so last time we started some polynomial inequalities so things like Q cubed or maybe let's do it in x x cubed minus 4X is less than zero say for example and basically what we have to do is we had to factor one side or factor the side with a polynomial leaving we always want a zero to be on one side and once we have a full factorization we know that these this polynomial is identically zero this equals zero at x equals zero two and minus two so we break up the real line into sub intervals so the sub intervals are between negative two and zero between zero and two two to infinity and negative two up to our negative Infinity up to negative two and we plug in an individual point in each so x equals negative three x equals say minus one you can choose whichever point you like and plug it into the polynomial that is factored and you plug in one point so for example if you plug in 1 let's start with three if you plug in three you'll get positive times positive times positive get overall a positive value if you plug in 1 you get a positive and a negative and a positive so you overall get a negative plug in negative one you get a negative a negative and a positive so overall a positive if you plug in x equals minus three you get negative negative negative which leads you to be negative you want to find where it's less than zero um so you would have everything less than -2 or everything between 0 and 2. so this would be your solution in in with the less than notation you could also write it with interval notation in this form and that would be just as good okay so this is the kind of problems that we started off doing what we're going to do now is to move into uh rational inequalities so rational foreign qualities things like you know what if you had 2x plus 4 divided by x minus 1 to be less than one something like this well um you still want to bring everything over to one side so we'd bring this over to this side 2x plus 4 over x minus 1 minus 1 less than zero and now here you want to put a common denominator so essentially what we're doing here is putting x minus 1 itself is the common denominator this remains as two X plus four and then um we multiply the negative one by x minus one since we have distinct um uh so if you remember that little kind of formula if you have distinct factors p over Q Plus R over s this is p s plus RQ all over q s and um so that's what we're using here and if we simplify this this has to be less than zero we get two x plus four minus X plus one is less than zero so we get X plus five over x minus one less than zero and now kind of similar to what we were doing earlier we set remember in this earlier problem what did we do we set the polynomial to be equal to zero and found where it equals zero in the rational case we're going to set the numerator to be zero and the denominator to be zero these are going to be kind of what the your interval is going to divide up into so x equals minus 5 here x equals 1 here so our intervals divide up into negative 5 and then one so we have sub intervals X less than negative 5 x between negative five and one and the next bigger than one now here there's one little caveat what I'm going to do here is I'm going to solve this one and I'm going to separately do x minus 5 over x minus 1 less than or equal to zero because there's just a subtle thing here okay so notice um let's plug in some numbers now uh let's do this so let's say we plug in minus 6 here we plug in 0 here we plug in 2 here if you plug in x equals minus six into the polynomial um so let's let's maybe make a little so here's our polynomial here's the x value x equals minus six um and let's put this maybe in the next row so the polynomial is X plus five over x minus one if you plug in the following values what do you get you get a negative number divided by a negative number this is a positive number plug in zero you get a positive over a negative which is a negative plug in 2 you get positive over positive which is positive so we're looking for less than zero and so we're looking for this interval where zero is which is between negative five and one so you might think okay well my answer so the answer is the answer is X is between negative five and one it works perfectly if um you're in the strict less than case now things just change a little bit if you have a what if you had less than or equal to zero well earlier what we would have done is to just say oh well you just closed the interval instead of having an open interval negative five to one you would have the closed interval meaning you would include negative five and one and this is almost true why is it almost true because technically X can't equal one because that makes the denominator zero if you plug in x equals one here you got a zero in the denominator so you do include negative five but you don't include one so you have to be careful that you never include something that makes the denominator zero that's the only caveat in the whole process everything else is pretty much the same you set the numerator to be zero set the denominator to be zero get the values and um and then and then separate out okay so let's do one more example so we get a we'll do again a let's do a greater than equal to this time so let's say we had uh two x minus 1 over x squared minus 9 greater than or equal to zero okay so what do we want to do we want this to be great equals zero so we won't need to divide up our interval into and values that make the numerator zero and separately the denominator equal to zero so from the numerator equaling zero you get two more two x minus one equals zero two x equals one so x equals a half from the denominator is zero you get x squared is nine so x equals plus or minus three so we have three values we have negative three we have a half and then we have three so our sub intervals are everything that's in between less than negative 3 between negative three and a half and between a half and three and above three so let's plug in x equals minus four x equals say zero x equals one x equals four and we plug that into two x minus one over x minus three X plus three that's a factorization of x squared minus nine so what do we get foreign if we have negative four we're going to have a negative divided by a negative times another negative which is overall a negative so again we don't need to know exact exactly the value Just A Sign plug in 0 you get a negative Over N negative times a positive that's overall positive if we have x equals one you get a positive over a negative times a positive that's negative and if you have here you get a positive over positive times positive everything's positive here so plus minus plus minus and so then what you can do is you can say okay well we want the places where we're positive so we want this interval and we want this interval so again you don't you don't want to just jump and say okay it's going to be negative 3 to a half as well as 3 to the positive Infinity okay this is not what you want to do you don't you don't want to say exactly this it's close to this but it's not exactly this I'm just rewriting the same thing um in uh with the inequality form now notice the problem with this is you can't plug in x equals plus or minus 3. these are not allowed because those make the denominator zero so what you need to do is you need to delete the parts that make those zero so you need to delete plus and minus 3. so as long as you just delete plus or minus three and leave the other one the other parts closed you're in good shape so this is your um this is your final answer right here just getting rid of plus and minus three okay so this is how you do rational inequalities we're going to move forward from rational inequalities to a new topic in the next video