than m then the horizontal asymptote is y is equal to zero which is the x line right which is the case that we have right now gang thank you for the follows by the way and subs and donations and bits and stuff if i'm missing my apologies i just want to make sure we i don't make any mistakes no brain farts while i look here look here right so what we have here here is a horizontal asymptote that means why this is y equals to zero the line y equals to zero that means f of x can never be zero which should be intuitive right can you plug anything in for x to make f of x equal to zero no you can't if you set x is equal to zero that's one over two well when x is zero y is one over two right i really need to to get better with mathematics this just goes over my does it on charger days it's it once we start putting points on here you'll see how this plays out right now take a look at this thing now we need to graph this thing right we got our vertical asymptote that's a no go zone we got our horizontal asymptote because a no go zone right so what we have here okay is the boundaries of this function and that's the way you start graphing a function you find the areas that you can't go to right no go zone no go zone no go zone no go zone we got two of them right then what you do is you find you try to find some key points okay first of all we could do a table of values if you want x and f of x all right let's do a couple of table values now we don't have a no go to no go to zone when x is equal to zero so let's find out what the y intercept is when x is equal to zero because that's negative one that's zero that's one that's two and etc right so let's find out what f of zero is when x is zero f of zero is going to be one over zero plus two which is going to be one over two cool so when x is zero y is one over two so if that's one we're here all right that point is on this graph now what you do is you ask yourself what happens to the function as you approach this asymptote right now the limits what you can do is we'll do it by point right but that's where calculus comes in because calculus sort of gives you the behavior of a function defines it better right so let's assume we're here and we're going to start moving towards this asymptote so let's find out what f of negative one is right so what we're going to do we're going to go f of negative one is equal to one over negative one plus two negative one plus two is one so this just becomes one so when x is negative one y is one right when x is negative one y is one cool now one of the properties of a function is when you're graphing something is asymptote sort of act like magnets if this is my function coming towards an asymptote the function either does this because this sort of acts like a magnet pushing pushing it pushing it pushing right the function comes like this it either goes like this right because it can't touch it or it comes in it goes like this dives down or comes in and does a or right there's a couple other things it could do too right so all right so without calculus you can only precisely estimate what a value is at a certain point no we're we're actually finding the exact value right now right we're actually finding what y is exactly at when x is equal to negative one right and then what we do is we can get closer and closer to negative two we can just put points here points here points here but what's going to happen is basically this thing is going to shoot up right so what you can do is you can write it like this you can say f of x as x approaches negative two from the positive side right let me write this bigger so you see it right it's sort of terminology again mathematicians are the laziest people on the planet they create little symbols to figure things out right so if you got a function this function right here as x approaches negative two from the positive side so what we're saying is if we're on this function we're going to approach negative two from above negative two what is f of x equal to well y goes up up up up up up up it equals infinity it goes forever so it's going to be infinity and you can say positive infinity if you want because sometimes it might go down become negative infinity right now check this out what happens as you go this way as x gets bigger and bigger so let's say x is two if you sub in x is two f of two is equal to one over two plus two which is one over four so when x is two you're at a quarter and because an asymptote again even a horizontal asymptote acts like a magnet here let me put this down acts so if this is a horizontal asymptote and we're coming down to this this thing keeps on pushing up pushing up pushing up pushing up pushing up it'll go closer and closer to this asymptote but will never touch it right so what this means is this function goes like this why does it do that because no matter how big an x value you put in for x in this function the bottom is going to become bigger and bigger and one divided by a huge number is going to be really close to zero but it's never going to be touching zero and it will never be negative right so we're good on this side of the vertical asymptote and what you want to do whenever you're graphing functions is you want to find out what's happening on either side of vertical asymptotes or holes or whatever you have right so you're mainly concerned when you're graphing something the zones that you're going to look at are the x restrictions that you have right it's a vertical asymptote that you have so we know what the function is going to do on this side and we know the function is not going to be here otherwise it wouldn't be a function right hello chichou how do you how do you do i do well thank you i hope you're doing well as well happy new year for jamaica right on jamaica happy new year brother happy new year brother infinitesimal it becomes infinitesimal i hope you guys are having a nice warm christmas here you can hear the rain it's gotten wintery right so we know what's going on on this side let's see what happens on this side right and by the way the other way you could write for this is f of x what happens to f of x as x approaches zero right not sorry not zero as x goes to infinity right as x goes to infinity as x gets bigger and bigger right f of x approaches zero right now what happens as you approach negative two from the negative side from this side right is it going to go like this or is it going to go like this let's check it out so let's find out let's pick another integer closest the closest integer to negative two okay no mods in the chat elder god is here yes but how do we prove that using limits uh how do you prove it using limits this is sort of the proof i don't know if it's you call the proof uh what kind of proof oh you're thinking about doing calculus in terms of okay i'll show you okay i'll show you after we look at what the function does this way okay you're talking about calculus doing the first fundamental theorem of calculus so we'll do it okay now pick negative three so f of negative three is going to be one over negative three plus two which is going to be one over negative one which is negative one oh wow so f of negative three when x is negative three y is negative one so we're here cool right no no elder god it's all good no no we don't have any trolls or anything here right now everybody's into mathematics because mathematics makes everybody smarter stuff right yeah that's d of x stuff indeed let's know it right no let's just ask if there are any mods in the chat right so this point is on this function and there's an asymptote and there's an asymptote and we know that once the point is here you can't touch this line or that line you can't cross it and they act as magnets so from here we know the function does this and this right that's what it does okay quite cool we're about or you look at it you show i'm in uh west coast of canada vancouver victoria rainy we're in a temperate rainforest it's nice very green very green lots of rain lots of rain i think the stats is out of 365 days a year i think it rains here or it's overcast here it's like 220 days or something it's crazy a lot of people move here they go oh it's so beautiful so beautiful and then they move here they live here for a couple of years they go stir crazy they're like oh my god it's so so dark and overcast and cloudy and rainy they they don't handle it well some people right so that's what it looks like now let's use the first fundamental theory of calculus to do this look at the limits of this hopefully i can remember how to do it right are you okay with this and this is the way you will write this one right you would say f of x as x approaches negative two so we're the same thing here what's what happens to f of x as x approaches negative two but for this one we went from positive side going this way for this one we're gonna go from the negative side well if you approach negative two from bottom negative means from less than negative two you go this way you're gonna approach negative infinity negative infinity right cool i used to reside in germany when i studied there i moved to jamaica and now i sort of live off the land you could say oh man awesome awesome but take but take one two three living off the land sounds like a great thing to do you're growing your own food and stuff no rain here in erasora no you guys get very little rain one in 65 days i've been here and again don't forget free assange free assange free assange julien assange is a publisher and journalist that is being crucified for trying to bring transparency and accountability of capital as power to humanity for more information so your julien assange and wikileaks playlist let's do the fundamental theorem of calculus on this x plus two let's write that down x plus two now fundamental theorem calculus says this right what you're doing is i'm going to give you a quick little intro on it we have a video on there if you do calculus into an intro fundamental theorem and whatnot it'll pop up because we did a live stream on this right so basically you're doing this if you have a function right what you want to do is you want to let's say this is your function let's say that's your function you want to find the slope of a certain point right the rate of change at that point so what you do is you zoom in narrow down on that point so you first you find the slope between two two points and then you come closer closer closer closer you do this right and the fundamental theorem of calculus says this if you have a function f of x and you want to take the limit let me find the proper terminology i'm going to write it down and see if i have any mistakes on it so you want to find the limit of f of x as x approaches a certain number h let's say right i think it's h then what you do is you go f of x plus h oops h minus f of x over h right i think that's what it is ice then snow storm here in southern lp of minnesota put it yes yes sir i came here on vacation with my wife and realized wow we have to live here i was originally born in the southern part of italy so i have the beach in my blood ah nice went on vacation and decided to stay that's cool or you must have come back and organize things and then came back again right given the all liver breakage yes thanks this is what i was referencing okay also marco we do think it should be limit as h approaches zero oh that's h approaches zero that's right thanks dice power my bad so this is limit as h approaches zero and we're going to put down is equal to limit as h approaches zero and this is the derivative f of x limit do i have that right you put the limit on this side seriously let's we do this right f of x prime the derivative is limit as h approaches zero right pitra how are you doing hello hello chicho i think some of your neighbors might be vampires come why i love your type of stream awesome awesome picture love doing it right so take a look at this thing what's the restriction we have in mathematics what was the restriction we had in mathematics the restriction we had in mathematics is we can't divide by zero so if h approaches zero if we set h is equal to zero we can't do that because then we're dividing by zero oh how do we deal with this well we substitute in what we have here right yes yes i did return i was originally a police detective in italy so it took some time to get off the grid back on oh man a great decision getting out of the policing to go into jamaica live off the land was so much better i'm italian but i want to hear you so take a look at this thing and welcome italy by the way salutations right so let's do this when you're doing this do this right and this basically means and by the way gang thank you for the follows thank you for the subs appreciate them and the bits and the donations and the tokens and all that jazz right so f of x plus h means just substitute x plus h for here right so what we're gonna do is hopefully have enough room because this gets messy okay whoa guy how are you doing chicho a cholo i didn't know you do math omis of course or ale carnal on a graph if you are on the x-axis x is zero right on the graph if you're on the x-axis no y is zero on a graph if you're on the x-axis y is zero and if you're on y y is zero no no no the other way around if you're on a on a graph x and f of x or if you want to call this y right if you're here this is y is equal to zero that's y is one y is two y is negative one y is negative two so y is zero and focus focus focus right and the other way around right it's a common mistake by the way it's a common mistake okay it just feels like you want to say x is zero but it's not it's y is zero right so substitute this in for x right so f of i'm just going to do it here and then we're going to erase it so we're basically we're doing f of x plus h well f of x plus h just means substitute this in for x so that becomes one over x plus h plus two you're okay with that this is some breaking bad man reminds me of eisenberg i don't make the blue stuff though chicho honestly i highly uh recommend this to anybody despite of their social status or age i have never been this happy in my life i've had so much more time to focus on my relationship with my partner and certain parts of me as an example as an example my skin have become so clear and oh brother i doubted not i seriously i believe you i believe you everybody has to get out of a rat race and one of the ways you get out of a rat race is learn mathematics right like the number line indeed right so let's plug that in so that becomes f of x plus h is one over let me write this down oh we've got to do the limit i've got to write down the limit too right we can't forget this part limit as h approaches zero of one over x plus h plus two minus one over and that's just f of x x plus two all over h right well what do we do now we can't we still can't set h is equal to zero because we get a division by zero well crunch crunch crunch add these two fractions how do we add these two fractions well we've got to find a common denominator common denominator is x plus h plus two times x plus two right so what i'm going to do is i'm going to write this limit as h approaches zero i'm going to do a division over here divided by h right because i'm going to do this part so common denominator for adding these two guys and it's just adding fractions right is x plus h plus two times x plus two well what did you multiply this by to give you this when you multiply by x plus two so you multiply top by x plus two so that just becomes one times x plus two which is x plus two minus what did you multiply x plus two by to give you this you multiply by x plus h plus two right so you go minus one times that x plus h plus two right but yeah it's nice nobody bothers me my bills are dirt cheap i have garden everything is local i buy my meat at a local farm and i can see the animals and food i am eating when they are alive i've never used barbeque so much nice nice van dutchie been four years since i completed my math degree and i can maybe remember 30 percent of brother me too i had to go back and relearn all this right this is the same type of stuff we did a while back yeah so expand this well this is just x plus two minus minus in front of bracket means you subtract all these right all the sides change minus x minus h minus two so we just instead of rewriting it i'm doing this right now we're going to simplify this okay if we simplify this this becomes x minus x will that kill each other two minus two they kill each other so in top limit as h approaches zero at the top we got negative h over x plus h plus two times x plus two divided by h means times one over h right that's what it means so chicho what can you do with these math i feel like this would be helpful for one uh in the business for sure in business and engineering and philosophy and sociology and data analysis and like everywhere really uh have i do i use calculus in my daily life no but i do look at the rate of change i do look at graphs to see to get a visual of how function behaves right over time and then you can do the calculus i.e doing a calculus means looking into the future or looking into the past looking at the rate of change to see how a system behaves so it's really analyzing systems right just doing these sort of problems elevates your mind it shows your ability to think yeah and it's problem solving it it's critical thought is and you're you're able to if you know mathematics really here's one thing i tell everybody when you know math like it can't do harm first of all if you know something you learn mathematics it's not going to do damage like it's one of the few things when you acquire knowledge over certain thing it's not going to or tool it could only benefit you right but one thing mathematics does it allows you to see bs come in a mile away right when people start throwing numbers at you and stuff like this you can you can do the calculus you can do the mathematics and go hey wait a second those numbers don't make sense right so it's a great bs detector right so it prevents you from falling for propaganda very important cool what is your current occupation my friend you seem like an interest i i do i create content online and i've been teaching mathematics for like 20 years privately i can't function in an institution i figured that one out a long time ago right i would get fired or i would i would not be happy so i sort of do what you did i took myself off the grid but i like i don't mind living i love living here actually living in a rainforest for me is beautiful right i love the greenery right all computers use a system of open closed zero and one to make a graph computers yeah until we get to all the quantum computing right big o notation is the efficiency of the graph made by the data so take a look at this thing we're multiplying fractions number one rule for multiplying fractions reduced before you multiply so take a look at this thing i'm a little tangent right what if we had this 25 over 15 times 28 over 36 times 18 over 14 times i don't know 9 over 15 right how do you simplify this how do you multiply this well if you're multiplying fractions i've had people do this i go okay multiply these guys multiply multiply multiply what is this equal to i've had students take a calculator when they first start working with and they start punching in numbers and i let them do this by the way they punch in numbers this times this times this times this and they get one gigantic number right they got a gigantic number and then they go this times this times this times this and they get another gigantic number and then they go okay that's the answer they tell me i go well you can't leave it like that you got to simplify reduce it and then they try to reduce this fraction plus smaller form right and then let them do this and they do this i go okay uh do you think there's an easier way to do this they go like well i just use a calculator i just do this i go well here's another easier way simplify before you multiply my government failed calculus i'm sure i don't know god simplify before you multiply anything from the top can kill anything from the bottom right as long as there's no plus and minus between five goes into 25 five times five goes into 15 three times three goes into nine three times three goes into 15 five times five kills five 14 goes into 28 twice six goes into 36 six times six goes into 18 three times two goes into six three times three kills three oh the answer is one one that's cool answers one isn't that a lot easier than multiplying the numbers and then trying to figure out what they are afterwards you have to simplify the fraction first right so keep that in mind all right well we got negative h over this times one over h h kills h so we got negative one up top we got rid of an h what can we do now well what we can do is multiply this out right yes chicho but now solve for the napkin ring paradox like now what you can do is foil this out or you got limit as h approaches zero right of negative one over x plus h plus two times x plus two well now it's just you can just sub in h is equal to zero because you you're not going to get a division by zero so if you sub in h is equal to zero this is what you get negative one over x plus two times x plus two which is really equal to negative one over x squared plus four x plus four so as for this function as f of f of x as h approach for the limit of h approaches zero of this thing that's your function your equation of the tangent for this function right as that makes sense so if you want to find out what the tangent line is for any point of x you just plug it in so here let's find out what f of zero is so f of zero is going to be one over two because you sub in x is equal to zero you got one over two what's the slope of the graph at x is equal to zero well you just plug in because what we have is our function here now is f prime of x the derivative the derivative of this function is negative one over x squared plus four x plus four right so f prime of zero the slope of this function at x is equal to zero you just plug in zero here so it becomes one negative one over zero square plus four times zero plus four it's just negative one over four negative one over four that's the slope of the function at x is equal to zero right let's see you'll be ready also make it ring o in big o notation is that okay i hope that answer your question uh marco it's basically using the fundamental theorem of calculus apologies for not writing not properly at first anyway i gotta pop off pickle gang i hope you have good food you're eating whoops homemade pickles whoops hey come here you i'm gonna pop this guy i'm gonna pop another garlic think about it you choke what are we thinking about glyphonoid or make it make it ring o in big o notation ring o and ring zero and big o notation ring zero and big o notation and onion is great too pickled onion is amazing that's a lot of garlic oh i'll read more than that garlic is good for you keeps you healthy the clove of garlic and it keeps the COVID away it's pickled garlic so it's not going to burn your tongue homemade pickled garlic delicious should we do another vampire and keep the vampires away should we do another one of these guys let's do another one here's another rational expression okay simplify this and again it's fractions right x plus two x plus two over five divided by x minus three over x and this is just homework and i'm checking that a student of mine has written right so i just want to make sure she's doing it right because she's hand in hand in test they gave them right hi fan jane's bond how are you doing if you're doing this if you've got fractions over fractions right i'm sideways so this is x plus two over five divided by x minus three over x restrictions what are your restrictions restrictions your restrictions is here's a fraction here x can't equal zero okay now you have a division here now if you got a division here it changes to multiplication x plus two over five times and you flip it right x over x minus three oh you just got another restriction right and this is multiplication but i'm going to use a dot so we don't get confused with the x's so over here we've got another restriction x minus three can't equal zero so x can't equal two restrictions okay chicho won't get covid or vampire advice that's an all all around great medicine that's an all around great medicine oh i've done that and in france it's called the hospital rule hospital rule yeah no it needs to on certain things i use ginger root for my house yeah ginger root all ginger is amazing so now that's we got this can we simplify this we can't simplify this so those are restrictions so this is just straight up multiplication so the answer to this is just x times x plus two over five times x minus three and that's it you can't simplify this you can multiply it in you could go like this x squared plus two x over five x minus 15 right but i i prefer it like this factored form okay and my student got it right so yeah yeah yeah this is a silly question that they gave her right too easy it's just multiplying fractions right let's do another one another one is the next one is solving rational equations i'm drinking ginger lemon tea right now nice with honey nice health boost activated indeed play it more i'm drinking Persian black tea with uh uh spearmint that we picked from our garden that we dried salute everyone hope you have good drinks good food so let's solve the so the other two were just simplifying uh rational expressions right let's solve a rational equation so this would be solved and solving just means get x by itself right so four over two x plus one or minus one one minus one plus two over x plus three is equal to negative six negative six over two x squared or plus five x plus five x minus three so name of the game is okay thank you for the follows game name of the game is factor everything first right we mentioned this so factor everything first this is already factored that's already factored we need to factor this one right so let's factor this guy two x squared plus five x minus three i'm going to use the four step method right and we've got videos online for this four step method and this is a complex trinomial if you just do chichu complex trinomial the videos will pop up okay take this you get x squared plus five x minus six factor this guy x x two numbers are multiplied to give me negative six add to give you positive five what are they positive six and negative one right so plus six minus one and then you pop the two back in front so you get two x plus six times two x minus one and then factor whatever you can and dump it two comes out of that so this guy factored is x plus three and two x minus one right so this is factored is x plus three and two x minus one do you notice something when you're in school they try to make things work out nicely not like the real world right so x plus three two x minus one right so that's your check to see how things are working out right so this guy factored is this guy we don't need that guy no more right what do you do next when you factor everything you find your restrictions what are your restrictions restrictions restrictions is the denominators can't equal zero so x plus three can't equal zero which means x can't equal negative three x can't equal negative three two x minus one can't equal zero so x can't equal one over two right so x can't equal one over two this like you're doing this you do this x minus one cannot equal zero bring the negative one over two x cannot equal one divide by two so x can't equal one over two right that's all you're doing okay and these are the same as those so that's what it is then what you do if you have an equal sign what you can do is an equal sign says whatever you do on one side you can do to the other side so what you can do is multiply both sides of the equation by the common denominator and that kills the denominators right so we're going to multiply this whole equation by x plus three and x minus two so multiply this whole equation by multiply by x plus three and two x minus one so what's four here i'll do this and then do show you the step and then we're going to do it speedy goes Alice on the rest of them right what's four over two x minus one times x plus three times two x minus one that's over one anything from the top can kill anything from the bottom reduce your fractions before you multiply this kills this the reason we multiply by a common denominator because it kills all the denominators so this guy becomes four times x plus three two over x plus three times that the x plus three kills the x plus three so it's just two x minus one that multiplies the two two x minus one is equal to and this times this that kills that so it's just negative six oh that was easy nice let's erase these guys i'll erase this too now what you do you expand and you multiply you expand and simplify right combine your like terms so this becomes four x plus 12 plus two x minus two is equal to negative six combine your like terms four x plus two two x is six x 12 minus two is plus 10 is equal to negative six grab this do we keep bringing it over six x is equal to negative 16 divide by six so x is equal to common you can simplify this fraction negative 16 over six is negative eight over three is that what we got is that what she got i don't think she got that how come how come how come she didn't get the right answer did we do it wrong i don't think so oh check it out she got the answer she got x minus two she did it wrong i gotta make a little note here wrong gotta look at her work now to see where she made the mistake right unless you guys can see a mistake i did yeah yeah yeah i think we did it right okay so that's one mistake she made so i have to look at that mistake and tell her to do it again and that's the way i end up teaching i look at their work students work and if i find they got they got the wrong answer i look at their work and i say listen is this really what you do try it again and if they do the same mistake again and then we go over it with a different type of question and i get them to do it again right repetition is one way you learn right unless i did something wrong i don't think i did anything wrong i got i'm gonna write down the answer x is equal to negative eight over three that's what i got okay so let's do the next one here's another rational rational equation to solve right another rational equation to solve let me make sure we're still hopefully we're still live streaming we should be sort of like really raining hard here if it rains hard if it gets windy which is a little windy sometimes we lose the connection but i don't think we've lost the connection while this is popping up i'm going to write this down uh three x so solve this three x over so i want to solve this again so three x over x squared plus five x plus six plus two over this page is not loading up well stream says we're still going obs let's reload again yeah it looks like we're live still good good good good good good good good good good now two over x squared plus x minus two x squared plus x minus two is equal to five x over x squared plus two x minus three yeah minus three so what we do factor everything two numbers a multiply to give you six add to give you five x plus two x plus three two numbers a multiply to give you negative two add to give you one x plus two x minus one two numbers a multiply to give you negative three add to give you two x plus three x minus one right do you notice x plus two x plus two x plus three x plus three x minus one x minus one what's the common denominator for this common denominator for this is x plus two x plus three x minus one so you multiply the whole thing by those three three multiply by we're going to put a little dot x plus two x plus three x minus one what do you multiply this by to give you that well it's just x minus one right so you just the missing component here is what you multiply the top by right so this becomes three x times x minus one plus two times x plus three is equal to five x times x plus two right it's just whatever was missing in the bottom here from those three is what you multiply the top by okay i hope that's clear really straightforward and simplifies your equation greatly right you don't have to deal with the fractions no more right so multiply this out three x squared minus three x plus two x plus six is equal to five x squared plus ten x now because you have an x squared and an x term bring everything over to one side okay so i'm going to bring because i want my x squares to be positive i can combine these guys negative three x plus two x is negative x so those are gone and then i'm going to grab this whole thing and bring it over so this is negative three x squared plus x minus six okay right so what we end up having is two x squared plus eleven x minus six is equal to zero so what we need to do now is factor this duvaki right oh yeah what was our restrictions forgot the restrictions restrictions x cannot equal one negative two and negative three okay always write down your restrictions gang now factor this guy take the two factoring complex trinomials x squared plus eleven x minus 12 is equal to zero two numbers that multiply to give you negative 12 add to give you positive 11 is plus 12 and minus one drop the two back in take out the gcf and dump it two comes out of that so this is x plus six and this is two x minus one so your solution is x plus six is equal to zero and two x minus one is equal to zero so x is equal to negative six and x is equal to one over two are any of those as soon as you get the answer check your restrictions are any of those one of your restrictions if they are you kill them they're not solutions but they're not our restrictions are one negative two and negative three and our answers is negative six and half it's all legit so they're both work they're both good and she got the right answer rock and roll so good good and that was good I think I just wrote that one down wrong so that's not bad should we do one more let's do one more okay let's check this out for some reason I think the stream might be having major hiccups but we'll find out check it out check it out let's do the next one solve for this x minus four solve x minus four over x minus three plus x minus two over x minus three is equal to x minus three wow this one I think is an easy one what's the common denominator here it's just x minus three right so we multiply both sides of the equation by x minus three I'm going to type something in the chat just to make sure we're still doing good or is there hiccups there's some internet problems all over the place so what should we do let's do a command free assange oh I think we've lost it because the chat's not popping up definitely definitely but we're gonna finish this unfortunately it looks like the stream might have died but good thing we're recording this and we're gonna upload this right so let's do this x minus four oh we did this so we multiply both sides of the equation by x minus three so this times that the x minus three kills the x minus three so this is just x minus four times x minus three plus oh no the x minus three kills the x minus three we don't need the x minus three right so it's just x minus four so this is just x minus four plus x minus three kills x minus three so that's just x minus two and this is just x minus three times x minus three foil that guy combine your like terms okay we got x plus x is 2x negative four minus two is negative six is equal to foil this out you get x squared minus 3x minus 3x plus nine combine your like terms because minus 6x right grab these guys bring them over so this becomes x squared this becomes minus 2x this becomes plus six so this becomes negative 8x right plus 15 is equal to zero two numbers are multiplied to give you 15 add to giving negative eight x x minus five minus three because if you multiply negative five and negative three it gives you positive 15 add them you get negative eight so now you got x is equal to x minus five is equal to zero and x minus three is equal to zero so x is equal to five and x is equal to three cool those are the answers right now take a look at this thing we've got two answers and we always have to check it against our restrictions but we forgot to write down the restrictions what are the restrictions restriction is x can't equal three if x can't equal three that one is a bogus answer so the answer is only x equals five and she got the right answer she got the right answer sweet sweet sweet sweet sweet sweet sweet i'm not sure if anybody caught this it could be just the chat that's got killed it could be because the stream continues to say it's streaming and i was able to see myself on the stream but but we're not getting any chat action and we're like i wasn't even able to post my own chat on there so there's a hiccup somewhere okay i hope that works out we're going to upload this video to both bit shoot and youtube okay and since we're recording this it should work out fine and just in case there's people watching on the live stream and for the recorded version that's going to go up first of all thank you for being here gang thank you for the follows thank you for the subs thank you for the donations thank you for the bits thank you for the conversations thank you for the questions mods thank you for being here oh the god okay aside from that gang uh ask for what this is all about who i am i am on patreon patreon.com forward slash gcho chych oh if you want to follow this work if you want to support this work if you want to know what this is all about patreon is a great way to do so everything's layered on mathematics i don't put anything behind paywalls everything's great at commons you can follow the work and if you think this work deserves your support patreon is a great way to do so and for those of you who've been supporting this work on patreon thank you very much for the support gang very much appreciated we are live streaming on twitch twitch.tv forward slash gcho live chycholive if you want to participate in the chat when it's working twitch is where you want to be at and again gang thank you for the follows thank you for the subs thank you for donations thank you for being here okay i do announce these live streams 30 minutes before we go live on parlor hello mines vk gav and twitter and we do share additional content there and if our chat was working let's see if it's gonna work you can go to our chat on this on twitch anytime and type an exclamation mark social and all the links will pop up including our discord page where there's a lot of people who are joined who are sharing information for live streams where we don't have any visuals we do upload the audio to soundcloud.com forward slash chycholive chycholive as a podcast and it should be available in your favorite podcasting platform and we will be uploading this live stream this video to both betute and youtube and if you're on those platforms you can support this work by joining subscribing turning on notifications guaranteed to get your notifications through betute not so much through youtube and if you're on youtube you can support this work by joining youtube membership and for those of you who are supporting this work through youtube membership thank you very much for your support gang very much appreciate it and thank you for being here i hope you enjoyed the content the mathematics our first live stream for the new year we're going to do a lot more of these aside from that we got another five six live streams lined up for the next five days i believe tomorrow oh tomorrow we're going to do a comic book reading monday morning julien sange live stream monday night meditation live stream tuesday we're going to do music lyrics wednesday we're doing current events i believe i think and on thursday chicho salvia donor chronicles gang i hope you have a fantastic fantastic day i hope 2021 brings you lots of joy and happiness and uh we'll be here for a long time to come and there's a lot of live streams we're going to be doing this year i hope you join us bye everyone