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Current time:0:00Total duration:10:34

We're now on problem
number 28. And the question is,
is x an integer? Statement one tells us that
x over 2 is an integer. So if you think about it, x has
to be an integer, because x is divisible by 2. This tells us that x is
divisible by 2, because when I divide by 2 it's not
5.5 or something. I get an even-- 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 2, then x could be
6, 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 2. 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 1. So that 2x is an integer,
but x is not an integer. But if x is equal to one,
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. 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 0. 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 1, 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 ask 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. You'd have to set up a bunch
of different equations. The easiest thing to
do actually would be to expand this. So what's x plus 1 squared? What's x plus 1 times
x plus 1? So that's x squared plus 2x plus
1 is equal to x squared. And 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 information
do they give us? So they tell us that k plus
r is equal to 20. It's not obvious to me how to
figure out from this, what 1 over k plus 1 over r is. And just to experiment, if
we were to find a common denominator and 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 r's and
get 1 over k plus-- and 1 over r is the same thing
as k over kr. 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 one just
gives us the top. So that by itself
isn't enough. What does statement
two give us? kr is equal to 64. So statement two gives us
this information, right? Statement one 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 C. 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. So statement one tells us that
1/4 is less than x, which is less than 1/2. So if x is greater than 1/4,
statement one immediately tells us that x is not 1/4. Let's see. And x has to be less than 1/2. Well both of these numbers
are less than 1/2, right? Statement one. Let me do this in a color. Statement one tells us that
our answer is one of these two, right? That's what statement
one tells us. Because both of those-- that's
0.4, that's less than 1/2. 3/8 is what? Actually, no. This is interesting. 3/8 is actually greater-- no,
3/8 is obviously less than 1/2, right? What am I thinking. 4/8 is exactly 1/2. So 3/8 is also less than 1/2. So statement one tells us it's
one of those two choices. And statement two 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. Let's see. 3/8 is greater than
1/3, right? Because 3/9 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 1/3. Let's see. Is 3/8 less than 3/5? Well, sure, right? 3/8 is definitely
less than 3/5. And 2/5 is definitely
less than 3/5. So it seems that statements
two and one are actually giving you the same
information. And when you get both of them,
they actually don't clarify. 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 in real time. I don't know if-- it's very
possible that I make an error. Problem 32. Triangle PQR. Let me draw that. If PQ is equal to x, so this
distance is equal to x. And QR is equal to x plus 2. And 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 1, 2, 3. You need to be able to figure
out which of these angles has the greatest degree measure. And it might be intuition
to you. But in order to figure out which
angle has the greatest degree measure, you essentially
have to figure out which angle is opposite
the largest side. That's 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, then we'd be all set. So first of all, we know that
x plus 2 is greater than x. So our answer, 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 do they say? They say 1y is equal
to x plus 3. So if y is equal to x plus 3,
this is the largest side of the triangle, right? This is the largest side
of the triangle. This is the second largest. And
then this will be-- if I'm thinking about this
right, this will be the largest angle. Let me think about that. I mean, you could go back
to a little bit of our trigonometry, but this would
be the largest angle. And let's see. So I think one alone
is enough. It tells us this is
the longest side. The triangle would look
something like this. This is the longest side. This is the second
longest side. And then this would be
the shortest side. It would look something
like that. And then angle three would
be the largest angle. Let's see. Statement number two tells
us that x is equal to 2. x is equal to 2 gives us
no information about y. If we say that x is equal to 2,
then we know that x plus 2 is equal to 4. And we have no information
about y. So that really still
doesn't help us. Because it could easily be
a triangle like this. 2, and then the 4 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 4 could be like
that, if y was a really large number. Let's see. If you have the combination of
the two, well-- we know that one alone is sufficient. Two alone is not sufficient. And I'm just doing-- let me
think about it 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 2. Yeah. So unless I'm wrong,
the answer is a. Statement one alone
is sufficient. I'll see you in the
next video.