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we're on 121 121 it's as if x over 2 x over 2 is equal to 3 over Y is X less than Y so let's just try to so this is essentially saying that x times y x times y is equal to 6 I just cross multiplied or you could do that step by step multiply both sides by 2 then multiply both sides by Y and you get X Y is equal to 6 and they want to know whether X is less than Y statement one says y is greater than y is greater than or equal to 3 and gave y is greater than or equal to 3 so what does that mean so let's think about this x is equal to 6 divided by Y right if Y is greater than or equal to 3 so if it's three that's the largest that X can be right if Y is 3 if Y is any larger number then X will only be smaller so X will be less than or equal to 6 over 3 or X will be less than or equal to 2 and and just I want you to understand why i said that think about this it says y is greater than equal to 3 if y is 3 x is going to be too if y is 4 x is going to be one and a half if y is 6 x is going to be 1 so the highest value for x happens when y is equal to three and then as Y gets larger and larger X just gets smaller so X Y being greater than or equal to 3 says that X is less than or equal to 2 so statement one alone is sufficient to say that X is less than Y all right if x is less than 2 and Y is greater than three then X is definitely less than y statement two tells us y is less than or equal to 4 y is less than or equal to 4 so this is interesting this is it's really just the inverse of of this statement right here you now you're saying instead of Y being greater than certain number y is less than a certain number so if you think about it this way so let's go back to this equation so X is going to be if Y is 4 X is going to be X is going to be so when y let's do a little table y X if Y is equal to 4x is equal to 1.5 if Y is equal to 3x is equal to 2 if Y is equal to 2 then x is equal to 3 so actually this constraint I can pick a bunch of different why's that meet this constraint but it still doesn't tell me about anything whether X is greater than or less than Y here x is less than Y and here x is greater than Y so this statement is useless so statement one alone is sufficient to answer this question and statement to doesn't help us much 122 if V and W are different integers does v equals 0 does v equal 0 different integers remember this is different okay they say that V W is equal to is equal to V squared V W is equal to V squared now let's think about this if this doesn't tell me for sure that v is equal to 0 and in fact as long as well let's think about this the only way that this could be true is if v is equal to 0 that's one case now v does not equal to 0 so if we assume that's one case so if V does not equal to 0 then this case implies what that that W equals V think about it if V doesn't equal 0 we can divide both sides by V because we know we're not dividing by 0 you divide both sides by V you get on the left-hand side w because this vehicles out and then V squared divided by V you get V right and so you are left with well I want to be here though because because right now w would have to equal V right and they already told us that W and V are two different numbers so we know that this this situation right if W equals V then V does not equal zero right but they tell us in the problem statement that W and V are two different integers they tell us that W does not equal V so we know that V that the only way that this could be satisfied if W does not equal V is if V is equal to zero so statement one alone is sufficient to answer this question statement 2 W is equal to 2 W is equal to 2 well that doesn't do anything for me I mean w could be 2 and V could be 0 or V could be three they all they told us was that V and W are different integers and that they don't eat equal each other so this doesn't help me at all in determining whether V is equal to 0 so statement one alone is sufficient I just want to make sure I got that first part right I mean the only way that this is true this is true V is equal to 0 or it's true if w is equal to V either of those cases are true right that's the only way it works and we know W does not equal V so V done has to be equal to 0 State what are we on 123 123 what is the value of 36 thousand five hundred thirty six thousand five hundred times 1 point 0 5 1 point 0 5 to the nth power alright statement number one tells us n squared minus 5n plus 6 is equal to zero so we could solve for n there'll be two solutions so this is n minus 2 times n minus 3i just factored I said what two numbers when i multiply them equal positive 60 i said- 2-3 and when you add them become minus that's why I pick the minuses as opposed to + 2 + + 3 or some other combination equals 0 so statement one tells us that n is equal to 2 or n is equal to 3 so statement 1 by itself isn't enough because I don't know whether I should put a two here or three here so I can't determine this value just yet statement two tells us n n minus 2 does not equal 0 this statement by itself is fairly useless because it just tells me this just tells me that n does not equal to so and could be any other number so I don't know what number to put here but used in conjunction these two statements statement one says n is either two or three statement 2 says well it's not too so if you use both of these together we know that n is equal to three and then we could obviously solve this problem taking one point oh five to the third power and multiplying by 36,500 which I won't do because it would be a waste of time 124 124 and the rectangular coordinate system above let me draw that see they have the y-axis they have the x-axis and they draw a triangle I'll do it in a different color like that like that and the letters are o P and Q and the rectangular coordinate system above if opie is less than P Q so o P is less than P Q let me write that down o P is less than P Q is the area of o P Q greater than 48 so we want to know is area greater than 40 80 p is less than P Q so the way I drew this actually isn't isn't right it's probably a little bit shifted to the left a little bit but fair enough we can visualize we can imagine that so let's see statement number one the coordinates coordinates of point P are 6 comma 8 so this is 6 comma 8 let's see that tells us that tells us the hide right when we figure out the area of a triangle we need to know the base times the height and then multiply that times one-half this tells us that the height because the height of this triangle is the y-coordinate which is eight right so the altitude of this triangle is eight so we have the height but we still don't know the distance from 0 to Q that's so statement 1 by itself isn't isn't much information so that tells us that the point was 68 that P was 68 statement 2 says the coordinates of point Q are 13 comma 0 13 well this is helpful 13 comma 0 so this distance right here is 13 so by itself that's useless because if I didn't have the height I still wouldn't be able to figure out the area of this triangle but now if i use both statements I know the base times the height so the area is going to be the base times the height times one half so eight times one half is four times 13 is 52 and so 52 is clearly greater than 48 let me make sure that because i didn't use this information i didn't use that information at all but it doesn't let's see let me think about it let me think about actually no i think i'm wrong i think statement one alone might be enough let's assume that we don't have these are tricky let's assume that we don't have statement to there let me see if I can figure out whether the area is greater than 48 without it let me see yeah I think I'm wrong let me I caught myself okay so if this is if this is 6 comma 8 the area of this triangle right here well this distance right here is going to be six right there this distance right here is six 6 comma 8 right okay so the area of this triangle 6 times 8 48 times one half so the area of this triangle is 24 right but they told us that o P is less than P Q that this distance is less than this distance maybe I should draw it like that so you can kind of see it and the only way that this distance is less than this distance is if this distance is less than this distance this is a tricky problem right this has to be less than this right q has to be further away because that's the only way you're going to get this altitude to be longer than this altitude right because you're assuming that Q is along the axes so that's the only thing that you can shift back and forth so if this distance is larger than this distance then the area of this triangle is going to be at least 24 so the area is going to be greater than 48 so actually all you needed was statement one to solve this problem that's all you needed statement one gave you enough information statement 2 does not by itself give you enough information because you don't know how high p is it could be really low down here icity area it's going to be less than 48 it could be really high in which case the air would be greater than 48 but anyway see I I almost got that one wrong but I'll leave you there i'll see you in the next video that statement one alone was sufficient for this one see you soon