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Current time:0:00Total duration:9:53

We're on problem 119. Is 2x minus 3y less
than x squared? 2x minus 3y less than x
squared they ask us. All right, let's see if
we can figure it out. Statement 1, 2x minus 3y
is equal to minus 2. So they're telling us that the
left-hand side of this original question is
equal to minus 2. So now this question, if we
take statement 1 into consideration, it boils
down to is minus 2 less than x squared? So first you might be saying,
oh, I don't know. I don't know what x is. But think about it. Can x be a negative number? At worst case, x is 0. That's the worst case. If x was anything else, x
squared is going to be a positive number. So we actually know, just from
statement 1, that negative 2 has to be less than x squared,
unless x was some type of weird, imaginary number
or something. I think that doesn't
occur on the GMAT. So statement 1 alone
is sufficient. x squared is going to be at
least 0 or some positive number larger than 0. So statement 1 is sufficient. Statement two, x is greater
than 2 and y is greater than 0. So let me just try to come up
with two contradictory cases using these conditions to see. Let's say if x was-- and they
don't tell us that anything is an integer here. So if x was 2.1, x is equal to
2.1 and y is equal to-- I don't know-- y is
greater than 0. So let's say that y
is equal to 0.1. Let's see what happens here. So 2 times 2.1. You get 4.2 minus 3 times y. So minus 0.3 is less
than x squared. It's what? It's like 4 point something. So it's greater than 2. This is interesting. Actually this is interesting. So actually, you don't have
to try examples out. Let's just write the
equation down. 2x minus 3y is less
than x squared. So if y is greater than 0,
which they tell us in statement 2, then the quantity
on the left-hand side is always going to be 2x. It's always going to be
less than 2x, right? Because this'll have some
positive value. It might have an insignificantly
small value. We could make it 0.000001. But it's still going to have
some positive value. So no matter what y we choose,
as long as it's greater than 0, 2x minus 3y will be
greater than 2x. So we could say that if 2x is
less than x squared for any x we pick, than 2x minus 3y is
definitely going to be less than x squared. Why is that? Because minus 3y is going to
subtract from the 2x, because we know that y is
greater than 0. So let me ask you a question. Is 2x always going to be less
than x squared if x is greater than 2? Sure. I mean whatever x is, on the
left-hand side it's going to be 2 times-- pick a random
number-- 2 times x. It could be 2.5. But on the other side, on the
right-hand side, you're going to have 2.5 squared. So if you think about
it, you're going to have 2 times a number. That's always going to be less
than that number, which happens to be larger
than 2 squared. The smallest possible
is 2 times 2.00001. You could make a lot
of 0's there. But I think you get the point. That's still going to be less
than 2.0001 squared. Because you only have a 2 here
instead of a 2.0001. So actually, statement 2
alone is sufficient. Hopefully that made sense. You just have to see oh, if y is
greater than 0, then the 3y is definitely going to take
away from the 2x. Then as long as x is greater
than 2, 2x is always going to be less than x squared. If that makes sense to you, then
you should realize that statement 2 alone is sufficient
for this. That was interesting. Oh, no sorry. Either of them alone
were sufficient. We did that in the
first one, right? Statement 1 or statement 2,
independently are sufficient to answer that question. 120. A report consisting of 2,600
words is divided into 23 paragraphs. So 2,600 words and it
has 23 paragraphs. A two-paragraph preface is
then added to the report. Is the average number of words
per paragraph for all 25 paragraphs less than 120? So this is interesting. So the average number of words
per paragraph for all 25 paragraphs less than 120. So if we knew the total
number of words in the document we'd be set. We would just take the total
words divided by the total paragraphs they have now. They have the 23 original plus
the 2 from the preface, you have 25 total paragraphs. Now it told you the average
words per paragrah. Now, we don't know the total
words, because the total is going to be equal to the 2,600
words in the body of the document plus the words from the
preface-- I'll call that words sub preface-- all
of that divided by 25. This is the average. This is what we really have to
figure out, to figure out if it's less than 120. That's what they want to know. So statement number 1. Each paragraph of the preface
has more than 100 words. So I guess we could say that
the preface, the words from the preface are greater
than 200. That's another way of saying
that the total is going to be greater than 2,800. Now if the total is greater
than 2,800, what does that tell us about the total divided
by-- excuse me. So then the total divided by 25
is going to be greater than 2,800 divided by 25. What's 2,800 divided by 25? Let's see, 25 goes into 2,800. 1, 25, 30, 1, 1 times
25, 25, 50, 12. So it tells us that the
average is going to be greater than 112. So that doesn't help us. If it told us that the average
was going to be greater than 120, we would be done. We would say oh, we have enough
information to answer the question. Actually, the answer is no, that
the average is going to be greater than 120. But here, this tells us the
average is greater than 112. It could be 113 or
it could be 121. So it doesn't answer our
question of whether the average is less than 120. So statement one alone
is not that useful. Statement number 2 tells us each
paragraph of the preface has fewer than 100 words. So the total preface has
less than 300 words. They say each paragraph of the
preface, and there's 2 of them, has fewer than
150 words. So the total preface has
less than 300 words. So we could do the same logic. So that means that the total is
going to be less than what? The total is going to be less
than the original number of words plus the preface. So it's going to be less
than 2,600 plus 300. It's going to be less
than 2,900. Then if you divide both sides
by 25 to get the average, 25 goes into 2,900 really
four more times. That's right. The average is going to be less
than-- let me make sure I'm getting that right. So 4, 25 goes into 2,900. 1, 25, 40, 1, 25. Then we have 150. 25 goes into 150 six times. So we have the average
is less than 1/16. So statement 2 alone
is sufficient. Statement 2 tells us that the
average of the words per paragraph is less than
116, which is definitely less than 120. So statement 2 alone is
sufficient, and statement 1 doesn't help us much. I've run out of time. See you in the next video.