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we're now on problem number 28 and the question is is X an integer X an integer statement one tells us that x over two is an integer so X if you think about it has to be an integer because X is divisible by two this tells us that X is divisible by two because when I divide by two it's not like you know five point five or something I get an even I get us I get an integer so X has to be an integer and if you want to use an example well if X is divisible by two then you know X could be X could be six which is clearly well anyway that's almost too obvious I don't think I have to explain this too much but this alone is enough to know that X is an integer because it's actually divisible by two statement number two tells us that 2x is an integer this doesn't give me a lot of information because if X is equal to 1/2 then 2x is equal to one so X would be in so that 2x is an integer but X is not an integer but then if X is equal to 1 then 2x is equal to 2 and so 2x being an integer doesn't necessarily tell me whether or not X is an integer so this is useless and this is useful so the answer is one that only statement number one is necessary a problem 29 problem 29 I should write smaller so I could save space what is the value of x simple enough the first one says 2x plus 1 is equal to zero like I said before you could just look at this and say this is eighth grade algebra I can solve for X this is all I need and you could solve for it if you want you could say 2x is equal to minus one X is equal to minus 1/2 but that would be a waste of time because they're not asking you what X actually equals they're just asking you whether you could solve for it two says this is interesting X plus 1 squared is equal to x-squared now you might be tempted to just take the square root of both sides but that actually becomes complicated because you could have the positive or negative square roots and you would have to set up a bunch of different equations the easiest thing to do would actually to be expand this so let's X plus 1 squared well it's X plus 1 times X plus 1 so that's x squared plus 2 X plus 1 is equal to x squared and then if you subtract 2x if you subtract x squared from both sides of this equation you get 2x plus 1 is equal to 0 so you actually end up getting this again so both of these pieces of information are equivalent and each of them independently is enough to solve for X so the answer is d each statement alone is sufficient problem number 30 what is the value of 1 over K plus 1 over R and what did they give us what information do they give us so they tell us that K plus R is equal to 20 K plus R is equal to 20 not obvious to me how to figure out from this what 1 over k plus 1 over r is and just experiment if we were to find a common denominator add these together what is another way of writing this expression well the common denominator would be kr and 1 over K is the same thing as R over kr right you can cancel out the ours and get 1 over k plus and 1 over r is the same thing as K over K R so this statement is equivalent to this statement so we're trying to figure out what R plus K over K times R is statement 1 just gives us the top so that by itself isn't enough what a statement to give us kr is equal to 64 so statement 2 gives us this information right statement 1 gives us this information up here so we actually need both of them to figure out what this is equal to so the answer is see both statements together are sufficient but individually they're not that useful problem 31 if X is equal to one of the numbers 1/4 3/8 or 2/5 what is the value of x so X is one of these and we have to figure out which of they are ok so statement 1 tells us that 1 4 is less than X which is less than 1/2 so it immediately cancels if X is greater than 1/4 statement want immediately tells us that X is not 1/4 and it also let's see an X has to be less than 1/2 well both of these numbers are less than 1/2 right statement 1 let me do this in a color statement 1 tells us that our answer is either is one of these - right that's what statement 1 tells us because those both of those right that's 0.4 that's less than 1/2 3/8 3/8 is what actually no this is interesting 3/8 is actually greater no 3/8 is obviously less than 1/2 all right what do I think 4 eighths is exactly 1/2 so 3/8 is also less than 1/2 so statement 1 tells us is one of those two choices and then statement 2 says that 1/3 is less than X which is less than 3/5 so this tells us that X has to be greater than 1/3 so let's see 3/8 is greater than 1/3 right because 3 nights is 1/3 so 3/8 is greater than 1/3 2/5 is also greater than 1/3 right it's 0.4 so that's greater than 1 4/3 I see 3/8 is 3/8 less than 3/5 well sure right 3/8 definitely less than 3/5 and 2/5 are definitely less than 3/5 so this problem it seems like problem that statements 2 and 1 are actually giving you the same information and you get both of them they both actually don't clarify they don't they don't give you any clarity whether X is 3/8 or 2/5 so they just leave you hanging and so the answer is e together they're still not sufficient unless I missed something once again remember I'm just doing this real time I don't know if if I am it's very possible that I make an error problem 32 problem 32 triangle PQR let me draw that triangle P Q R if PQ is equal to X so this distance is equal to X and Q R is equal to X plus 2 and it's equal to X plus 2 and PR is equal to Y PR is equal to Y which of the three angles of triangle PQR has the greatest degree measure so they want to know which of these angles so angle one they don't say it two or three but they want to you need to be able to figure out which of these angles has the greatest degree measure and and you might be intuition to you but they're essentially in order to figure out which angle has the greatest degree measure you essentially have to figure out which angle which angle is opposite the largest side is is one way to think about it and that should get you which is the largest degree measure so if we could figure out whether X X plus 2 or Y is larger than then we'd be all set so first of all we know that X plus 2 is greater than Y is greater than X so our answer is as far as the longest side is either going to be Y or X plus 2 and the shortest side is either going to be Y or X but anyway what did they say whether what information they say one Y is equal to X plus three so if Y is equal to X plus three this is the largest side of the triangle right this is the largest side of the triangle this is the second largest and so that and then this will be this will vd if I'm if I'm thinking about this right this will be the largest angle let me think about that right I mean we could go back to a little bit of our trigonometry but right this this would be the largest angle and let's see so I think one alone is enough because it tells us that this is the longest side this is the longest side the triangle would look something like this it's the longest side that's the second longest side and then this would be the shortest side it would look something like that and then angle 3 would be the longest would be the largest angle see statement number two tells us that X is equal to two X is equal to two gives us no information about Y if we say that X is equal to two then we know that X plus two is equal to four and we have no information about Y so that really still doesn't help us because it could easily be a triangle like this two two and then the four could come back this way and be like that in which case this would be the largest angle or Y could be a really big distance so that four could be like that if Y was a really large number let's see if you have the combination of the two well I mean we know that one an alone is sufficient to alone is not sufficient and I'm just doing let me think about a little bit let me make sure that I'm right that one alone is sufficient yeah I'm pretty sure I mean I can't think of an example where I have to know that X is equal to two yeah so unless I'm wrong the answer is a statement one alone is sufficient I'll see you in the next video