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we're on problem 119 is 2x minus 3y less than x squared 2x minus 3y less than x squared they asked us all right let's see if we can figure it out statement 1 2x minus 3y is equal to minus 2 2x minus 3y is equal to minus 2 so they're telling us that the left-hand side of this original equation is equal to minus 2 so now this question if we take statement 1 into consideration it boils down to is minus 2 less than x squared so first time I have you say oh I don't know I don't know what X is but think about it can X be a negative number at worst worst case X is 0 right that's the worst case if X was anything else x squared is going to be a positive number right so we actually know just from statement 1 that negative 2 has to be less than x squared unless X was some type of weird imaginary number or something and I think that doesn't occur on the GMAT so statement 1 alone is sufficient x squared is going to be at least zero and it's probably at or some positive number larger than 0 so statement one is sufficient statement 2x is greater than 2x is greater than 2 and y is greater than 0 and Y is greater than 0 so let me just try to come up with two contradictory cases using these conditions to see let's say if X was and they don't tell us that anything is an integer here right so if X was 2 point 1 X is equal to 2 point 1 and Y is equal to I don't know just say Y is greater than 0 so let's say that Y is equal to 0.1 right let's see what happens here so 2 times 2 point 1 you get 4 point 2 minus 3 times y so minus point 3 is less than x squared so 20 this is what it's like four point four point something so it's greater than two this is interesting because I think it right because if Y is greater than zero actually this is interesting so I actually you don't have to try examples out because just let's just write the equation down 2x minus 3y is less than x squared so if Y is greater than zero which they tell us in statement two then the quantity on the left hand side is always going to be 2x there's always going to be less than 2x right because this will have some positive value it might have a in significantly smaller value why could we can make it point 0 0 0 0 0 1 but it's still going to have some positive value so no matter what Y we choose as long as it's greater than 0 2x minus 3y will be greater than 2x right so we could say that as long as 2x if 2x is less than x squared for any X we pick if 2x is less than x squared for any X we pick then 2x minus 3y is definitely going to be less than x squared why is that because minus 3y is going to subtract from the 2x because we know that Y is greater than 0 so let me ask your question is 2x always going to be less than x squared if X is greater than 2 sure I mean whatever X is it's going on the left hand side it's going to be 2 times you know pick a random number two x well X you know it could be 2.5 right but on the other side on the right hand side you're gonna have two point five squared so if you think about you're gonna have two times a number that's always going to be less than that number which happens to be larger than two squared right the small as possible is you know two times 2.000 one you could you know make that in it you can make the lot of zeros there but I think you get the point that's still going to be less than 2.000 one squared right because you only have a two here instead of a 2.000 one so actually statement two alone is sufficient hopefully that made sense you just have to see off y is greater than 0 then the 3y is definitely gonna take away from the 2x and then as long as X is greater than 2 2 X is always going to be less than x squared and if that makes sense to you then then you should realize that statement 2 alone is sufficient for this that was interesting oh no sorry not statement either of them alone were sufficient we did that in the first one right statement 1 or statement 2 independently are sufficient to answer that question 120 120 a report consisting of 2600 words is divided into 23 paragraph so 2600 words and it has 23 paragraphs a 2 paragraph preface is then added to the report so 2 paragraph preface is the average number of words per paragraph for all 25 paragraphs less than 120 less than 120 so this is interesting so the average number of Seawork words per paragraph for all twenty five paragraphs less than 120 so if we knew the total number of words in the document we'd be said we would just take the total words total words divided by the total paragraphs they have now they have the 23 original plus the two from the preface and you have 25 and you do a total paragraphs and now told you the average words per paragraph now you we don't know the total words because the total is going to be equal to the 2,600 words in the body of the document plus the words from the preface I'll call that words sub preface all of that divided by 25 this is the average and this is what we really have to figure out to figure out if it's less than 120 that's what they want to know so statement number one each paragraph of the preface has more than a hundred words so each so each paragraph of the preface has more than a hundred words so I could I guess we could say that the preface the words from the preface are greater than 200 and that's another way of saying that is that the total total is going to be greater than 2,800 now if the total is greater than 2,800 what does that tell us about the total / excuse me so then the total divided by 25 is going to be greater than 2,800 divided by 25 I was 2800 divided by 25 let's see 28 2500 goes into 25 20,000 to 2800 125 31 1 times 25 25 50 12 so it tells us that the average the average is going to be greater than 112 so that doesn't help us if it told us that the average was going to be greater than 120 we would be done we would say oh we have enough information to the answer the question and actually the answer is no that the average is going to be greater than 120 but here this tells us the average greater 112 it could be 113 or it could be 121 so it doesn't answer our question of whether the average is less than 120 so statement 1 alone is not that useful statement number two tells us each paragraph of the preface has fewer than 100 words so the total preface has less than 300 words right they say each paragraph of the preface and there's two of them has fewer than 150 words so the total preface has less than 300 words so we could do the same logic the total the total is going to be that means that the total is going to be less than what the total is going to be less than the original number of words plus the preface it's going to be less than 2,600 plus 300 is going to be less than 2,900 and then if you divide both sides by 25 to get the average 25 goes into 2,900 really really four more times right four more times right that's right so it's going to be the average is going to be less than let me make sure I'm getting that right so four on a 25 goes into 2,900 125 41:25 and we have 150 25 goes in 150 six times so we have the average is less than 1/16 so statement two alone is sufficient statement two tells us that the average of the words per paragraph is less than 1/16 which is definitely less than 120 so statement two alone is sufficient and statement 1 doesn't help us much and I've run out of time see in the next video