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before I go on with problem 143 I actually have a bone to pick with problem 142 and I did it I we we actually went through it correctly although I probably did a couple of more steps than I had to because when I do them in real time I tend to my brain wanders but now that I've had time to sit and look at a little bit I realized that I think the GMAT people may have made a mistake so what was the problem so the problem itself was 9 to the X plus 9 to the minus X is equal to B and the problem is do we have enough information to answer this question that's how they phrased it they don't say do you have enough information to say that this is true they say do we have enough information to answer this question so if I have enough information to say no this is not true that should be and you know that statement should still be good so we could ignore statement number one in the last video I showed that that was sufficient and I still agree that that is completely sufficient let's look at statement number two according to the GMAT people they say that statement number two is not sufficient to answer this question and I'll argue that it is can they say that statement number two says that X is greater than zero which is equal to B I take this to mean that X is greater than zero and zero is equal to B or B is equal to zero maybe I'm misunderstanding that but that's the only way I could think of interpreting this thing so if B is equal to zero what is this what is the statement that we're trying to see if we can answer or the question that we're trying to answer what does that boil down to well then the question boils down to 9 to the X plus 9 to the minus X is equal to zero and my question to you is if I'm taking 9 to any power to any power whatsoever can I ever get zero or a negative number in fact the only way I can get to zero as if I do you know 9 to the minus infinity power right because that equals 1 over 9 to the infinity so that might approach zero but even if I said X was infinity here on this side I'd have 9 to the infinity here so this would over this would be infinity plus zero so it would still approach infinity so there's actually no way and you know that's the limit at all that I don't to get confused but there's no way some people think oh if I'm taking to a negative exponent maybe that becomes a negative power no a negative exponent just means an inverse so this can be written as 1 over 9 to the X is equal to 0 so if X becomes a very large number this becomes very positive and this approaches zero if X becomes a very very very very small fraction this number this number becomes a small number but then this is the inverse of a small number so this becomes a large number so no matter what I do this stays and actually if X becomes negative if X becomes negative these just switch places right this becomes negative you know this becomes 1 over 9 to a positive X you get the point these are just negatives of each other these exponents and if X was 0 then both of these are equal to 1 so then you'd get 2 is equal to 0 so actually I don't know the way I see it statement 2 answers our question but the answer is no 9 to the X plus 9 of the minus X does not equal B because statement 2 says B is equal to 0 but anyway I don't want to harp on this too much but I think they actually made a mistake because statement 2 and my mind answers the question it just doesn't say that the the question is true let's move on to the next question don't want to wait but it's good I think discussions like that even though some poor chap because apparently these are real GMAT questions some poor chap might have gotten it wrong but we can use that as a piece of instruction so anyway 143 they say if M is greater than 0 and n is greater than 0 is M plus X over N plus X greater than M over N interesting so they're essentially saying if I have M over N and then I add the same number we call that X to both the numerator in the denominator does that make then does that make the whole fraction bigger and I'll just give you a little intuition right now if I add really if I had if I had a very small number for X relative to M and n it doesn't change the fraction much although well let me put it this way no matter if I add anything whatever X is and the larger X is the more that this fraction is going to approach 1 and you think about it if M is 1 and n is 2 and X is a million and X is a million then you're going to have you know that million is going to overpower the MD and you probably remember that a little bit from your limits but I just want to give you that intuition regardless of M and n RF x gets suitably large then this will approach one but I don't know if that's going to help us let's get let's look at the statements statement one M is less than n M is less than n so if M is less than n this says that this is that this is going to be less than one so it's going to be a fraction and I just said that if I add the necks to the numerator and the denominator or something you're going to approach one but but we have to be careful because they didn't tell us that X is positive so I might be subtracting X from the you know maybe this X is a negative a million let me give you that example so let's say M is less than n so let's say that M is equal to 1 and is equal to two so we could say if X is a Millions listen something similar let's say X is 100 so M plus X would be a hundred and one and n plus X would be 102 and that is gray easily greater than 1/2 right say Oh excellent statement one is sufficient but what if we were to and first of all that doesn't prove sufficiency but it's easy to prove that it's not sufficient because what if I were to add what if X were minus 100 then it would be 1 minus 100 would be minus 99 and then 2 minus 100 would be minus 98 and is that greater than 1/2 well no this is something this is a little over minus 1 so it's actually it's a negative number is going to be less than 1/2 so statement 1 alone is not sufficient because I can pick an M and then using statement number one and then an X and I could pick different X's where the statement is either true or not true this one is not true I'm trying to cross that out let's look at statement number two statement number two tells us statement number two tells us that X is greater than zero so let's think about just X being greater than zero alone does that help us well it does help us if M is less than n right because I just gave the example that if M is less than n so if you're starting with some fraction that's less than 1 right if you start with some fraction that's less than 1 and you add a positive number to both the numerator and the denominator you are going to approach you're just gonna get a little bit closer to 1 so you're actually going to increase the value of that fraction and you could test it out on a bunch of different numbers right if you add 1 to the numerator and the denominator of 1 half you get 2/3 if you got another one you get 3/4 if you got another one you get 4/5 did you see how this continuously as you add more and more to the numerator the denominator it approaches 1 and this X greater than 0 satisfies our problem with the first statement because our first name it says well what if X is negative then it doesn't work but let's think about whether X but X greater than 0 by itself is enough to to answer this question it works when M is less than n so it works in conjunction with statement 1 but what if n is greater than M what if the kids what if we start with an equation like what if we start with equation like 4 over 2 so if we is 4 over 2 is that let me make sure I get that is that less than is that less than 4 plus x over 2 plus X for any X greater than 0 well let's add I don't know let's add 4 to the numerator in the denominator so then this would be equal to 8 over 6 this number right here is 2 this number right here is what 1 and 1/3 so it's not true you need if X is equal to 0 this statement still doesn't work if we don't if n is greater than M so in order for this statement to be true or in order to answer this question you actually need both pieces of information so both statements together are sufficient to answer this question next problem 1 144 if N is a positive integer so they tell us n is greater than 0 is 1/10 to the N less than 0.01 so they're saying is 1 over 10 to the N less than 0.01 what's point oh one let me do let me know I'll write it down so this is the statement that's another way of saying is 1 over 10 to the N less than 0.01 is 1 over a hundred right which is the same thing as 1 over 10 squared right so well let's just think about it a little bit when is well let me look at the statements before I go off on my own tangent so statement days 1 they say n is greater n is greater than 2 so if n is greater than 2 as n increases is that going to make this whole number smaller or bigger well if I take a fraction especially 1/10 as I increase it to more and more powers I actually end up going more and more behind the decimal point right if n is greater than 2 at N equals two you get 0.01 and then if n is equal to three although it doesn't have to be an integer actually they tell us it's an integer so the N equals three what is it if n is three so they say n has to be at least three right and it's because it's a positive integer so what's one ten to the third power is 0.001 and that's definitely less than 0.01 what's one tenth to the fourth power that's point 0:01 and that's definitely less than 0.01 so this statement alone is sufficient to answer the question statement to statement two another way you could view this is this this inequality could be written as 10 to the minus n is less than 10 squared and then this is only going to happen if - you know actually that's not a good way to go about it I think the way I explained it the first time it's probably this inner way alright the second statement they say is 1 over 10 1 over 10 to the N minus 1 is less than 0.1 so what we could do here is just multiply both sides of this equation by 1 over 10 and what happens you get 1/10 times 1/10 to the N minus 1 is less than what's 1/10 times point one well it's one it's 1.001 right it's one-tenth of that and what does this simplify to you have 1/10 to the first power times 1/10 to the N minus one you add the exponents right you could put a first power there well this is equal to 1/10 to the N is less than 0.01 which is the statement that we originally had once at the N is less than 0.01 so statement 2 is actually equivalent to the original problem statement so each of these statements independently are sufficient to answer the question see in the next video