hi guys it's me teacher going in today's video we will talk about solving rational inequalities in our previous videos we talked about solving rational equations and right now our focus is on how to solve rational inequalities so without further ado let's do this topic here's our example for this video we're in we are asked to solve the rational inequality x squared minus 3x minus 4 over x squared minus 8x plus 16 is less than 0. so here you have a messenger data first is to secure whether your inequality is in general form meaning you add in bone expressions and as a left side or necessary while the other side is zero so let's start let's general format first thing i need to do here is to find first the critical values or the zeros but i need to factor out my numerator and denominator because these are trinomials the factors of x squared minus three x minus four are x minus four times x plus one so remember hello factors and trinomials next for your denominator this one is definitely x square minus 8x plus 16 is definitely a perfect square trinomial meaning the factors are x minus 4 times x minus four is less than zero now to solve for the critical value critical value what you need to do is to get the factors x minus four so i'm gonna tie x plus one i'm going to is i will equate each factor in x minus four not in equate this by zero okay so transpose habilito it will become x is equal to four mainly because next amino acid is this x plus one transpose x plus one c equal to zero is transpose to negative 1. okay so what's next after this actually guys there is the zeros of the numerator are negative four and negative one while the zeros of your denominator is equal to four estimates is we will plug in okay we will plug in this critical value satin graph so i'll be using okay i'll be using a different color okay political value nothing which is first is equal to foreign number line into different regions and the other critical value is negative one so they tie a negative one so so you can see you add the number line is already divided into three different regions so what will happen here now parama nathaniel mismo um solution sets possible values of x getting first underneath anything first region we need to assign or we need to get some representative from each [Music] as you can see the numbers are here are zero one two and three we can choose x is equal to zero to represent this region determine i will simply choose um which number negative two asymptoma negative to the bar i have x is equal to negative two and here on the other side the meeting only five x is equal to five so yes we will plug in all this representative satin original so let's try i will start with negative two so instead of using this okay so if negative this is x minus four meaning there is negative 2 i'm replacing your x being a negative 2 minus 4 times negative 1 plus one over negative two minus four times negative two minus four is less than zero simplifying attento [Music] negative six times atomic negative one over atom again negative six times negative six as you can see my canceling takan we have negative one over negative six again one over six nine again okay so as you can see less than zero and then less than zero so try not then one over six is less than zero is it true or false okay one over six is greater than zero mean meaning this is false faucet so it only genetond partner at institution because your x is equal to negative two negative false answer now let's move on with the next one simplifying minus four times zero plus one again i'm still using this eternal manganese x minus four squared bringing zero minus four squared simplify is less than zero sorry that then i hope the magazine is a part tata we have negative four times zero plus one is one over zero minus four is negative four squared then less than zero negative four times one that is negative four over negative four squared that is sixteen so it is higher next simplifying negative 4 over 16 nothing is negative 1 over 4 less than zero is negative 4 less than 0 yes so that is 5 5 5 4 times 5 plus one over five minus four squared is less than zero okay simplifying that until five minus four is one plus at times five plus one which is six over um five minus four is one so square so one times six is six over one so less than zero less than zero so six is less than zero as you can see this one is a false statement so when okay now we identified that the middle region is your is are the possible values of x after the representative whether the critical values are included okay we need to know if the this critical value is negative one four are included three not ten so here so dito um using the same equation the first critical value is negative one so we have negative one minus 4 times negative 1 negative 1 plus 1 over negative one minus four squared so as you can see again zero negative five times zero is less than zero over negative five squared so automatic guys negative five times zero individual again zero meaning you know zero is less than zero so false in case okay [Music] open circle yen because negative one is not included so what i'm not even for so for fundamental as your next critical value using this we have four minus four times four plus one over four minus four squared is less than zero and defined it's not included notation okay we have negative one comma four so it looks like foreign set builder notation but nothing converts a balance at builder now x such that x or but i'm gonna want it so that builder notation attendant x such that x is greater than negative one but less than four so okay okay that x is greater than we will start with here so middle middle middle variable x x is greater than so negative 1 negative 4. so i hope guys to learn something from this video on how to solve rational inequality medium having video nothing but for sure my attitude because this one is discussed detailed so if you're new to my channel 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