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we're on problem 154 154 they ask us is X negative so the question is is X less than 0 statement number one tells us let's see they say X to the third times one minus x squared is less than 0 now let's simplify let's see X to the third minus X to the fifth is less than 0 let's see we could add X to the fifth to both sides X to the third is less than X to the fifth let's divide both sides by x squared and that won't change the inequality since x squared is definitely going to be positive any integer squared even if it's negative is positive so you divide both sides by x squared you get X is less than X to the third now let's see does this answer that question or let's see what X's would satisfy this condition so this is definitely if you think of so X we can already say that X cannot be equal to zero right we anything where this would have been inequality doesn't work X can't be equal to zero it can't be equal to 1 because 1 is equal to 1 2/3 and it can't be equal to negative 1 so we we know that immediately we know that for regular positive numbers for numbers greater than 1 for numbers greater than 1 taking it to the third power is always going to be greater than the number itself you know if you took two to the third power that's greater than 2 3 to the third power is greater than 3 you know one and a half to the 3rd power is greater than 1 and a half so we know that this this statement right here implies that X is greater than 1 and let's see are there any is there any negative is there in a situation where X can be negative well let's see if X first off X is between 0 and 1 X if X is between 0 & 1 let's see if you have one half one half is not less than 1/8 so it doesn't work between 0 & 1 but what if you went between X is what if X were between negative 1 & 0 let's try an example if you have negative 1/2 is is less than negative 1/8 it's more negative right negative 1/2 is negative 1/2 is less than negative 1/8 right now you could try that with any fraction because when you take it 1/3 power it becomes a smaller fraction recut but it becomes a smaller negative fraction so X will be less than it cuz it's more negative so this is so this is true so right now this statement applies if this is true then X is either greater than 1 or it's between negative 1 and 0 and let's see does it work if X is less than negative 1 negative 2 is negative 2 less than negative 8 nope right so this is these are the only ranges where it works this statement implies this but this still isn't enough information to tell us whether X is less than 0 X could be greater than 1 which is definitely greater than 0 or X could be less than 0 so we don't know yet just from statement one statement to statement 2 tells us x squared minus 1 is less than 0 so add 1 to both sides that tell us that x squared is less than 1 so when because something squared going to be less than 1 what's going to have to be between negative 1 it's going to have to be negative 1 and 1 right it's going to sense you're gonna have to be a positive or a negative fraction less than 1 that's the only way that when you square it you get a number less than 1 right if it was greater than 1 you're definitely gonna get a number greater than 1 and it's going to be positive either way and if you get if it's 1 it's going to be 1 and then zeros in our range so statement 2 tells us this but it includes both positive and negative numbers so it doesn't answer our question whether X is less than 0 but if we use both statements combined right what is the intersection what is the overlap of this and this well if X has to has to be between negative 1 and 1 and it has to be one of these right the only one that it overlaps with is this one you cannot have X being greater than 1 and X being in this range you can't have X being in this range and X being in this range since this is a subset of this so this is actually the more restrictive and so since we go to the that actually tells us that X has to be negative so both statements combined are sufficient to answer this question but individually they're not the last question 1:55 Marcia's bucket can hold a maximum of how many litres of water so we want to know the capacity of her bucket the capacity of her bucket statement number one says the bucket currently contains nine litres of water that's useless we don't know how I mean you know that could be the bucket nine litres could be that or maybe nine litres is the full capacity that tells us nothing about what the capacity of the bucket is you know right if I told you I had a bag of the sandwich that tells you nothing as far as how large of a sandwich I can eat so statement one is doesn't seem that useful statement two and three liters of water three liters of water are added to the bucket when it is half-full of water so one half times the capacity right when it's half full of water you add three liters so if three liters of water are added to the bucket when it is half-full of water the amount of water in the the amount of water in the bucket will increase by one-third so that is equal to so to increase so that is equal to to increase something by one-third that is like multiplying it by 1 and 1/3 right that's one in 1/3 times the capacity that's what they're saying right right the amount of water in the bucket will increase by one so they're essentially saying if you started off with oh no no sorry you're gonna have one in 1/3 times what you started off with which was half your capacity which was half your capacity that makes sense we say we're starting with half of our capacity and we're adding 3 litres we're adding 3 litres and they're saying that that will increase the amount we have by one in 1/3 let me write that let's say we start with we start with I don't know X and we're adding 3 litres you're saying that will equal 1 in 1/3 or 4/3 times X right this this shows that X is increasing by one third you could view this as one plus 1/3 times X this is an increase of 1/3 and they tell us that our starting point is 1/2 of capacity so that's where I get the 1/2 see that's our starting point plus 3 is equal to 4/3 times the starting point or where we're adding water to 1/2 times C and we could easily solve for C here so you get a 2 this cancels out and then you get 1/2 C plus 3 is equal to 2/3 C and let's see we could subtract C from but 1/2 C from both sides and you get 3 is equal to 2/3 minus 1/2 C and then you could just do that fact fraction and you get C is equal to 3 over well what is this 1/2 so if you do it over 6 4 minus 3 so this is 1/6 so this turns into 1/6 so C is equal to 18 but anyway you didn't have to do all that you should just realize that as soon as you can write this algebraic equation down it's a linear equation with one unknown and that one unknown is what we're trying to solve for you have enough information to solve the problem so statement C sorry statement 2 alone is sufficient to solve this problem and we're all done with data sufficiency see y'all in the next section