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Current time:0:00Total duration:11:46

Before I go on with problem 143,
I actually have a bone to pick with problem 142. We actually went through it
correctly, although I probably did a couple of more steps
then I had to. Because when I do them in real
time my brain wanders. But now that I've had time to
sit and look at it a little bit, I realize that I think
the GMAT people may have made a mistake. So what was the problem? The problem itself was 9 to the
x plus 9 to the minus x is equal to b. The problem is, do we have
enough information to answer this question? That's how they phrase 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 statement should
still be good. So we can ignore statement
number 1. In the last video I showed
that that was sufficient. I still agree that that is
completely sufficient. But let's look at statement
number 2. According to the GMAT people,
they say that statement number 2 is not sufficient to
answer this question. I'll argue that it is. Because they say that statement
number 2 says that x is greater than 0, which
is equal to b. I take this to mean that x is
greater than 0 and 0 is equal to b or b is equal to 0. Maybe I'm misunderstanding that,
but that's the only way I can think of interpreting
this thing. So if b is equal to 0, 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 0. My question to you is, if I'm
taking 9 to any power, to any power whatsoever, can I ever
get 0 or a negative number? In fact, the only way I can get
to 0 is if I do 9 to the minus infinity power, right? Because that equals 1 over
9 to the infinity. So that might approach 0. But even if I said x was
infinity here, on this side I would have 9 to the
infinity here. This would be infinity plus 0. So it would still approach
infinity. So there's actually no way-- and
that's the limit and all that, I don't want to get
confused-- but there's no way. Some people think, oh, if I'm
taking it to a negative exponent, maybe that becomes
a negative power. No, a negative exponent
just means an inverse. So this could 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 0. If x becomes a very, very, very,
very small fraction, this number becomes
a small number. But then this is the inverse
of a small number, so this becomes a large number. Actually, if x becomes
negative, these just switch places. This becomes 1 over 9
to the positive x. You got the point. These are just negatives of each
other, these exponents. If x was 0, then both of
these are equal to 1. So then you get 2
is equal to 0. The way I see it, statement
2 answers our question. But the answer is no, 9 to the
x plus 9 to the minus x does not equal b because statement
2 says b is equal to 0. But anyway, I don't harp on
this too much, but I think they actually made a mistake. Because statement 2, in my mind,
answers the question. It just doesn't say that
the question is true. Let's move on to the
next question. 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 and the denominator
does that make the whole fraction bigger? I'll just give you a little
intuition right now. If I had a very small number for
x, relative to m and n, it doesn't change the
fraction much. Well, let me put it this way. If I had anything, whatever x
is, and the larger x is, the more that this fraction is
going to approach 1. You think about it, if m is
1, and n is 2, and x is 1 million, and x is 1 million,
then that million is going to overpower the m and the n. You probably remember that a
little bit from your limits. But I just want to give
you that intuition. Regardless of what m and n are,
if x gets suitably large, than this will approach 1. But I don't know if that's
going to help us. Let's look at the statements. Statement 1, m is less than n. So if m is less than n, this
says that this is going to be less than 1. So it's going to
be a fraction. I just said that if I add an
x to the numerator and the denominator of something, you're
going to approach 1. But we have to be careful
because they didn't tell us that x is positive. So I might be subtracting x. Maybe this x is a negative
1 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, n is equal to 2. So we could say if x is 1
million-- let's say something similar, let's say x is 100-- so
m plus x would be 101 and n plus x would be 102. That is easily greater
than 1/2. You say, oh, excellent. Statement 1 is sufficient. First of all, that doesn't prove
sufficiency, but it's easy to prove that it's
not sufficient. Because what if x
were minus 100. Then it would be 1 minus
100 would be minus 99. Then 2 minus 100 would
be minus 98. Is that a greater than 1/2? Well, no. This is a little over minus 1. So it's actually a
negative number. It's going to be
less than 1/2. So statement 1 alone
is not sufficient. Because I can pick an m and an
n using statement number 1, and then an x. 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 2. Statement number 2 tells us
that x is greater than 0. So let's think about just x
being greater than 0 alone. Does that help us? Well, it does help us
if m is less than n. 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-- and you add a positive
number to both the numerator and the denominator, you're just
going to get a little bit closer to 1. So you're actually going to
increase the value of that fraction You could test it out
on a bunch of different numbers, right? If you add 1 to the numerator
and the denominator of 1/2, you get 2/3. If you add another
1, you get 3/4. If you add another
1, you get 4/5. So you see how this continues. As you add more and more to
the numerator and the denominator, it approaches 1. This x greater than 0 satisfies
our problem with the first statement. Because our first says, oh,
what if x is negative? Then it doesn't work. But let's think about whether x
greater than 0 by itself is enough 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 we start with an
equation like 4 over 2? 4 over 2, is that-- let me make
sure I get that-- is that less than 4 plus x over 2 plus
x for any x greater than 0? Well, let's add 4 to the
numerator and the denominator. So then this would be
equal to 8 over 6. This number right here is 2. This number right
here is 1 1/3. So it's not true. If x is equal to 0, this
statement still doesn't work 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, 144. If n as a positive integer-- so
they tell us n is greater than 0-- is 1/10 to the
n less than 0.01? So what they're saying is 1/10
to the n less than 0.01. Well, what's 0.01? Well, let me write it out. So this statement, that's
another way of saying 1/10 to the n less than-- 0.01 is 1 over
100, which is the same thing as 1/10 squared. Well let's just think about
it a little bit. Let me look at the statements
before I go off on my own tangent. So statement 1, they say
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, and I increase it to more and more
powers, I actually end up going more and more behind
the decimal point. If n is greater than 2, at
n equals 2, you get 0.01. Then if n is equal to 3,
although it doesn't have to be an integer. Actually they tell us
it's an interger. So if n equals 3, what is it? So they say n has to be at
least 3 because it's a positive integer. So what's 1/10 to
the third power? It's 0.001. That's definitely
less than 0.01. What's 1/10 to the
fourth power? That's 0.0001. That's definitely
less than 0.01. So this statement alone
is sufficient to answer the question. Statement 2. Another way you view this is,
this inequality could be written as 10 to the minus n
is less than 10 squared. Then this is only going to
happen if minus-- no, actually that's not a good way
to go about it. I think the way I explained it
the first time is probably the better way. All right. The second statement they say
is 1/10 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. What happens? You get 1/10 times 1/10 to the
n minus 1 is less than-- what's 1/10 times 0.1? Well, it's 1.01, right? It's 1/10 of that. What does this simplify to? You have 1/10 to the first
power times 1/10 to the n minus 1. You add the exponents. 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, 1/10 to 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 you in the next video.