Algebra II: Simplifying Polynomials

WEBVTT
Kind: captions
Language: en

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We're on problem 17.

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And it asks us which expression
shows the complete

00:00:06.850 --> 00:00:15.310
factorization of 12x
squared minus 147?

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Let's see, first of all,
can we factor out?

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147 is a strange
looking number.

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Let's see if we can factor
out something.

00:00:21.690 --> 00:00:23.390
It's definitely not divisible
by 12, because

00:00:23.390 --> 00:00:25.490
12 goes into 144.

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Is it divisible by 3?

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1 plus 4 is 5, 5 plus 7 is 12,
and 12 is divisible by 3.

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And I don't know if you've
seen that before.

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You just add up the digits and
if that is divisible by 3,

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then the number's
divisible by 3.

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So 147 is the divisible by 3.

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Let's see what it is.

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Let's work it out.

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It's 4 times 3 is
12, 27, it's 49.

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So we can factor
3 out of this.

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So then it becomes 3 times--
what's 12 divided by 3-- 4x

00:00:55.761 --> 00:01:00.410
squared minus-- 147 divided
by 3, which we just

00:01:00.410 --> 00:01:03.070
figured out, was 49.

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And now this, once again, looks
just like a squared

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minus b squared, where a would
be 2x and b would be 9.

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So now we can factor
that more.

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And I'll just switch
colors arbitrarily.

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That becomes 3 times a plus b,
so that would be 2x plus 7

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times a minus b.

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2x minus 7.

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And that is choice D.

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In choice D, they wrote the 2x
minus 7 first, but it's the

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same difference.

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So it's choice D.

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Problem 18.

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Let me see if I can get
problem 18 going.

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Problem 18.

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Well, maybe I should
cut and paste it.

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Why not?

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So I copied it and then
let me paste it here.

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That's the problem.

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I'll rewrite it because I don't
if that's big enough for

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you to see.

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So it says x plus 3 over x plus
5 plus 6 over x squared

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plus 3x minus 10.

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So when you add fractions,
whether you're doing it with

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algebraic fractions or regular
fractions, you have to find a

00:02:17.030 --> 00:02:18.510
common denominator.

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And let's see, we have to find
the least common multiple of

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the denominators here, but I
have a suspicion that this x

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plus 5 goes into this.

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So let's see if I
can factor this.

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What two numbers, when I
multiply them, equal minus 10,

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and when I add them
equal plus 3?

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Let's see.

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Well, 5 times what
is minus 10?

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It's 5 times minus 2 and 5 plus
minus 2 is 3, so that

00:02:41.730 --> 00:02:44.870
works. x minus 2.

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So this is actually our
least common multiple.

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This expression right here.

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So let me write it that way.

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So that is equal to-- I'm just
going to rewrite this as x

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plus 5 over x minus 2.

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Now, x plus 3 over x plus 5, if
we were to multiply both of

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those times x minus 2 to get
something in this form, this x

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plus 3 over x plus 5 is the same
thing as x plus 3 times x

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minus 2 over x plus 5 times
x minus 2, right?

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You could just cancel this
out right here and

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you'd get back to that.

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And now we're adding that.

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We're adding this term
to this term.

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6 over this, well, this is the
same thing as this, so that's

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just plus 6.

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And now we just get into
simplification mode.

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So x times x is x squared.

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x times minus 2 is minus 2x.

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3 times x is plus 3x.

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3 times minus 2 is minus 6, so
that's this term right here.

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Got us that.

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And then we have the plus 6.

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And then all of that
is over this stuff.

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And I look at the choices and it
seems like they have it in

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this form, so I'll just
write it in that form.

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x squared plus 3x minus 10.

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And let's see, the minus 6 and
the plus 6 cancels out, and

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we're left with minus
2x plus 3x.

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So that's x squared-- minus 2
plus 3-- plus x over x squared

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plus 3x minus 10.

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And that is choice A.

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Choice A.

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Next problem.

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I'm almost out of space.

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I'll draw a line here, just so
you don't get distracted.

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What is the simplified form of,
and they write 3a squared

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b to the third, c to the minus
2, all of that over--

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interesting-- a to the minus 1
b squared c, and all of that

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is to the third power.

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So let's get in simplification
mode.

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So this bottom part, we can
re-simplify as-- let's see,

00:05:05.820 --> 00:05:06.770
maybe I wrote too big.

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3a squared, b to the third,
c to the minus 2.

00:05:11.630 --> 00:05:15.610
All of that, it's a to the
negative 1, b squared, c to

00:05:15.610 --> 00:05:18.220
the third power, that's
each of these items

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to the third power.

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So a to the minus 1 to
the third power.

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You can multiply
the exponents.

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That becomes a to the minus 3.

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I just took the negative
1 times the 3.

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b squared to the third power.

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That's b to the 2 times 3.

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That's b to the sixth power.

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And then finally c.

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Well, that's just c to the first
of the third power, so

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that's c to the third.

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And now we can just say, well,
this is the same thing.

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Let me switch colors.

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This color's getting mundane.

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This is equal to 3 times a to
the 2-- we're dividing by a to

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the negative third-- so
it's 2 minus minus 3.

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Let me write that, just so you
understand-- 2 minus minus 3

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power-- that's where I got the
2; that's where I got the

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minus 3, and I subtracted
because I'm dividing-- times b

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to the 3 minus 6, times c
to the minus 2 minus 3.

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Once again, if we were
multiplying these two, I would

00:06:21.200 --> 00:06:22.890
add the exponents.

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But anyway, let's
simplify this.

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This equals 3a, 2 minus minus,
so that's plus, 3a to the

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fifth, b to the minus
3, c to the minus 5.

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And this is the same thing
as 3a to the fifth.

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b to the minus 3 is the same
thing as 1 over b to the

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third, so over b to the third.

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And this is the same thing
as 1 over c to the fifth.

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And that is choice A.

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Choice A.

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Next problem.

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Next problem.

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Oh, we already finished
that page.

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All right.

00:07:08.670 --> 00:07:09.990
Let me copy and paste
what they wrote.

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Let me put it at the top of this
right there and paste it.

00:07:19.730 --> 00:07:21.600
That's what they're asking
and I'll write it.

00:07:21.600 --> 00:07:24.630
20x to the minus fourth
over 27y squared

00:07:24.630 --> 00:07:27.590
divided by this fraction.

00:07:27.590 --> 00:07:29.620
So first of all, when you divide
by a fraction, that's

00:07:29.620 --> 00:07:32.120
the same thing as multiplying
by the inverse, right?

00:07:32.120 --> 00:07:33.520
So let's do that.

00:07:33.520 --> 00:07:36.130
I want to get rid of this
pesky-looking division sign.

00:07:36.130 --> 00:07:44.130
So let's rewrite this as 20x to
the minus fourth over 27y

00:07:44.130 --> 00:07:47.770
squared times-- instead of
dividing by this, let's

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multiply by the inverse.

00:07:49.070 --> 00:07:50.600
So what's the inverse?

00:07:50.600 --> 00:07:57.570
15y to the minus 5 over
8x to the minus 3.

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I just flipped it.

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All right?

00:08:00.500 --> 00:08:02.600
So let's see what
we can do here.

00:08:02.600 --> 00:08:05.890
It seems like a lot of these
numbers have common factors.

00:08:05.890 --> 00:08:11.530
Let's see, if we divide 15
by 3 and 27 by 3, so

00:08:11.530 --> 00:08:13.910
this becomes 5.

00:08:13.910 --> 00:08:16.960
27 divided by 3 becomes 9.

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And just so you can think,
you can view this as one

00:08:18.380 --> 00:08:20.760
continuous denominator or
one continuous fraction.

00:08:20.760 --> 00:08:24.020
20x to the minus 4 times-- well,
now it's 5y to the minus

00:08:24.020 --> 00:08:26.780
fifth divided by 9y squared.

00:08:26.780 --> 00:08:28.440
Because when you multiply
fractions, you just multiply

00:08:28.440 --> 00:08:30.230
the numerator times the
numerator divided by the

00:08:30.230 --> 00:08:32.140
denominator times
the denominator.

00:08:32.140 --> 00:08:33.950
Anyway, let's just
keep simplifying.

00:08:33.950 --> 00:08:39.669
If you divide this
by 4, you get 5.

00:08:39.669 --> 00:08:42.190
If you divide this
by 4, you get 2.

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Let's see, I don't want to do
too many steps all at once.

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You get 5x to the minus fourth
times 5y to the minus fifth,

00:08:51.480 --> 00:08:58.040
all of that over 9y squared
times 2x to the minus 3.

00:08:58.040 --> 00:08:58.290
So let's see.

00:08:58.290 --> 00:08:59.420
Let's get all the numbers out.

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So that is equal to 5 times 5
is 25 over 9 times 2 is 18

00:09:09.220 --> 00:09:15.590
times x-- let's do the x-- x to
the minus 4-- minus because

00:09:15.590 --> 00:09:17.950
we're dividing-- minus 3.

00:09:17.950 --> 00:09:29.050
Minus minus 3 times y to the
minus 5 minus 2, because we're

00:09:29.050 --> 00:09:30.420
dividing by that one.

00:09:30.420 --> 00:09:34.850
And that is equal to 25/18.

00:09:34.850 --> 00:09:39.010
Let's see, a minus minus,
so that becomes a plus.

00:09:39.010 --> 00:09:44.010
Minus 4 plus 3, x to the
negative 1, and then minus 5

00:09:44.010 --> 00:09:47.170
minus 2, y to the minus 7.

00:09:50.600 --> 00:09:53.270
This is the same thing
as 1 over x to the 1.

00:09:53.270 --> 00:09:54.830
This is the same thing
is 1 over y to the 7.

00:09:54.830 --> 00:10:01.006
So this is equal to
25 over 18x, 1

00:10:01.006 --> 00:10:04.460
over x, y to the seventh.

00:10:04.460 --> 00:10:06.050
y to the seventh in the
denominator is the same thing

00:10:06.050 --> 00:10:08.300
as y to the minus seventh
in the numerator.

00:10:08.300 --> 00:10:11.390
Anyway, that is choice--
let's see, 25 over 18--

00:10:11.390 --> 00:10:13.630
that is choice D.

00:10:13.630 --> 00:10:15.020
Choice D.

00:10:15.020 --> 00:10:16.080
And I'm out of time.

00:10:16.080 --> 00:10:17.800
See you in the next video.


CA Standards:  Algebra II

WEBVTT
Kind: captions
Language: en

00:00:00.710 --> 00:00:02.520
I copied and pasted the next
view because they had a little

00:00:02.520 --> 00:00:04.600
bit of a drawing, so I thought,
better than me

00:00:04.600 --> 00:00:06.420
drawing it myself.

00:00:06.420 --> 00:00:09.140
So we're on problem number 13
and I cut it off a little bit.

00:00:09.140 --> 00:00:11.360
What is the volume of
the figure below?

00:00:11.360 --> 00:00:13.670
Well, the volume is just
the width times the

00:00:13.670 --> 00:00:15.550
height times the depth.

00:00:15.550 --> 00:00:21.580
So the volume is just going to
be-- the width is x plus 4

00:00:21.580 --> 00:00:27.340
times x plus 1 times x plus 6.

00:00:27.340 --> 00:00:28.510
X plus 4 times x plus 1.

00:00:28.510 --> 00:00:32.810
We can do this in our head, but
I always find it easier to

00:00:32.810 --> 00:00:35.780
do it this way. x plus
1 times x plus 1.

00:00:35.780 --> 00:00:37.260
1 times 4 is 4.

00:00:37.260 --> 00:00:39.420
1 times x is x.

00:00:39.420 --> 00:00:44.630
And then you have the x place.
x times 4 is 4x, x

00:00:44.630 --> 00:00:48.250
times x is x squared.

00:00:48.250 --> 00:00:49.010
Fair enough.

00:00:49.010 --> 00:00:54.270
So that equals x squared
plus 5x plus 4.

00:00:54.270 --> 00:00:56.630
Now, we have to multiply that
times x plus 6, right?

00:00:56.630 --> 00:00:59.690
We did x plus 4 and x plus 1,
and this was a 6. x plus 6,

00:00:59.690 --> 00:01:00.850
that's this side.

00:01:00.850 --> 00:01:05.730
So now we multiply this
times x plus 6.

00:01:05.730 --> 00:01:08.270
6 times 4 is 24.

00:01:08.270 --> 00:01:12.900
6 times 5x is 30x.

00:01:12.900 --> 00:01:17.315
6 times x squared
is 6x squared.

00:01:17.315 --> 00:01:19.910
I hope you can read
what I'm writing.

00:01:19.910 --> 00:01:20.770
Scroll down a little bit.

00:01:20.770 --> 00:01:21.920
Now we do the x's place.

00:01:21.920 --> 00:01:23.130
I'll do it in a different color
because I think I'm

00:01:23.130 --> 00:01:29.921
going to run out of--
so x times 4 is 4x.

00:01:29.921 --> 00:01:32.020
x times 5x is 5x squared.

00:01:34.860 --> 00:01:37.795
x times x squared is
x to the third.

00:01:37.795 --> 00:01:40.210
Have to run into
the white area.

00:01:40.210 --> 00:01:42.760
So this is x to the third.

00:01:42.760 --> 00:01:43.750
What's 6 plus 5?

00:01:43.750 --> 00:01:49.580
Plus 11x squared plus
30x-- no, no, sorry.

00:01:49.580 --> 00:01:53.720
30 plus 4 is plus 34x plus 24.

00:01:53.720 --> 00:01:55.610
So x to the third, all of these
choices are x to the

00:01:55.610 --> 00:01:57.820
third, plus 11x squared,
so it's either b

00:01:57.820 --> 00:02:01.000
your d, plus 34x.

00:02:01.000 --> 00:02:01.730
That's that one.

00:02:01.730 --> 00:02:04.570
Plus 34x plus 24.

00:02:04.570 --> 00:02:07.330
So the choice is b.

00:02:07.330 --> 00:02:12.235
Next problem All right.

00:02:17.200 --> 00:02:19.670
8a to the third plus
c to the third.

00:02:19.670 --> 00:02:20.770
Which of these choices?

00:02:20.770 --> 00:02:21.760
So that's interesting.

00:02:21.760 --> 00:02:25.960
This is not a trivial thing
to just factor.

00:02:25.960 --> 00:02:28.290
We'll have to do some
intuition here.

00:02:28.290 --> 00:02:29.950
Well, let's think about it.

00:02:29.950 --> 00:02:37.010
In order to have a positive
number for the a term, every

00:02:37.010 --> 00:02:38.800
time we're multiplying, we have
to have a positive, so

00:02:38.800 --> 00:02:41.710
this is 2a times 2a times 2a.

00:02:41.710 --> 00:02:43.810
That would get us to
8a to the third.

00:02:43.810 --> 00:02:45.490
But there's something
interesting about choice a,

00:02:45.490 --> 00:02:49.210
why I already feel like
canceling out choice a.

00:02:49.210 --> 00:02:51.930
Choice a is essentially
2a plus c to

00:02:51.930 --> 00:02:53.570
the third power right?

00:02:53.570 --> 00:02:57.580
Choice a is 2a plus c
to the third power.

00:02:57.580 --> 00:03:00.280
And every time you multiply
this out, every time you

00:03:00.280 --> 00:03:03.410
multiply 2a plus c times itself,
you're going to add

00:03:03.410 --> 00:03:05.810
more terms. No terms are
going to cancel out.

00:03:05.810 --> 00:03:08.900
I mean, if you were to square
this, you're going to get,

00:03:08.900 --> 00:03:11.240
just to show you, you
get 4a squared

00:03:11.240 --> 00:03:15.570
plus 4ac plus c squared.

00:03:15.570 --> 00:03:17.800
And then to get the third power,
you'd multiply that

00:03:17.800 --> 00:03:19.790
times 2a plus c.

00:03:19.790 --> 00:03:20.990
So you're going to
get this thing.

00:03:20.990 --> 00:03:25.420
You're actually going to get
four terms. Choice a is

00:03:25.420 --> 00:03:27.240
definitely not the case.

00:03:27.240 --> 00:03:29.795
In order for one of these to
simplify to this, we're going

00:03:29.795 --> 00:03:31.950
to have to have some positives
and negatives there to cancel

00:03:31.950 --> 00:03:34.462
out terms as you expand
these polynomials.

00:03:34.462 --> 00:03:36.700
So let's look at the other
choices and see if

00:03:36.700 --> 00:03:37.540
they make any sense.

00:03:37.540 --> 00:03:41.890
2a times-- and I like looking
at the a terms and what

00:03:41.890 --> 00:03:44.190
generates the c cubed terms and
maybe we can cancel out

00:03:44.190 --> 00:03:45.040
some things.

00:03:45.040 --> 00:03:49.370
2a times 4a squared, that's
8a to the third.

00:03:49.370 --> 00:03:54.200
But minus c times c squared,
that's minus c to the the

00:03:54.200 --> 00:03:55.590
third, right?

00:03:55.590 --> 00:03:57.840
So I don't even have to worry
about multiplying

00:03:57.840 --> 00:03:58.590
this whole thing out.

00:03:58.590 --> 00:04:02.290
I know because the c cubed term
is going to come only

00:04:02.290 --> 00:04:05.090
from this term multiplied
by this term.

00:04:05.090 --> 00:04:07.450
There'll be other terms that
hopefully will cancel out, but

00:04:07.450 --> 00:04:10.280
we know that choice b cannot be
the choice because minus c

00:04:10.280 --> 00:04:13.660
times c squared is minus c to
the third, and this is a plus

00:04:13.660 --> 00:04:14.480
right here.

00:04:14.480 --> 00:04:17.190
So we know b can't be
the right answer.

00:04:17.190 --> 00:04:18.779
Let's see, choice c.

00:04:18.779 --> 00:04:19.810
I can use the same logic.

00:04:19.810 --> 00:04:22.260
We have a minus c and
we have a plus c.

00:04:22.260 --> 00:04:23.280
We have a plus c here.

00:04:23.280 --> 00:04:25.770
We would have to have two plus
c's in order for this to work,

00:04:25.770 --> 00:04:27.370
so this can't be the answer.

00:04:27.370 --> 00:04:29.900
So just by deductive reasoning,
I can say it's

00:04:29.900 --> 00:04:32.730
probably choice d, but I'm going
to prove it to you since

00:04:32.730 --> 00:04:37.170
we're here to learn, not to just
answer things correctly.

00:04:37.170 --> 00:04:42.060
So if I were to multiply those
two numbers, I have 4a squared

00:04:42.060 --> 00:04:47.720
minus 2ac plus c squared,
and I'm multiplying that

00:04:47.720 --> 00:04:51.580
times 2a plus c.

00:04:54.730 --> 00:04:57.510
So I'm going to mix up all of
the terms a little bit, but

00:04:57.510 --> 00:04:58.920
you will get the idea.

00:04:58.920 --> 00:05:00.890
So the important thing when
you're multiplying, you could

00:05:00.890 --> 00:05:02.700
do this in your head, is that
you multiply each of these

00:05:02.700 --> 00:05:04.390
terms times each of those.

00:05:04.390 --> 00:05:07.800
So c times all of that plus
this times all of of that.

00:05:07.800 --> 00:05:11.750
So c times all of that
is c times c squared.

00:05:11.750 --> 00:05:14.100
I'm not doing it in the right
place notation like I did last

00:05:14.100 --> 00:05:16.090
time because it gets a little
more confusing now.

00:05:16.090 --> 00:05:19.010
But c times c squared
is c to the third.

00:05:21.700 --> 00:05:28.650
c times minus 2ac is
minus 2ac squared.

00:05:28.650 --> 00:05:31.570
c times 4a squared
is 4a squared c.

00:05:35.170 --> 00:05:37.940
And now we can do this one.

00:05:37.940 --> 00:05:39.380
2a times c squared.

00:05:42.860 --> 00:05:44.730
I'm going to put it in this
place because it turns into ac

00:05:44.730 --> 00:05:46.910
squared, right? ac squared
and I have a 2.

00:05:46.910 --> 00:05:49.166
So 2a times c squared
is 2ac squared.

00:05:51.950 --> 00:05:58.920
2a times minus 2ac, you get a
minus 4a squared c, so it's

00:05:58.920 --> 00:06:02.980
minus 4a squared c.

00:06:02.980 --> 00:06:04.170
This is a plus here.

00:06:04.170 --> 00:06:08.520
And then 2a times 4a squared
is 8a to the third.

00:06:08.520 --> 00:06:11.510
And luckily enough, let's see,
they get 8a to the third.

00:06:14.250 --> 00:06:16.750
This cancels out, this cancel
out, and you get

00:06:16.750 --> 00:06:18.130
plus c to the third.

00:06:18.130 --> 00:06:21.210
So we were right by doing our
deductive reasoning, that

00:06:21.210 --> 00:06:24.830
choice d was the answer.

00:06:24.830 --> 00:06:26.170
Next problem.

00:06:26.170 --> 00:06:27.420
I've run out of space.

00:06:31.740 --> 00:06:36.210
Let me see what the next
problem even is.

00:06:36.210 --> 00:06:37.740
Actually, where was I?

00:06:37.740 --> 00:06:38.990
OK, I see.

00:06:38.990 --> 00:06:41.500
OK, problem 15.

00:06:41.500 --> 00:06:44.234
The total area of a rectangle
is 4x to the

00:06:44.234 --> 00:06:47.205
fourth minus 9y squared.

00:06:49.720 --> 00:06:53.660
Which factors could represent
the length times the width?

00:06:53.660 --> 00:06:57.060
OK, so essentially you're saying
the length times the

00:06:57.060 --> 00:06:59.765
width is going to be equal to
the area, which is 4x to the

00:06:59.765 --> 00:07:04.610
fourth minus 9y squared.

00:07:04.610 --> 00:07:08.350
So to some degree, we have to
factor this into multiple

00:07:08.350 --> 00:07:09.290
expressions.

00:07:09.290 --> 00:07:11.730
And the first thing you should
always look at when you have

00:07:11.730 --> 00:07:14.790
to factor any type of an
expression, especially if

00:07:14.790 --> 00:07:18.060
there's only two terms in the
expression, does it fit the

00:07:18.060 --> 00:07:22.080
form a squared minus
b squared?

00:07:22.080 --> 00:07:25.980
Because if it does, you can
factor this into a plus b

00:07:25.980 --> 00:07:27.900
times a minus b.

00:07:27.900 --> 00:07:29.580
And if you don't trust that
these equal each other, I've

00:07:29.580 --> 00:07:32.250
made multiple videos about it,
but just multiply them. a plus

00:07:32.250 --> 00:07:35.390
b times a minus b is equal to
a squared minus b squared.

00:07:35.390 --> 00:07:39.630
And luckily, this fits
that format, right?

00:07:39.630 --> 00:07:46.650
If a squared was equal to 4x to
the fourth, then what's a?

00:07:46.650 --> 00:07:50.410
a would be equal
to 2x squared.

00:07:50.410 --> 00:07:57.325
And if b squared is equal to 9y
squared, then b is equal to

00:07:57.325 --> 00:08:00.620
the square root of that,
which is 3y.

00:08:00.620 --> 00:08:02.670
And so this definitely
is equal to a

00:08:02.670 --> 00:08:05.300
squared minus b squared.

00:08:05.300 --> 00:08:08.200
So this expression right here
can be factored into a plus b

00:08:08.200 --> 00:08:09.170
times a minus b.

00:08:09.170 --> 00:08:17.160
So a plus b is 2x squared
plus 3y times a minus b.

00:08:17.160 --> 00:08:22.250
2x squared minus 3y.

00:08:22.250 --> 00:08:23.810
Let's see, which
choice is that?

00:08:23.810 --> 00:08:25.920
2x squared plus 3y?

00:08:25.920 --> 00:08:26.870
OK, they switched the order.

00:08:26.870 --> 00:08:32.510
Choice a says 2x squared minus
3y times 2x squared plus 3y,

00:08:32.510 --> 00:08:33.179
which is the same thing.

00:08:33.179 --> 00:08:34.929
You can switch orders when
you're multiplying, so the

00:08:34.929 --> 00:08:38.140
answer is a.

00:08:38.140 --> 00:08:41.610
Next problem, problem 16.

00:08:41.610 --> 00:08:44.400
I'll switch colors
arbitrarily.

00:08:44.400 --> 00:08:51.850
Problem 16: Which product of
factors is equivalent to-- let

00:08:51.850 --> 00:08:55.680
me copy and paste this because
this is an interesting-- so

00:08:55.680 --> 00:08:58.860
let me copy and paste
this problem.

00:08:58.860 --> 00:09:04.450
OK, I've copied it, and now
let me paste it here.

00:09:04.450 --> 00:09:08.570
OK, which product of factors
is equivalent to that thing

00:09:08.570 --> 00:09:09.010
over there?

00:09:09.010 --> 00:09:10.340
So let's simplify that.

00:09:10.340 --> 00:09:12.090
That's my first impulse.

00:09:12.090 --> 00:09:15.060
No, no, no, we don't even
have to simplify it.

00:09:15.060 --> 00:09:17.960
This fits our same pattern,
right, if we can fit the

00:09:17.960 --> 00:09:20.730
pattern a squared
minus b squared.

00:09:20.730 --> 00:09:21.550
So look at this thing.

00:09:21.550 --> 00:09:27.060
This has x plus 1 squared
minus y squared.

00:09:27.060 --> 00:09:32.560
So if we said that a is equal to
x plus 1 and b is equal to

00:09:32.560 --> 00:09:34.950
y, then we fit the pattern.

00:09:34.950 --> 00:09:37.530
This is a squared
minus b squared.

00:09:37.530 --> 00:09:41.320
So then this thing can just
factor into a plus b

00:09:41.320 --> 00:09:43.620
times a minus b.

00:09:43.620 --> 00:09:44.950
So what's a plus b?

00:09:44.950 --> 00:09:49.440
It's x plus 1 plus y.

00:09:49.440 --> 00:09:50.530
That's a plus b.

00:09:50.530 --> 00:09:56.320
And then a minus b is
x plus 1 minus y.

00:09:56.320 --> 00:09:59.030
And that is choice d.

00:09:59.030 --> 00:10:01.890
That problem was easier than I
thought I was going to be.

00:10:01.890 --> 00:10:06.510
Choice d, right there.

00:10:06.510 --> 00:10:07.710
And I'm out of time again.

00:10:07.710 --> 00:10:09.990
I'll see you in the
next video.


CA Standards: Algebra II (Algebraic Division/Multiplication)

WEBVTT
Kind: captions
Language: en

00:00:00.620 --> 00:00:01.030
All right.

00:00:01.030 --> 00:00:02.000
We're on problem number 9.

00:00:02.000 --> 00:00:06.440
At least for the next few, I
don't think I have to copy and

00:00:06.440 --> 00:00:08.020
paste these problems.
I can just write it.

00:00:08.020 --> 00:00:10.670
It looks like a little bit of
algebraic long division.

00:00:10.670 --> 00:00:18.140
Problem 9: They want us to
divide 2x plus 7 into-- it's

00:00:18.140 --> 00:00:22.310
pretty long so I'll give it some
space-- 2x to the fourth

00:00:22.310 --> 00:00:34.110
plus 21x to the third plus 35x
squared minus 37x plus 46.

00:00:34.110 --> 00:00:35.670
This might look daunting,
but once you

00:00:35.670 --> 00:00:37.560
start, it's not so bad.

00:00:37.560 --> 00:00:39.520
So like any long division,
you kind of look

00:00:39.520 --> 00:00:40.560
at the biggest place.

00:00:40.560 --> 00:00:43.890
And the largest degreed place,
at least in this polynomial,

00:00:43.890 --> 00:00:45.330
is the 2x place.

00:00:45.330 --> 00:00:46.290
And that's what we work with.

00:00:46.290 --> 00:00:50.820
So we say, how many times does
2x go into 2x to the fourth?

00:00:50.820 --> 00:00:51.160
Well, let's see.

00:00:51.160 --> 00:00:53.900
The 2 goes into the 2 one time,
and x goes into x to the

00:00:53.900 --> 00:00:56.050
fourth x to the third times.

00:00:56.050 --> 00:00:57.415
What's x to the fourth
divided by x?

00:00:57.415 --> 00:00:58.570
It's x to the third.

00:00:58.570 --> 00:01:02.570
So we say it goes into it
x to the third times.

00:01:02.570 --> 00:01:04.604
And if we were right, when we
multiply, we should get 2x to

00:01:04.604 --> 00:01:04.994
the fourth.

00:01:04.994 --> 00:01:10.630
So x to the third times 2x
is 2x to the fourth.

00:01:10.630 --> 00:01:18.700
And then x to the third times
7 is plus 7x to the third.

00:01:18.700 --> 00:01:23.000
If this was some other-- well,
we'll worry about it when we

00:01:23.000 --> 00:01:24.940
get to it, but you would want
to put it under the

00:01:24.940 --> 00:01:27.190
appropriate place
in this number.

00:01:27.190 --> 00:01:29.640
It just happened to be that x to
the third was the next one.

00:01:29.640 --> 00:01:32.600
Now you subtract just like you
do with regular long division.

00:01:32.600 --> 00:01:33.450
These cancel out.

00:01:33.450 --> 00:01:35.715
21 minus 7 is 14x
to the third.

00:01:38.800 --> 00:01:41.470
And then you could bring down
all of this, but we can worry

00:01:41.470 --> 00:01:43.280
about that in a second.

00:01:43.280 --> 00:01:46.940
So how many times does 2x go
into 14x to the third?

00:01:46.940 --> 00:01:52.840
Well, 2 goes into 14 seven
times, plus 7, and x goes into

00:01:52.840 --> 00:01:55.480
x to the third x
squared times.

00:01:55.480 --> 00:01:57.980
If we're right, when we
multiply, it should work out.

00:01:57.980 --> 00:02:03.960
So 7x squared times 2x is equal
to 14x to the third,

00:02:03.960 --> 00:02:05.310
like we thought it should.

00:02:05.310 --> 00:02:08.090
7x squared times 7 is what?

00:02:08.090 --> 00:02:15.130
That's plus 49x squared.

00:02:15.130 --> 00:02:17.870
And now we subtract.

00:02:17.870 --> 00:02:20.840
14x to the third minus
14x to the third.

00:02:20.840 --> 00:02:22.170
That's 0.

00:02:22.170 --> 00:02:24.650
Now we have 35-- we could bring
this down, just because

00:02:24.650 --> 00:02:26.050
it makes it easier.

00:02:26.050 --> 00:02:28.490
35x squared.

00:02:28.490 --> 00:02:31.580
35x squared minus 49x squared.

00:02:31.580 --> 00:02:31.990
Let's see.

00:02:31.990 --> 00:02:34.120
So it's minus 14, right?

00:02:34.120 --> 00:02:35.810
49 minus 35.

00:02:35.810 --> 00:02:39.840
So it's minus 14x squared.

00:02:39.840 --> 00:02:42.640
And then 2x goes into
minus 14x squared.

00:02:42.640 --> 00:02:48.570
So 2 goes into minus 14 minus 7
times, and then x goes into

00:02:48.570 --> 00:02:51.230
x squared x times.

00:02:51.230 --> 00:02:54.690
When you multiply this minus 7x
times 2x, so you get minus

00:02:54.690 --> 00:02:58.262
7 times 2 is minus 14.

00:02:58.262 --> 00:03:00.610
x times x is x squared.

00:03:00.610 --> 00:03:01.390
Of course, you have
to multiply it

00:03:01.390 --> 00:03:02.690
times both terms, right?

00:03:02.690 --> 00:03:04.430
We're multiplying it times
this whole expression.

00:03:04.430 --> 00:03:13.670
Minus 7x times 7 is minus 49x.

00:03:13.670 --> 00:03:15.640
Now we're ready to subtract
again and I'm

00:03:15.640 --> 00:03:17.140
running out of space.

00:03:17.140 --> 00:03:18.560
These cancel out.

00:03:18.560 --> 00:03:24.210
We could bring this minus
37x down, so minus 37x.

00:03:24.210 --> 00:03:27.040
Remember, we're subtracting
this, right?

00:03:27.040 --> 00:03:29.170
We could almost add
the negative.

00:03:29.170 --> 00:03:31.350
Let's make the negative.

00:03:31.350 --> 00:03:36.250
So this is minus 37x minus minus
49x is the same thing as

00:03:36.250 --> 00:03:39.220
minus 37x plus 49x.

00:03:39.220 --> 00:03:41.360
I made both of those
positive just to

00:03:41.360 --> 00:03:43.620
simplify it in my brain.

00:03:43.620 --> 00:03:43.900
Oh, whoops.

00:03:43.900 --> 00:03:45.880
I didn't want to go
down that much.

00:03:45.880 --> 00:03:46.320
OK.

00:03:46.320 --> 00:03:52.560
So these cancel out and then
this is equal to 12x.

00:03:52.560 --> 00:03:55.320
And 2x plus 7 goes into 12x.

00:03:55.320 --> 00:03:55.950
Well, now, let's see.

00:03:55.950 --> 00:04:01.010
2 goes into 12 six times, x goes
into x one time, so then

00:04:01.010 --> 00:04:04.530
this is equal to plus 6.

00:04:04.530 --> 00:04:07.980
6 times 2x plus 7.

00:04:07.980 --> 00:04:11.800
6 times to 2x-- I know you
can't see it-- is 12x.

00:04:11.800 --> 00:04:19.029
And then 6 times the 7 is 42.

00:04:19.029 --> 00:04:20.709
We have a 46 up there.

00:04:20.709 --> 00:04:21.720
I should have written smaller.

00:04:21.720 --> 00:04:23.500
So you get this 46.

00:04:23.500 --> 00:04:25.390
And now we're ready
to subtract again.

00:04:25.390 --> 00:04:27.340
12x minus 12x is 0.

00:04:27.340 --> 00:04:32.030
46 minus 42 is 4, so we
have a remainder of 4.

00:04:32.030 --> 00:04:35.360
So let's see, our answer
is all of this

00:04:35.360 --> 00:04:36.430
with a remainder 4.

00:04:36.430 --> 00:04:40.380
So all of that plus the
remainder over what we're

00:04:40.380 --> 00:04:44.190
using to divide into:
2x plus 7.

00:04:44.190 --> 00:04:46.620
So our answer, let's see
what choice it is.

00:04:46.620 --> 00:04:53.050
It's x to the third plus 7 x
squared minus 7x plus 6 plus 4

00:04:53.050 --> 00:04:54.750
over 2x plus 7.

00:04:54.750 --> 00:04:56.990
We just took our remainder and
said, OK, whatever this is,

00:04:56.990 --> 00:04:58.410
that's also there.

00:04:58.410 --> 00:05:05.100
So that is choice-- let's see, x
to the third plus 7x squared

00:05:05.100 --> 00:05:15.340
minus 7x plus 6 plus
4 over 2x plus 7.

00:05:18.500 --> 00:05:20.850
So it's actually choice
d is our answer.

00:05:20.850 --> 00:05:23.940
I was just reading choice a.

00:05:23.940 --> 00:05:25.300
I should have shown you that.

00:05:25.300 --> 00:05:27.360
So choice d is our answer.

00:05:27.360 --> 00:05:28.310
Next question.

00:05:28.310 --> 00:05:31.530
Problem number 10.

00:05:31.530 --> 00:05:34.000
That one took up more space
than I expected.

00:05:34.000 --> 00:05:35.510
Problem number 10.

00:05:35.510 --> 00:05:38.900
Which polynomial represents-- so
they want us to multiply--

00:05:38.900 --> 00:05:45.330
this how they write it-- 3x
squared plus x minus 4 times

00:05:45.330 --> 00:05:48.230
2x minus 5.

00:05:48.230 --> 00:05:50.520
I find that we should just
multiply these the way I would

00:05:50.520 --> 00:05:51.500
multiply any number.

00:05:51.500 --> 00:05:58.110
So I would rewrite it as 3x
squared plus x minus 4 times,

00:05:58.110 --> 00:06:01.250
and I put the same degree under
the same degree, times

00:06:01.250 --> 00:06:04.560
2x minus 5.

00:06:04.560 --> 00:06:07.650
And now let's take the
0th degree place.

00:06:07.650 --> 00:06:08.590
I guess you could
view that way.

00:06:08.590 --> 00:06:13.470
Minus 5 times minus
4 is plus 20.

00:06:13.470 --> 00:06:17.310
Minus 5 times x minus 5x.

00:06:17.310 --> 00:06:25.370
Minus 5 times 3x squared
is minus 15x squared.

00:06:25.370 --> 00:06:29.380
2x times minus 4 is minus 8x.

00:06:29.380 --> 00:06:33.800
I'll put it in this place
because that's our x's space.

00:06:33.800 --> 00:06:35.400
2x times x is 2x squared.

00:06:38.940 --> 00:06:41.110
And then I should have had all
of this-- actually, I think I

00:06:41.110 --> 00:06:42.040
can do that.

00:06:42.040 --> 00:06:44.830
Let me see if I can move all
of this to the right.

00:06:44.830 --> 00:06:47.740
The magic of computers.

00:06:47.740 --> 00:06:50.720
I can move it all to the right,
although I should color

00:06:50.720 --> 00:06:52.300
that back in.

00:06:52.300 --> 00:06:54.020
Let's see if I can color that.

00:06:54.020 --> 00:06:55.060
Look at that!

00:06:55.060 --> 00:06:57.830
All right, back in business.

00:06:57.830 --> 00:06:58.170
All right.

00:06:58.170 --> 00:07:00.930
So I did 2x times minus
4 is minus 8x.

00:07:00.930 --> 00:07:03.500
2x times x is 2x squared.

00:07:03.500 --> 00:07:10.950
2x times 3x squared is
6x to the third.

00:07:10.950 --> 00:07:12.070
This is a plus.

00:07:12.070 --> 00:07:15.390
Now we just add up everything.

00:07:15.390 --> 00:07:21.672
So we get 6x to the third minus
15 plus 2 minus 13x

00:07:21.672 --> 00:07:28.130
squared minus 13x plus 20.

00:07:28.130 --> 00:07:31.550
So 6x to the third minus 13x
squared-- I'm just checking

00:07:31.550 --> 00:07:35.200
all the choices-- minus
13x plus 20.

00:07:35.200 --> 00:07:37.880
That's choice b.

00:07:37.880 --> 00:07:41.340
Problem 11.

00:07:41.340 --> 00:07:42.360
Problem 11.

00:07:42.360 --> 00:07:49.770
Let's see, they wrote-- this
is problem 11, just so you

00:07:49.770 --> 00:07:50.600
know what we're doing.

00:07:50.600 --> 00:08:01.380
Minus 2x squared plus 6x plus
1 minus 2 times 4x squared

00:08:01.380 --> 00:08:06.350
minus 3x plus 1 is equal to--
well now, we just have to

00:08:06.350 --> 00:08:08.590
simplify this.

00:08:08.590 --> 00:08:09.590
Let me rewrite this part.

00:08:09.590 --> 00:08:14.595
This is minus 2x squared
plus 6x 1.

00:08:17.510 --> 00:08:18.940
Let's distribute a minus 2.

00:08:18.940 --> 00:08:25.250
So minus 2 times 4x squared
is minus 8x squared.

00:08:25.250 --> 00:08:26.440
That's that.

00:08:26.440 --> 00:08:32.200
Minus 2 times minus 3x,
that's plus 6x.

00:08:32.200 --> 00:08:35.710
Minus 2 times 1 is minus 2.

00:08:35.710 --> 00:08:36.409
And now, let's see.

00:08:36.409 --> 00:08:38.730
We can add the like terms up.

00:08:38.730 --> 00:08:39.630
Let's see what this
is equal to.

00:08:39.630 --> 00:08:40.789
Let's do the x squared term.

00:08:40.789 --> 00:08:47.250
So first I'll do the minus 2x
squared and minus 8x squared,

00:08:47.250 --> 00:08:52.120
so that is minus 2 minus 8,
that's minus 10x squared.

00:08:52.120 --> 00:08:54.270
Let me do another color
for each of them.

00:08:54.270 --> 00:08:57.610
Then I have the x terms. I
have 6x, and then I have

00:08:57.610 --> 00:08:58.930
another 6x.

00:08:58.930 --> 00:09:04.240
You add them together, you
get 12x, plus 12x.

00:09:04.240 --> 00:09:07.960
And then you have your ones
terms or x to the 0th power,

00:09:07.960 --> 00:09:08.570
you can view it.

00:09:08.570 --> 00:09:14.270
You have plus 1 and minus
2, so that's minus 1.

00:09:14.270 --> 00:09:19.430
And then that is choice-- let's
see, let me scroll down.

00:09:19.430 --> 00:09:24.130
That is choice d. minus 10x
squared plus 12x minus 1.

00:09:24.130 --> 00:09:26.270
Choice d.

00:09:26.270 --> 00:09:29.370
Problem number 12.

00:09:29.370 --> 00:09:30.570
Problem 12.

00:09:30.570 --> 00:09:31.820
We're on a roll.

00:09:35.070 --> 00:09:37.645
Problem 12: Which expression
is equivalent to-- so now

00:09:37.645 --> 00:09:39.380
they're multiplying.

00:09:39.380 --> 00:09:47.950
They write 6y squared minus
2 times 6y plus 2.

00:09:47.950 --> 00:09:50.230
You could do this in your head,
and you should practice

00:09:50.230 --> 00:09:51.810
doing it in your head, but if
you never want to make a

00:09:51.810 --> 00:09:53.850
mistake, you just write
it like this.

00:09:53.850 --> 00:09:55.900
6y squared.

00:09:55.900 --> 00:09:57.680
And I have nothing
in the x's spot.

00:09:57.680 --> 00:09:59.470
And I say minus 2.

00:09:59.470 --> 00:10:01.590
And then this one is 6y.

00:10:01.590 --> 00:10:03.520
I write that in the y spot.

00:10:03.520 --> 00:10:06.070
This is y squared, this is y,
this is y to the 0, which is

00:10:06.070 --> 00:10:08.850
just 1, plus 2.

00:10:08.850 --> 00:10:10.960
And now I multiply them.

00:10:10.960 --> 00:10:13.670
2 times minus 2 is minus 4.

00:10:13.670 --> 00:10:18.690
2 times 6y squared
is 12 y squared.

00:10:18.690 --> 00:10:21.520
So I write that in that
spot, 12y squared.

00:10:21.520 --> 00:10:25.380
Now 6y times minus
2 is minus 12y.

00:10:25.380 --> 00:10:26.630
So that'll go here.

00:10:29.520 --> 00:10:36.340
And then 6y times 6y squared
is 36y to the third.

00:10:36.340 --> 00:10:38.520
So we have our answer.

00:10:38.520 --> 00:10:42.747
It is-- you just add them up--
36y to the third plus 12y

00:10:42.747 --> 00:10:48.450
squared minus 12y minus 4.

00:10:48.450 --> 00:10:51.270
And that is choice d.

00:10:51.270 --> 00:10:53.030
And you could do this in your
head, and that's a good thing

00:10:53.030 --> 00:10:53.660
to practice.

00:10:53.660 --> 00:10:57.480
But I just wanted to make sure
it was clean when I did it.

00:10:57.480 --> 00:11:01.980
And let's see, should I continue
with the next video

00:11:01.980 --> 00:11:03.340
or should I stop there?

00:11:03.340 --> 00:11:04.230
Because I'm at 11 minutes.

00:11:04.230 --> 00:11:07.510
I'll stop there and I'll see
you in the next video.


California Standards Test: Algebra II (Graphing Inequalities

WEBVTT
Kind: captions
Language: en

00:00:00.710 --> 00:00:03.210
We're on problem number six and
I actually figured out how

00:00:03.210 --> 00:00:07.780
to cut and paste from the PDF
file to my little writing

00:00:07.780 --> 00:00:10.020
board, so we actually have
access to the problem.

00:00:10.020 --> 00:00:13.460
So it says: What system of
inequalities best represents

00:00:13.460 --> 00:00:15.270
the graph shown below?

00:00:15.270 --> 00:00:18.130
So we want to figure out what
inequalities represents this

00:00:18.130 --> 00:00:19.670
shaded area.

00:00:19.670 --> 00:00:22.580
So a good place to start is just
to figure out what these

00:00:22.580 --> 00:00:26.320
dotted line graphs, what the
equations of them look like.

00:00:26.320 --> 00:00:28.790
So this one, let's do the
hard one first, right?

00:00:28.790 --> 00:00:29.810
This is at an angle.

00:00:29.810 --> 00:00:31.960
So whenever I look at a line,
I like to think what's it's

00:00:31.960 --> 00:00:34.180
slope and what it's y-intercept,
right?

00:00:34.180 --> 00:00:37.860
And then you could just write
it in the form y is equal to

00:00:37.860 --> 00:00:42.790
mx plus b, where b is
the y-intercept

00:00:42.790 --> 00:00:45.170
and m is the slope.

00:00:45.170 --> 00:00:47.680
So let's try to figure out the
equation of this line up here,

00:00:47.680 --> 00:00:48.510
this dotted line.

00:00:48.510 --> 00:00:49.670
So what's its slope?

00:00:49.670 --> 00:00:53.730
So when I move over 1 and x,
when I move over to the right,

00:00:53.730 --> 00:00:56.970
when x increases by 1, or let's
say when x increases by

00:00:56.970 --> 00:00:59.920
2, how much does y increase?

00:00:59.920 --> 00:01:02.070
Well, y increases
by 2 as well.

00:01:02.070 --> 00:01:03.950
Or whatever we increase x by.

00:01:03.950 --> 00:01:07.230
If x increases by 1, then
y increases by 1.

00:01:07.230 --> 00:01:08.580
So the slope is 1, right?

00:01:08.580 --> 00:01:15.520
Slope is just change in
y over change in x.

00:01:15.520 --> 00:01:19.350
So the slope of this
first line is 1.

00:01:19.350 --> 00:01:22.950
So m is equal to 1, for at least
this first line up here.

00:01:22.950 --> 00:01:24.170
And what's it's y-intercept?

00:01:24.170 --> 00:01:28.055
The y-intercept is just what
does the graph equal when x is

00:01:28.055 --> 00:01:30.110
equal to 0 or where does it
intersect the y-axis?

00:01:30.110 --> 00:01:34.810
Well, it intersects the y-axis
right here at y is equal to 1,

00:01:34.810 --> 00:01:37.140
so b is equal to 1.

00:01:37.140 --> 00:01:44.720
So the equation of this dotted
line up here is y is equal to

00:01:44.720 --> 00:01:48.100
mx, the slope times x, we
figured out the slope was 1,

00:01:48.100 --> 00:01:51.450
so that's x plus the
y-intercept, which we also

00:01:51.450 --> 00:01:53.430
figured out was 1.

00:01:53.430 --> 00:01:55.770
So that's the graph of
that line, right?

00:01:55.770 --> 00:01:59.750
But notice, the shaded region
is below that line.

00:01:59.750 --> 00:02:02.760
And it also does not include
the line, right?

00:02:02.760 --> 00:02:03.960
That's why they drew
a dotted line.

00:02:03.960 --> 00:02:06.200
If it was a solid line, that
means that we're including the

00:02:06.200 --> 00:02:07.290
line itself.

00:02:07.290 --> 00:02:09.720
But the shaded region does
not include the line.

00:02:09.720 --> 00:02:10.680
But it's below it.

00:02:10.680 --> 00:02:15.260
Think of it this way: For any
given x-- let's pick this x.

00:02:15.260 --> 00:02:16.450
Let me do it in a
different color.

00:02:16.450 --> 00:02:17.940
Let me do it in blue.

00:02:17.940 --> 00:02:22.680
If I were to pick x is equal to
1, this is the point y is

00:02:22.680 --> 00:02:24.310
equal to x plus 1.

00:02:24.310 --> 00:02:25.470
That's the equation.

00:02:25.470 --> 00:02:30.420
Now, my question to you is y
above y is equal to x plus 1?

00:02:30.420 --> 00:02:32.900
Is the shaded region y is
greater than x plus 1, which

00:02:32.900 --> 00:02:34.270
would be that region?

00:02:34.270 --> 00:02:36.990
Or would it be y is less
than x plus 1,

00:02:36.990 --> 00:02:39.370
which is this region?

00:02:39.370 --> 00:02:41.350
Well, clearly, it is y is
less than x plus 1.

00:02:41.350 --> 00:02:44.450
You pick any x, x
is equal to 3.

00:02:44.450 --> 00:02:47.795
This is what y would be equal
to. y would be equal to 3 plus

00:02:47.795 --> 00:02:50.330
1 if you picked this choice.

00:02:50.330 --> 00:02:53.620
But the y's that satisfy the
grey area are always less than

00:02:53.620 --> 00:02:54.730
this point.

00:02:54.730 --> 00:02:57.940
So at least the grey area
relative to this equation, we

00:02:57.940 --> 00:03:05.030
can write the inequality y
is less than x plus 1.

00:03:05.030 --> 00:03:06.630
And we know it's not less
than or equal to.

00:03:06.630 --> 00:03:08.730
It would be less than or equal
to if they had filled in this

00:03:08.730 --> 00:03:11.660
line, if this was part of
the area, but it's not.

00:03:11.660 --> 00:03:12.610
They drew a dotted line.

00:03:12.610 --> 00:03:14.750
So that's why we know it's
less than, right?

00:03:14.750 --> 00:03:16.820
If the grey area was up here
it would be greater than.

00:03:16.820 --> 00:03:19.430
You just pick an x, you say,
oh, the y's less than the

00:03:19.430 --> 00:03:20.980
line, not the y is greater.

00:03:20.980 --> 00:03:22.450
But we're not done, right?

00:03:22.450 --> 00:03:27.780
Because this would be the
solution set if all of this

00:03:27.780 --> 00:03:28.520
area was grey.

00:03:28.520 --> 00:03:30.500
But it's only grey up
to a certain point.

00:03:30.500 --> 00:03:34.070
It's only grey up to this
line down here.

00:03:34.070 --> 00:03:34.930
This line is easier.

00:03:34.930 --> 00:03:37.000
This line has no slope, right?

00:03:37.000 --> 00:03:39.996
No matter what my change in
x, my change in y is 0.

00:03:39.996 --> 00:03:41.530
So what's the equation
of this?

00:03:41.530 --> 00:03:44.100
The equation of this line
is just slope is 0.

00:03:44.100 --> 00:03:46.770
So y is equal to 0x plus
the y-intercept.

00:03:46.770 --> 00:03:49.060
Well, the y-intercept
is minus 2.

00:03:49.060 --> 00:03:52.250
So the equation of this line
is y is equal to minus 2.

00:03:52.250 --> 00:03:54.940
Let me do it in the purple.

00:03:54.940 --> 00:03:56.540
I'll just write it here.

00:03:56.540 --> 00:04:00.740
That equation is y is equal to
minus 2, Which makes sense.

00:04:00.740 --> 00:04:03.690
For all x, y is equal
to minus to 2.

00:04:03.690 --> 00:04:06.810
Now, my question to you
is this grey area.

00:04:06.810 --> 00:04:13.130
For any given x, the y is
greater than the line or the y

00:04:13.130 --> 00:04:15.020
is less than the line?

00:04:15.020 --> 00:04:18.450
Well, let's pick this x.

00:04:18.450 --> 00:04:20.680
When x is equal to 3, y is
equal to negative 2.

00:04:20.680 --> 00:04:21.980
That's this line.

00:04:21.980 --> 00:04:25.670
Now, is the grey area-- the
y's for that x-- above the

00:04:25.670 --> 00:04:27.220
line or below it?

00:04:27.220 --> 00:04:28.790
Well, it's above the
line, right?

00:04:28.790 --> 00:04:31.640
So if we look at it from the
point of view of this

00:04:31.640 --> 00:04:36.430
boundary, we say that y has to
be greater than minus 2.

00:04:36.430 --> 00:04:38.330
That's the shaded area.

00:04:38.330 --> 00:04:41.110
y is equal to negative 2 is this
dotted line's equation.

00:04:41.110 --> 00:04:45.230
The shaded area, at least
relative to this line, is y is

00:04:45.230 --> 00:04:46.480
greater than negative 2.

00:04:50.030 --> 00:04:52.790
If I just had this by itself,
I would shade up everything

00:04:52.790 --> 00:04:54.980
above y is equal to
negative 2, right?

00:04:54.980 --> 00:04:58.070
But I want just the combination
of both of these.

00:04:58.070 --> 00:05:00.460
So essentially, we have to say
that y-- let me do it in a

00:05:00.460 --> 00:05:03.650
nice, vibrant color.

00:05:03.650 --> 00:05:06.400
Since the shaded area is below
this line, we know that y is

00:05:06.400 --> 00:05:08.740
less than x plus 1.

00:05:08.740 --> 00:05:11.850
And since the shaded area is
above this line, we know that

00:05:11.850 --> 00:05:16.060
y is greater than negative 2.

00:05:16.060 --> 00:05:16.990
Where could I write that?

00:05:16.990 --> 00:05:18.240
I'll write it right here.

00:05:21.010 --> 00:05:23.700
So which of the choices
satisfy that?

00:05:23.700 --> 00:05:25.050
Let's see.

00:05:25.050 --> 00:05:27.290
Whoops, I went to the
next problem.

00:05:27.290 --> 00:05:30.530
So we know that y is greater
than negative 2.

00:05:30.530 --> 00:05:33.030
So these can't-- let me
just cross those out.

00:05:33.030 --> 00:05:35.270
That and that can't
be the answer.

00:05:35.270 --> 00:05:38.520
And then we also know that
y is less than x plus 1.

00:05:38.520 --> 00:05:42.090
So we know that this
is the answer.

00:05:42.090 --> 00:05:44.520
And frankly, if you just looked
at the choices, you

00:05:44.520 --> 00:05:47.450
really didn't have to figure out
the slope and y-intercept.

00:05:47.450 --> 00:05:49.480
They're essentially giving you
the equations of the two

00:05:49.480 --> 00:05:50.080
lines, right?

00:05:50.080 --> 00:05:52.950
You say, like, oh, one is x plus
1 and one is negative 2,

00:05:52.950 --> 00:05:55.230
and I just have to figure out
whether y is greater than or

00:05:55.230 --> 00:05:56.730
less than each of those lines.

00:05:56.730 --> 00:05:59.170
And you can say, OK, relative
to the y equals negative 2,

00:05:59.170 --> 00:06:01.600
we're definitely greater than
negative 2, so I'm one of

00:06:01.600 --> 00:06:04.730
these two choices, and relative
to y is equal to x

00:06:04.730 --> 00:06:07.630
plus 1, we're definitely less
than that line, so we have to

00:06:07.630 --> 00:06:09.200
be choice b.

00:06:09.200 --> 00:06:12.360
Next problem.

00:06:12.360 --> 00:06:16.460
Which point lies in the solution
set for the system?

00:06:16.460 --> 00:06:18.350
So you essentially just have
to say which coordinates

00:06:18.350 --> 00:06:20.460
satisfy both of these
equations?

00:06:20.460 --> 00:06:22.370
And frankly, you could graph
them and you could draw the

00:06:22.370 --> 00:06:24.930
area like we did in the last
one, or you could just try out

00:06:24.930 --> 00:06:25.870
the numbers, right?

00:06:25.870 --> 00:06:27.270
Let's just try the numbers.

00:06:27.270 --> 00:06:29.970
That never hurts.

00:06:29.970 --> 00:06:31.060
So minus 4, 1.

00:06:31.060 --> 00:06:32.190
Let's see if it satisfies it.

00:06:32.190 --> 00:06:34.510
So remember, the first
coordinate is x, the

00:06:34.510 --> 00:06:35.370
second one is y.

00:06:35.370 --> 00:06:41.900
So 2y would be 2 times
minus 1 minus x.

00:06:41.900 --> 00:06:45.240
So minus minus 4.

00:06:45.240 --> 00:06:48.480
Now, is that greater than
or equal to minus 6?

00:06:48.480 --> 00:06:53.270
Well, this is minus 2 plus 4.

00:06:53.270 --> 00:06:54.590
And this is what?

00:06:54.590 --> 00:06:59.120
Minus 2 plus 4 is equal to 2,
which is definitely greater

00:06:59.120 --> 00:07:00.980
than or equal to minus 6.

00:07:00.980 --> 00:07:02.900
So it satisfies the
first equation.

00:07:02.900 --> 00:07:05.480
Let's see if it satisfies
the second equation.

00:07:05.480 --> 00:07:07.720
I'll do it in another color and
I'll do it up in the black

00:07:07.720 --> 00:07:08.140
area up here.

00:07:08.140 --> 00:07:09.620
So 2 times y.

00:07:09.620 --> 00:07:14.980
So it's 2 times minus 1--
y is minus 1-- minus

00:07:14.980 --> 00:07:16.835
3 times minus 4.

00:07:19.550 --> 00:07:21.780
And they want to know is
that less than minus 6?

00:07:21.780 --> 00:07:23.490
So you get minus 2.

00:07:23.490 --> 00:07:32.070
3 times 4 is 12, so
it's plus 12.

00:07:32.070 --> 00:07:34.700
Negative times a negative is a
positive, so that equals plus

00:07:34.700 --> 00:07:37.890
10, which is not less
than negative 6.

00:07:37.890 --> 00:07:39.590
So the first one
does not work.

00:07:39.590 --> 00:07:41.020
That is not the answer.

00:07:41.020 --> 00:07:42.400
Let's try the second one.

00:07:42.400 --> 00:07:44.030
3 comma 1.

00:07:44.030 --> 00:07:45.220
So y is 1.

00:07:45.220 --> 00:07:50.950
So it becomes 2 times 1, which
is just 2, minus 3.

00:07:50.950 --> 00:07:53.430
Is that greater than or
equal to negative 6?

00:07:53.430 --> 00:07:56.730
So you get negative 1 is
definitely greater than

00:07:56.730 --> 00:07:59.450
negative 6, so this satisfies
the first equation.

00:07:59.450 --> 00:08:01.140
Does it satisfy the
second one?

00:08:01.140 --> 00:08:02.150
2 times y.

00:08:02.150 --> 00:08:09.295
2 times 1 is 2, minus
3 times 3.

00:08:12.890 --> 00:08:14.600
Is that less than minus 6?

00:08:14.600 --> 00:08:17.520
So you get 2 minus 9.

00:08:17.520 --> 00:08:18.780
Is that less than minus 6?

00:08:18.780 --> 00:08:21.350
You get minus 7 is less
than minus 6.

00:08:21.350 --> 00:08:22.750
So that is true.

00:08:22.750 --> 00:08:23.830
So choice b works.

00:08:23.830 --> 00:08:26.390
You don't have to
go any further.

00:08:26.390 --> 00:08:29.060
That point lies in the solution
set, which just means

00:08:29.060 --> 00:08:32.630
it satisfies both of
these inequalites.

00:08:32.630 --> 00:08:34.950
Next problem.

00:08:34.950 --> 00:08:37.970
Scroll down a little bit.

00:08:37.970 --> 00:08:40.570
Which system of linear
equalities is represented by

00:08:40.570 --> 00:08:41.179
this graph?

00:08:41.179 --> 00:08:44.059
So now we can use some of the
skills we used the last time.

00:08:44.059 --> 00:08:46.120
I don't have much space
to draw all of them.

00:08:46.120 --> 00:08:48.980
So let's just to figure out
the-- let's see, do they gives

00:08:48.980 --> 00:08:50.900
us-- no, they don't even give
us the equations because all

00:08:50.900 --> 00:08:52.110
the equations are different, so
we have to figure out the

00:08:52.110 --> 00:08:54.740
equations of these two graphs.

00:08:54.740 --> 00:08:58.450
So let's do this top
one first. So let's

00:08:58.450 --> 00:08:59.460
figure out its slope.

00:08:59.460 --> 00:09:01.920
When you increase-- and you
can just eyeball this one.

00:09:01.920 --> 00:09:07.350
When you increase x by
3, y increases by the

00:09:07.350 --> 00:09:09.250
same amount, right?

00:09:09.250 --> 00:09:11.630
So change in y over change
in x would be 3 over 3.

00:09:11.630 --> 00:09:14.860
Whatever you change x by, y
increases the same amount.

00:09:14.860 --> 00:09:17.890
So the slope of this first one
at least-- let me do it in a

00:09:17.890 --> 00:09:21.530
color you can see over black--
the slope is equal to 1.

00:09:21.530 --> 00:09:23.870
And what's its y-intercept?

00:09:23.870 --> 00:09:25.720
Well, you just look at where did
it intercept the y-axis?

00:09:25.720 --> 00:09:26.650
Right there.

00:09:26.650 --> 00:09:28.860
So the y-intercept is
equal to minus 2.

00:09:28.860 --> 00:09:33.770
So this first equation
right here is y is

00:09:33.770 --> 00:09:37.370
equal to x minus 2.

00:09:37.370 --> 00:09:38.000
Fair enough.

00:09:38.000 --> 00:09:40.480
And this grey area, is
it above or below it?

00:09:40.480 --> 00:09:41.150
It's below it.

00:09:41.150 --> 00:09:42.480
You pick any x.

00:09:42.480 --> 00:09:44.940
For any x, this is y is
equal to x minus 2.

00:09:44.940 --> 00:09:47.090
The grey area is for all
the y's that are

00:09:47.090 --> 00:09:49.060
less than it, right?

00:09:49.060 --> 00:09:51.040
And notice, now the
line is shaded in.

00:09:51.040 --> 00:09:54.840
So the line is part of
the solution set.

00:09:54.840 --> 00:09:57.930
So at least relative to this, we
know that the grey area is

00:09:57.930 --> 00:10:02.990
all y's that satisfy y is less
than or equal to x minus 2.

00:10:02.990 --> 00:10:06.430
Less than because it's below it,
and it's equal to because

00:10:06.430 --> 00:10:08.000
they actually drew
in the line.

00:10:08.000 --> 00:10:09.500
But now we have to figure
out the lower bound.

00:10:09.500 --> 00:10:11.870
What's the equation
of this line?

00:10:11.870 --> 00:10:13.560
So let's think about
it a little bit.

00:10:13.560 --> 00:10:16.650
This one's a little
more interesting.

00:10:16.650 --> 00:10:27.220
If I increase x by 1, so if
I'm going from this point,

00:10:27.220 --> 00:10:28.310
what happens to y?

00:10:28.310 --> 00:10:30.910
y goes down by 2, right?

00:10:30.910 --> 00:10:32.680
1 minus 2.

00:10:32.680 --> 00:10:39.500
If I increase x by 2, y
goes down by 4, right?

00:10:39.500 --> 00:10:40.740
Minus 4.

00:10:40.740 --> 00:10:45.380
So at least in this case, the
slope, change in y, is minus 4

00:10:45.380 --> 00:10:47.550
whenever change in
x is positive 2.

00:10:47.550 --> 00:10:51.180
Or it could've been change in
y is minus 2 whenever change

00:10:51.180 --> 00:10:53.740
in x is plus 1, right?

00:10:53.740 --> 00:10:55.590
So it equals minus 2.

00:10:55.590 --> 00:10:56.330
And that makes sense.

00:10:56.330 --> 00:10:58.710
For every 1 you go over,
the slope goes down 2.

00:10:58.710 --> 00:11:01.280
And what's it's y-intercept?

00:11:01.280 --> 00:11:05.440
Let's see, y-intercept is just
right there. y is equal to 3.

00:11:05.440 --> 00:11:06.960
So b is equal to 3.

00:11:06.960 --> 00:11:14.720
So this second graph is y is
equal to minus 2x plus 3.

00:11:14.720 --> 00:11:18.630
And then, once again, the shaded
area includes this, but

00:11:18.630 --> 00:11:23.200
we have to be greater than
or equal to this graph.

00:11:23.200 --> 00:11:26.060
We have to be greater than,
because if you pick any x, the

00:11:26.060 --> 00:11:29.400
y's that satisfy the shaded area
are equal to the graph or

00:11:29.400 --> 00:11:32.350
greater than for any x.

00:11:32.350 --> 00:11:34.210
So there we have
our inequality.

00:11:34.210 --> 00:11:37.940
So this one, this area, you
could kind of say is y has to

00:11:37.940 --> 00:11:42.250
be greater than or equal
to minus 2x plus 3.

00:11:42.250 --> 00:11:44.420
That's from this boundary.

00:11:44.420 --> 00:11:47.300
And we also know that y
has to be less than or

00:11:47.300 --> 00:11:49.840
equal to x minus 2.

00:11:49.840 --> 00:11:51.450
Now, hopefully, that's
one of the choices.

00:11:51.450 --> 00:11:53.610
If we haven't a made
a careless mistake.

00:11:53.610 --> 00:11:55.400
Let me see.

00:11:55.400 --> 00:11:56.330
All right.

00:11:56.330 --> 00:12:02.350
y is greater than or
equal to-- OK, so

00:12:02.350 --> 00:12:03.530
this is not the answer.

00:12:03.530 --> 00:12:05.670
Neither of these are
the answers.

00:12:05.670 --> 00:12:06.890
That's not the answer.

00:12:06.890 --> 00:12:08.320
Let me cross that out.

00:12:08.320 --> 00:12:10.200
y is greater than or
equal to-- we have

00:12:10.200 --> 00:12:12.860
minus 2x plus 3, right?

00:12:12.860 --> 00:12:15.740
Let me make sure I
got that right.

00:12:15.740 --> 00:12:17.620
Minus 2x plus 3.

00:12:17.620 --> 00:12:19.520
Yeah, let me see.

00:12:19.520 --> 00:12:20.760
Right, so they're tricky.

00:12:20.760 --> 00:12:25.100
So the slope on that second
line was minus 2x plus 3.

00:12:25.100 --> 00:12:26.840
They got the second
part right.

00:12:26.840 --> 00:12:28.500
Maybe I made a careless mistake,
but I don't think

00:12:28.500 --> 00:12:29.660
this is right either.

00:12:29.660 --> 00:12:30.770
Let's see if we can rearrange.

00:12:30.770 --> 00:12:33.440
So these other two, they put
the x's and the y's on the

00:12:33.440 --> 00:12:34.830
same side, so let's do that.

00:12:34.830 --> 00:12:37.250
That top equation, we
can rewrite it.

00:12:37.250 --> 00:12:38.890
Add 2x to both sides.

00:12:38.890 --> 00:12:43.560
You get 2x plus y is greater
than or equal to 3.

00:12:43.560 --> 00:12:50.340
And the second equation, if we
were to subtract x from both

00:12:50.340 --> 00:12:57.860
sides, you get y minus x is less
than or equal to minus 2.

00:12:57.860 --> 00:12:59.390
And now this is interesting.

00:12:59.390 --> 00:13:01.490
So there's no minus 2 on the
right-hand side here,

00:13:01.490 --> 00:13:02.050
but what can we do?

00:13:02.050 --> 00:13:03.990
If we multiply both sides
of this equation by

00:13:03.990 --> 00:13:07.770
minus 2, what happens?

00:13:07.770 --> 00:13:11.040
Sorry, multiply both sides of
the equation by minus 1.

00:13:11.040 --> 00:13:14.585
You get minus y plus x.

00:13:14.585 --> 00:13:17.330
And when you multiply an
inequality by a negative

00:13:17.330 --> 00:13:20.240
number, multiply or divide, you
switch the inequalites.

00:13:20.240 --> 00:13:22.700
So it's greater than or equal
to-- and we're multiplying by

00:13:22.700 --> 00:13:24.260
negative 1-- 2.

00:13:24.260 --> 00:13:28.890
So it becomes x minus y is
greater than or equal to 2.

00:13:28.890 --> 00:13:32.190
So our two equations turn into
2x plus y is greater than or

00:13:32.190 --> 00:13:34.890
equal to 3, which is this one.

00:13:34.890 --> 00:13:37.460
And we get x minus y is greater
than or equal to 2,

00:13:37.460 --> 00:13:38.890
which is that one.

00:13:38.890 --> 00:13:39.440
And we're done.

00:13:39.440 --> 00:13:41.480
It's choice d.

00:13:41.480 --> 00:13:43.520
See you in the next video.


California Standards Test: Algebra II

WEBVTT
Kind: captions
Language: en

00:00:00.730 --> 00:00:04.980
So we're on problem number one
on the California Standards

00:00:04.980 --> 00:00:07.280
Test Problems for Algebra II.

00:00:07.280 --> 00:00:10.260
And so the theory being is if
you're able to do all or at

00:00:10.260 --> 00:00:13.100
least understand all 80 of these
questions, I think you

00:00:13.100 --> 00:00:15.610
have a pretty good understanding
of Algebra II,

00:00:15.610 --> 00:00:18.540
at least from the perspective
of California.

00:00:18.540 --> 00:00:19.820
Let's see, problem number one.

00:00:19.820 --> 00:00:23.850
What is the complete solution
to the equation the absolute

00:00:23.850 --> 00:00:31.920
value of 3 minus 6x
is equal to 15?

00:00:31.920 --> 00:00:32.689
So let's think about it.

00:00:32.689 --> 00:00:35.880
3 minus 6x could clearly be 15,
because the absolute value

00:00:35.880 --> 00:00:37.600
of 15 is 15.

00:00:37.600 --> 00:00:40.620
So we could say that 3
-- sorry that's a 3.

00:00:40.620 --> 00:00:45.490
3 minus 6x could
be equal to 15.

00:00:45.490 --> 00:00:48.430
But 3 minus 6x could also be
equal to minus 15, right?

00:00:48.430 --> 00:00:51.430
Because the absolute value
of minus 15 is also 15.

00:00:51.430 --> 00:00:52.620
So let's write that down.

00:00:52.620 --> 00:00:59.270
Or 3 minus 6x is equal
to minus 15, right?

00:00:59.270 --> 00:01:01.640
If this was equal to minus 15,
you'd take the absolute value,

00:01:01.640 --> 00:01:02.590
you'd get positive 15.

00:01:02.590 --> 00:01:04.239
So either of these
could be true.

00:01:04.239 --> 00:01:05.700
So let's solve for x.

00:01:05.700 --> 00:01:08.750
Subtract 3 from both sides,
you get minus

00:01:08.750 --> 00:01:12.400
6x is equal to 12.

00:01:12.400 --> 00:01:18.980
Divide both sides by minus 6,
you get x is equal to minus 2.

00:01:18.980 --> 00:01:22.250
And so here, if we subtract 3
from both sides, you get minus

00:01:22.250 --> 00:01:26.120
6x is equal to minus
15 minus 3.

00:01:26.120 --> 00:01:28.920
That's minus 18.

00:01:28.920 --> 00:01:32.800
Divide both sides my minus
6, you get x is equal

00:01:32.800 --> 00:01:34.300
to positive 3, right?

00:01:34.300 --> 00:01:36.990
A negative divided by a negative
, so x is equal to 3.

00:01:36.990 --> 00:01:41.240
So x could be minus 2 or
3, and that's choice b.

00:01:45.880 --> 00:01:46.790
Problem two.

00:01:46.790 --> 00:01:50.630
Let me switch colors to make
things interesting.

00:01:50.630 --> 00:01:51.840
And this is a similar
type of problem.

00:01:51.840 --> 00:01:57.820
What are the possible values of
x in the absolute value of

00:01:57.820 --> 00:02:03.320
12 minus 4x is equal to 2?

00:02:03.320 --> 00:02:04.190
Same logic.

00:02:04.190 --> 00:02:06.210
12 minus 4x could be 2, right?

00:02:06.210 --> 00:02:07.530
The absolute value of 2 is 2.

00:02:07.530 --> 00:02:13.980
So you you get 12 minus
4x is equal to 2.

00:02:13.980 --> 00:02:16.260
Or 12 minus 4x could
be minus 2, right?

00:02:16.260 --> 00:02:19.570
Because the absolute value
of minus 2 is 2.

00:02:19.570 --> 00:02:26.420
So then you get 12 minus 4x
could be equal to minus 2.

00:02:26.420 --> 00:02:28.590
So this is an or proposition.

00:02:28.590 --> 00:02:33.800
So let's see, subtract 12 from
both sides, you get minus 4x.

00:02:33.800 --> 00:02:36.626
2 minus 12 is minus 10.

00:02:36.626 --> 00:02:37.940
Then you get x.

00:02:37.940 --> 00:02:43.220
Divide both sides by minus 4
is minus 10 over minus 4.

00:02:43.220 --> 00:02:44.980
The negatives cancel out.

00:02:44.980 --> 00:02:51.665
And that's equal to 5/2,
or 2 and 1/2, for 2.5.

00:02:51.665 --> 00:02:54.720
And the only reason I wrote 2.5
is I was looking at the

00:02:54.720 --> 00:02:58.190
choices, and they have
everything as decimals.

00:02:58.190 --> 00:02:59.900
So let's look at the other
possibilities.

00:02:59.900 --> 00:03:04.110
Subtract 12 from both sides, you
get minus 4x is equal to

00:03:04.110 --> 00:03:08.634
minus 2 minus 12 is minus 14.

00:03:08.634 --> 00:03:14.670
x is equal to minus
14 over minus 4.

00:03:14.670 --> 00:03:19.230
Negatives cancel out, which is
equal to 7/2, which is equal

00:03:19.230 --> 00:03:22.810
to 3 and 1/2, which
is equal to 3.5.

00:03:22.810 --> 00:03:29.580
So x could be 2.5 or 3.5,
and that is choice d.

00:03:29.580 --> 00:03:32.620
Problem three.

00:03:32.620 --> 00:03:35.140
I'll switch to maybe green.

00:03:35.140 --> 00:03:36.120
Problem three.

00:03:36.120 --> 00:03:39.270
For weddings, Sherada bought
several dozen roses and

00:03:39.270 --> 00:03:41.720
several dozen carnations.

00:03:41.720 --> 00:03:48.562
The roses cost $15 per dozen,
so roses are $15 per dozen,

00:03:48.562 --> 00:03:52.490
the carnations cost $8 per
dozen, so the carnations are

00:03:52.490 --> 00:03:55.460
$8 per dozen.

00:03:55.460 --> 00:03:58.650
Sherada bought a total of
17 dozen flowers and

00:03:58.650 --> 00:04:00.535
paid a total $192.

00:04:00.535 --> 00:04:03.040
How many roses did she buy?

00:04:03.040 --> 00:04:05.230
All right, so let's set
some variables.

00:04:05.230 --> 00:04:09.080
Let R equal dozens of roses.

00:04:12.010 --> 00:04:13.285
I hope you can read
my handwriting.

00:04:15.940 --> 00:04:21.540
And let C equal to dozens
of carnations.

00:04:24.720 --> 00:04:26.700
All right, so they say
that she bought a

00:04:26.700 --> 00:04:28.600
total of 17 dozen flowers.

00:04:28.600 --> 00:04:35.200
So the dozens of roses she
bought plus the dozens of

00:04:35.200 --> 00:04:39.330
carnations she bought must be
equal to 17 because she bought

00:04:39.330 --> 00:04:41.060
a total of 17 dozens.

00:04:41.060 --> 00:04:44.270
It also tells us that
she spent $192.

00:04:44.270 --> 00:04:46.240
So how do we figure out how
much money she spent?

00:04:46.240 --> 00:04:49.420
So if she bought R dozens
of roses, how much did

00:04:49.420 --> 00:04:50.320
she spend on roses?

00:04:50.320 --> 00:04:53.800
Well, each dozen of
roses is $15.

00:04:53.800 --> 00:04:56.560
So she spent 15 times the
number of dozens.

00:04:56.560 --> 00:04:58.800
That's how much she
spent on roses.

00:04:58.800 --> 00:05:00.750
How much did she spend
on carnations?

00:05:00.750 --> 00:05:03.630
Well, she bought C dozens
of carnations.

00:05:03.630 --> 00:05:05.900
Each dozen was $8.

00:05:05.900 --> 00:05:08.900
She spent 8C dollars
on carnations.

00:05:08.900 --> 00:05:10.840
So her total amount of money she
spent was the amount she

00:05:10.840 --> 00:05:14.160
spent on roses plus the amount
she spent on carnations, and

00:05:14.160 --> 00:05:17.090
that is equal to 192.

00:05:17.090 --> 00:05:20.070
Now, we have two linear
equations and two unknowns.

00:05:20.070 --> 00:05:22.240
We can just solve them.

00:05:22.240 --> 00:05:23.880
We want to solve for the
number of roses.

00:05:23.880 --> 00:05:24.872
That was the question.

00:05:24.872 --> 00:05:28.790
So let's try to cancel out
the number of carnations.

00:05:28.790 --> 00:05:31.770
So if we multiply this top
equation by minus 8, we'll get

00:05:31.770 --> 00:05:32.780
a minus 8C.

00:05:32.780 --> 00:05:35.570
That'll cancel out with that,
so let's do that.

00:05:35.570 --> 00:05:38.250
So let's multiply that top
equation by minus 8.

00:05:38.250 --> 00:05:43.550
You get minus 8R minus 8C.

00:05:43.550 --> 00:05:45.230
What's minus 8 times 17?

00:05:45.230 --> 00:05:49.800
Let's see, 17-- I'll do it in
the corner here-- 17 times 8,

00:05:49.800 --> 00:06:00.120
7 times 8 is 56, 1 times 8 is
13, plus 5 is 1, 1 times-- oh,

00:06:00.120 --> 00:06:01.030
my brain was getting
ahead of me.

00:06:01.030 --> 00:06:03.460
1 times 8 is 8, plus 5 is 13.

00:06:03.460 --> 00:06:05.130
Sorry.

00:06:05.130 --> 00:06:07.930
It's equal to 136.

00:06:07.930 --> 00:06:09.910
Oh, and we would multiply it by
negative 8, so we'd put a

00:06:09.910 --> 00:06:10.900
negative there.

00:06:10.900 --> 00:06:14.100
And now we can add these
two equations.

00:06:14.100 --> 00:06:18.020
15 minus 8, 7R.

00:06:18.020 --> 00:06:20.760
8C minus 8C is 0-- and that
was the whole point behind

00:06:20.760 --> 00:06:23.640
multiplying this top equation
by minus 8-- is equal

00:06:23.640 --> 00:06:26.270
to 192 minus 36.

00:06:26.270 --> 00:06:26.940
Let's see.

00:06:26.940 --> 00:06:30.680
We can do little borrowing, or
regrouping, they call it now.

00:06:30.680 --> 00:06:36.220
That's 12, that's 8,
so it equals 56.

00:06:36.220 --> 00:06:40.220
7R is equal to 56, so
R is equal to 8.

00:06:40.220 --> 00:06:43.815
So Sherada bought 8 dozens of
roses, and that is choice C.

00:06:47.130 --> 00:06:48.380
Problem four.

00:06:50.940 --> 00:06:52.430
What is the solution
to the system of

00:06:52.430 --> 00:06:54.070
equations shown below?

00:06:54.070 --> 00:06:57.130
And there's three of them with
three variables, always the

00:06:57.130 --> 00:06:58.250
hairiest problem.

00:06:58.250 --> 00:06:59.160
Let's see.

00:06:59.160 --> 00:07:08.990
We have 2x minus y plus
3z is equal to 8.

00:07:08.990 --> 00:07:18.400
We have x minus 6y minus
z is equal to 0.

00:07:18.400 --> 00:07:26.670
And then you get minus 6x plus
3y minus 9z is equal to 24.

00:07:26.670 --> 00:07:28.390
Now there's a bunch of
ways to solve these.

00:07:28.390 --> 00:07:31.230
You could use matrices and
inverses of matrices and

00:07:31.230 --> 00:07:34.010
Gauss-Jordan elimination, or you
can just kind of use your

00:07:34.010 --> 00:07:37.140
traditional multiplying
equations by each other and

00:07:37.140 --> 00:07:40.240
adding them so that you can kind
of constrain them or take

00:07:40.240 --> 00:07:40.920
out variables.

00:07:40.920 --> 00:07:43.160
And we'll do the last one
because I don't want to

00:07:43.160 --> 00:07:46.570
introduce any other kind of
matrix manipulation here.

00:07:46.570 --> 00:07:49.680
So what I want to do is, let's
use this equation with both of

00:07:49.680 --> 00:07:55.300
these, and try to cancel
out the x's, right?

00:07:55.300 --> 00:07:56.390
Well, let's think of how
we could do that.

00:07:56.390 --> 00:07:59.280
Well, one thing I like to do,
just to simplify it, is to get

00:07:59.280 --> 00:08:03.050
all of the equations in what I
consider lowest integer from.

00:08:03.050 --> 00:08:04.690
I know that's not
a formal term.

00:08:04.690 --> 00:08:06.960
But look at all the terms
in this equation.

00:08:06.960 --> 00:08:09.510
They're all divisible
by 3, right?

00:08:09.510 --> 00:08:11.340
So let's divide all
of them by 3.

00:08:11.340 --> 00:08:14.520
So this last equation would
become-- if you divide

00:08:14.520 --> 00:08:21.720
everything by 3, you'd get minus
2x plus y minus 3z is

00:08:21.720 --> 00:08:24.070
equal to 8.

00:08:24.070 --> 00:08:25.220
So let's cancel that out.

00:08:25.220 --> 00:08:26.350
I just divided it by 3.

00:08:26.350 --> 00:08:27.110
It's the same thing.

00:08:27.110 --> 00:08:28.880
Remember, whatever you do to
one side of an equation you

00:08:28.880 --> 00:08:31.570
can do to the other side
of the equation.

00:08:31.570 --> 00:08:35.260
Anyway, so let's use this
equation and this equation and

00:08:35.260 --> 00:08:39.320
then this equation and this
equation to get two equations

00:08:39.320 --> 00:08:41.169
with just y's and z's.

00:08:41.169 --> 00:08:43.419
So essentially we want to
cancel out the x's.

00:08:43.419 --> 00:08:45.710
So to cancel out the x's with
this one and this one, we can

00:08:45.710 --> 00:08:46.550
just add them, right?

00:08:46.550 --> 00:08:48.670
Because 2x plus minus
2x, that's going to

00:08:48.670 --> 00:08:50.320
cancel out the x's.

00:08:50.320 --> 00:08:51.210
So we can just do that.

00:08:51.210 --> 00:08:52.710
And then when we do it with
this equation, we could

00:08:52.710 --> 00:08:56.220
multiply this equation times
minus 2 first and it'll cancel

00:08:56.220 --> 00:08:57.520
out the x's there.

00:08:57.520 --> 00:08:59.210
So let's do it.

00:08:59.210 --> 00:09:01.990
So let's just write it and
I'll write it in green.

00:09:01.990 --> 00:09:07.220
So if I add this equation to
this equation, 2x minus 2x is

00:09:07.220 --> 00:09:14.230
0x minus y plus y, that's 0y.

00:09:14.230 --> 00:09:16.220
These are all zero, but I'm just
writing them down to show

00:09:16.220 --> 00:09:18.230
that we're done.

00:09:18.230 --> 00:09:21.030
Well, they all cancel out.

00:09:21.030 --> 00:09:23.380
Plus 0z.

00:09:23.380 --> 00:09:28.030
These are equal to 8 plus
8 is equal to 16.

00:09:28.030 --> 00:09:33.060
So you get 0 is equal to 16,
which makes no sense.

00:09:33.060 --> 00:09:34.840
So there's actually
no solution here.

00:09:34.840 --> 00:09:37.030
And if you wanted to think
about visually what's

00:09:37.030 --> 00:09:40.590
happening is each of these
equations represents a plane

00:09:40.590 --> 00:09:42.060
in three dimensions.

00:09:42.060 --> 00:09:44.780
And the only way you get no
solution or no intersection

00:09:44.780 --> 00:09:49.070
between this equation and this
equation or this equation,

00:09:49.070 --> 00:09:51.000
which are essentially the same
equation, the only reason why

00:09:51.000 --> 00:09:53.730
you can't get the intersection
between them is that they must

00:09:53.730 --> 00:09:58.510
be parallel equations.

00:09:58.510 --> 00:10:01.680
You could get an infinite number
of solutions if they

00:10:01.680 --> 00:10:03.400
were actually the same plane.

00:10:03.400 --> 00:10:06.630
But these are parallel planes,
so they'll never intersect.

00:10:06.630 --> 00:10:07.390
So we saw.

00:10:07.390 --> 00:10:10.090
And it kind of makes sense
because this part-- well, I'm

00:10:10.090 --> 00:10:11.000
not going to go into
the details.

00:10:11.000 --> 00:10:14.270
We're just focused on solving
problems right now.

00:10:14.270 --> 00:10:16.410
But it's fair enough to say
that when you add this

00:10:16.410 --> 00:10:19.710
equation to this equation to
cancel out the x's, everything

00:10:19.710 --> 00:10:22.830
else cancels out and you got
a nonsensical statement, 0

00:10:22.830 --> 00:10:27.160
equals 16, so there's no
solution and that's choice c.

00:10:27.160 --> 00:10:30.900
And those are parallel planes
in three dimensions.

00:10:30.900 --> 00:10:34.360
Problem five.

00:10:34.360 --> 00:10:37.770
A restaurant manager bought
20 packages of bagels.

00:10:37.770 --> 00:10:41.720
Some packages contained 6
bagels each and the rest

00:10:41.720 --> 00:10:43.090
contain 12 bagels each.

00:10:43.090 --> 00:10:45.480
There were a 168
bagels in all.

00:10:45.480 --> 00:10:48.500
How many packets of 12 bagels
did the manager buy?

00:10:51.310 --> 00:10:54.090
So he bought some
packages of 6.

00:10:54.090 --> 00:10:58.930
So the packages of 6 plus the
packages of 12, that's just my

00:10:58.930 --> 00:11:01.700
random notation that
I invented.

00:11:01.700 --> 00:11:05.570
He bought a total of 20
packages, so that equals 20,

00:11:05.570 --> 00:11:08.860
and he bought a total of
168 bagels in all.

00:11:08.860 --> 00:11:12.050
So how many bagels did he
get from the 6-packs?

00:11:12.050 --> 00:11:15.520
Well, each of them had
6 bagels, right?

00:11:15.520 --> 00:11:18.160
So he got 6 times the
number of packs.

00:11:18.160 --> 00:11:19.680
So he got 6 times this.

00:11:19.680 --> 00:11:22.420
This is the number of bagels he
got from the 6-packs, and

00:11:22.420 --> 00:11:25.750
then this is the number
bagels he got from

00:11:25.750 --> 00:11:26.720
the 12-packs, right?

00:11:26.720 --> 00:11:33.500
The number of packs times 12,
and then that is equal to 168.

00:11:33.500 --> 00:11:35.680
Two equations, two unknowns.

00:11:35.680 --> 00:11:36.670
They're linear equations.

00:11:36.670 --> 00:11:39.360
We should be able
to solve this.

00:11:39.360 --> 00:11:42.020
So they want to know how
many packages of the 12

00:11:42.020 --> 00:11:42.710
bagels did he buy?

00:11:42.710 --> 00:11:45.735
So they want to know what p sub
12, or whatever you want

00:11:45.735 --> 00:11:47.250
to call that is.

00:11:47.250 --> 00:11:49.860
So if we want to do that, let's
cancel out the P6's.

00:11:49.860 --> 00:11:55.330
So if we multiply that top
equation by minus 6, you get

00:11:55.330 --> 00:12:02.695
minus 6 times the 6-packs, minus
6 times the 12-packs--

00:12:02.695 --> 00:12:05.540
I'm just multiplying this top
thing by minus 6-- is

00:12:05.540 --> 00:12:09.200
equal to minus 120.

00:12:09.200 --> 00:12:11.220
Now, we can add the
two equations.

00:12:11.220 --> 00:12:12.130
These cancel out.

00:12:12.130 --> 00:12:14.080
That was the whole point.

00:12:14.080 --> 00:12:17.430
12 times the 12-packs minus 6
times the 12-packs, that's

00:12:17.430 --> 00:12:20.085
equal to 6 times the 12-packs.

00:12:20.085 --> 00:12:24.950
It's equal to 168
minus 120 is 48.

00:12:24.950 --> 00:12:29.210
And so we get the 12-packs,
there were 48

00:12:29.210 --> 00:12:30.810
divided by 6 is 8.

00:12:30.810 --> 00:12:32.240
So there were 8 12-packs.

00:12:32.240 --> 00:12:33.610
The manager bought 8 12-packs.

00:12:33.610 --> 00:12:36.110
That is choice b.

00:12:36.110 --> 00:12:37.850
And I'll see you in the
next video because

00:12:37.850 --> 00:12:39.560
I'm all out of time.


GMAT: Math 50

WEBVTT
Kind: captions
Language: en

00:00:00.820 --> 00:00:02.970
We're on problem--
where are we?

00:00:02.970 --> 00:00:06.670
Problem 232.

00:00:06.670 --> 00:00:07.830
Only a few left.

00:00:07.830 --> 00:00:22.280
If 1/x minus 1 over x plus 1 is
equal to 1 over x plus 4,

00:00:22.280 --> 00:00:25.850
then x could be which
of the following?

00:00:25.850 --> 00:00:27.740
And you might have a temptation
to just invert both

00:00:27.740 --> 00:00:29.480
sides, but you can't
do that just yet.

00:00:29.480 --> 00:00:30.840
Because if you took the inverse
of this, it's not like

00:00:30.840 --> 00:00:32.850
you just get x minus x plus 1.

00:00:32.850 --> 00:00:35.680
You actually have to subtract,
or you could say add these two

00:00:35.680 --> 00:00:36.480
fractions together.

00:00:36.480 --> 00:00:37.590
So let's do that.

00:00:37.590 --> 00:00:40.320
So if we simplify this left-hand
side, a common

00:00:40.320 --> 00:00:44.700
denominator would be
x times x plus 1.

00:00:44.700 --> 00:00:48.500
And then a numerator would be,
let's see-- 1/x is the same

00:00:48.500 --> 00:00:51.950
thing as x plus 1 over this
thing, because that cancels

00:00:51.950 --> 00:00:56.610
out and you get 1/x, minus-- 1
over x plus 1, that's the same

00:00:56.610 --> 00:00:59.080
thing as x/x times x plus 1.

00:00:59.080 --> 00:01:01.960
The x's would cancel out,
you get 1 over x plus 1.

00:01:01.960 --> 00:01:07.380
So this left-hand side, the x's
cancel out and you get 1

00:01:07.380 --> 00:01:10.070
over-- well, we could
just make that x

00:01:10.070 --> 00:01:13.130
squared plus x, right?

00:01:13.130 --> 00:01:15.825
So now this simplifies to-- the
left-hand side is 1 over x

00:01:15.825 --> 00:01:17.670
squared plus x, and the
right-hand side is

00:01:17.670 --> 00:01:20.730
1 over x plus 4.

00:01:20.730 --> 00:01:22.690
And now we can invert both
sides or just set the

00:01:22.690 --> 00:01:23.880
denominators as equal
to each other.

00:01:23.880 --> 00:01:24.670
But let's just invert both.

00:01:24.670 --> 00:01:25.930
It's the same thing, actually.

00:01:25.930 --> 00:01:28.540
So if you take the inverse of
both sides and you get-- and

00:01:28.540 --> 00:01:33.130
I'll just switch colors just
randomly-- x squared plus x is

00:01:33.130 --> 00:01:36.080
equal to x plus 4.

00:01:36.080 --> 00:01:38.290
Let's subtract x from
both sides.

00:01:38.290 --> 00:01:41.260
You get x squared
is equal to 4.

00:01:41.260 --> 00:01:45.320
So x is equal to plus
or minus 2.

00:01:45.320 --> 00:01:48.210
And they say then x could be--
well, choice C is minus 2, so

00:01:48.210 --> 00:01:51.270
that's one of the values
that x could be.

00:01:51.270 --> 00:01:56.070
Next question, 233.

00:01:56.070 --> 00:02:04.490
All right, they have 1/2 to the
minus 3 power times 1/4 to

00:02:04.490 --> 00:02:11.130
the minus 2 power times
1/16 to the minus 1.

00:02:11.130 --> 00:02:13.760
The first thing I do when I see
negative exponents, just

00:02:13.760 --> 00:02:16.250
so they don't confuse
me, is that I just

00:02:16.250 --> 00:02:17.340
invert all of the numbers.

00:02:17.340 --> 00:02:18.910
So this is the same thing.

00:02:18.910 --> 00:02:22.960
This is the exact same thing as
equal to-- this is the same

00:02:22.960 --> 00:02:25.720
thing as 2 to the third
power, right?

00:02:25.720 --> 00:02:27.830
When you take something to a
negative exponent, that's the

00:02:27.830 --> 00:02:30.690
same thing as the inverse of
that thing to the positive

00:02:30.690 --> 00:02:32.830
exponent, so this is 2
to the third power.

00:02:32.830 --> 00:02:39.200
This is times 4 squared
times 16.

00:02:39.200 --> 00:02:46.510
And so 2 to the third is 8
times 16 times 16, right?

00:02:46.510 --> 00:02:50.830
That's 8 times 256.

00:02:50.830 --> 00:02:52.260
And then-- oh, wait.

00:02:52.260 --> 00:02:54.510
I'm going down the wrong path,
because they have it in

00:02:54.510 --> 00:02:55.880
completely different format.

00:02:55.880 --> 00:02:58.430
So actually, maybe I should look
at the choices before I

00:02:58.430 --> 00:03:02.300
continue these problems.
So let me rewrite it.

00:03:02.300 --> 00:03:06.090
So they all have it as -- you
know, 1/2 to some power, 1/8

00:03:06.090 --> 00:03:07.490
to some power, so let
me rewrite this.

00:03:07.490 --> 00:03:09.410
I went down the wrong road.

00:03:09.410 --> 00:03:11.760
So let's write all of these
as powers of 1/2.

00:03:11.760 --> 00:03:13.800
So 1/2 to the minus 1/3, fine.

00:03:13.800 --> 00:03:17.150
That's still 1/2 to
the minus 1/3.

00:03:17.150 --> 00:03:18.930
1/4, what is that?

00:03:18.930 --> 00:03:23.630
1/4 is the same thing
as 1/2 squared.

00:03:23.630 --> 00:03:26.110
We're raising that
to the minus 2.

00:03:26.110 --> 00:03:31.260
And then 1/16, that's the same
thing as 1/2 to the third

00:03:31.260 --> 00:03:33.205
power is 1/8, so it's
to the fourth power.

00:03:33.205 --> 00:03:35.420
We're taking that to
the negative 1.

00:03:35.420 --> 00:03:38.330
And then this is-- we'll just
use what we know about

00:03:38.330 --> 00:03:41.330
exponent rules, and we've proven
this in other Khan

00:03:41.330 --> 00:03:42.970
Academy videos.

00:03:42.970 --> 00:03:45.360
1/2 squared to the minus 2.

00:03:45.360 --> 00:03:48.230
You can just multiply these
exponents, so that's equal to

00:03:48.230 --> 00:03:51.650
1/2 to the minus 4.

00:03:51.650 --> 00:03:56.190
And then 1/2 to the 4 to the
minus 1, you multiply them and

00:03:56.190 --> 00:03:58.190
that's equal to minus 4.

00:03:58.190 --> 00:04:00.800
And now we have the same base,
and we're multiplying them, so

00:04:00.800 --> 00:04:02.010
we can just add the exponents.

00:04:02.010 --> 00:04:09.420
This is equal to 1/2 to the
minus 3 plus minus 4, minus 7,

00:04:09.420 --> 00:04:11.520
minus 4, is equal to minus 11.

00:04:11.520 --> 00:04:17.310
So it's 1/2 to the negative 11th
power, which is choice B.

00:04:17.310 --> 00:04:18.310
That was a good lesson there.

00:04:18.310 --> 00:04:20.540
Look at your choices before you
go down some path because

00:04:20.540 --> 00:04:22.280
I was actually going to evaluate
this thing, which

00:04:22.280 --> 00:04:24.990
would have taken a little
bit of extra time.

00:04:24.990 --> 00:04:30.000
Problem 234.

00:04:30.000 --> 00:04:33.800
In a certain game, a large
container is filled with red,

00:04:33.800 --> 00:04:40.320
yellow, green-- so let's see,
there's red, yellow, green,

00:04:40.320 --> 00:04:43.980
and blue beads, worth
respectively-- so respectively

00:04:43.980 --> 00:04:46.200
means the same order; we said
the colors are in the same

00:04:46.200 --> 00:04:52.450
order that they're worth -- so
worth respectively 7, 5, 3,

00:04:52.450 --> 00:04:54.840
and 2 points each.

00:04:54.840 --> 00:04:58.290
OK, a number of beads are then
removed from the container.

00:04:58.290 --> 00:05:03.460
If the product of the point of
values of the removed beads is

00:05:03.460 --> 00:05:10.010
147,000, how many red
beads were removed?

00:05:10.010 --> 00:05:10.850
So this is interesting.

00:05:10.850 --> 00:05:13.320
So there's something that
hopefully will jump out at you

00:05:13.320 --> 00:05:14.800
about these numbers.

00:05:14.800 --> 00:05:17.230
They're all prime
numbers, right?

00:05:17.230 --> 00:05:19.390
So if you look at any number,
you can tell how many of a

00:05:19.390 --> 00:05:21.790
certain prime number there is in
that number by just doing a

00:05:21.790 --> 00:05:23.310
prime factorization.

00:05:23.310 --> 00:05:28.320
So let's do that for 147,000.

00:05:28.320 --> 00:05:30.560
And you'd be surprised
how quickly a

00:05:30.560 --> 00:05:32.080
number can get smaller.

00:05:32.080 --> 00:05:33.330
So let's see.

00:05:33.330 --> 00:05:37.190
How many times-- let's
see, does 3 go 147?

00:05:37.190 --> 00:05:38.800
I think we've done
this before.

00:05:38.800 --> 00:05:41.050
Sure, 3 times -- let me
just do it up here.

00:05:41.050 --> 00:05:47.210
3 goes into 147 4,
12, 27, 49 times.

00:05:47.210 --> 00:05:51.410
So you get a 3 and you
get a 49, right?

00:05:51.410 --> 00:05:53.710
49,000, I should say.

00:05:53.710 --> 00:05:59.370
You could do a 7 and a 7,000.

00:05:59.370 --> 00:06:07.370
And then you can get another
7, and you can get a 1,000.

00:06:07.370 --> 00:06:09.520
And then you could do
2 a bunch of times.

00:06:09.520 --> 00:06:14.130
So you get a 2, you get a --
well, the main thing, just so

00:06:14.130 --> 00:06:16.490
we don't waste time is they want
to know-- what color do

00:06:16.490 --> 00:06:17.530
they want to know?

00:06:17.530 --> 00:06:22.660
They want to know how many red
beads were removed, right?

00:06:22.660 --> 00:06:24.910
So red corresponds to 7.

00:06:24.910 --> 00:06:26.060
So all the beads, you
multiply them

00:06:26.060 --> 00:06:28.240
together and you get 147,000.

00:06:28.240 --> 00:06:31.440
So we've essentially factored
out all the 7's that we can

00:06:31.440 --> 00:06:33.530
fit into 147,000.

00:06:33.530 --> 00:06:36.450
7 does not go into
1,000 at all.

00:06:36.450 --> 00:06:37.870
7 is not a factor.

00:06:37.870 --> 00:06:39.120
So we've already solved
our problem.

00:06:39.120 --> 00:06:43.700
There are exactly two 7 prime
factors in 147,000.

00:06:43.700 --> 00:06:44.770
You could keep going
with this.

00:06:44.770 --> 00:06:46.410
You know, you get a 2.

00:06:46.410 --> 00:06:47.820
2 times 500.

00:06:47.820 --> 00:06:48.820
You get another 2.

00:06:48.820 --> 00:06:49.990
2 times 250.

00:06:49.990 --> 00:06:51.620
There's a bunch of
2's in here.

00:06:51.620 --> 00:06:54.000
And then you'd probably get-- I
don't know what else you're

00:06:54.000 --> 00:06:54.370
going to get.

00:06:54.370 --> 00:06:56.480
You'd probably get a couple
of 5's in here, and

00:06:56.480 --> 00:06:57.280
then you'd be done.

00:06:57.280 --> 00:06:58.310
But these are all the 7's.

00:06:58.310 --> 00:07:01.550
There are two 7's in 147,000.

00:07:01.550 --> 00:07:04.640
Two 7 prime factors, so there
would be 2 red beads.

00:07:04.640 --> 00:07:06.610
And that is choice D.

00:07:09.860 --> 00:07:13.320
Problem 235.

00:07:13.320 --> 00:07:14.750
Switch colors.

00:07:14.750 --> 00:07:17.800
235.

00:07:17.800 --> 00:07:25.190
If 2 over 1 plus 2 over
y is equal to 1,

00:07:25.190 --> 00:07:27.750
then what does y equal?

00:07:27.750 --> 00:07:29.460
Let's take the inverse
of both.

00:07:29.460 --> 00:07:32.800
Let's multiply both sides of the
equation times 1 plus 2/y.

00:07:32.800 --> 00:07:38.340
And so we get 2 is equal
to 1 plus 2/y.

00:07:38.340 --> 00:07:42.260
And let's multiply both sides
of the equation by y.

00:07:42.260 --> 00:07:44.710
So you get 2y.

00:07:44.710 --> 00:07:47.090
And actually, you could probably
already eyeball this.

00:07:47.090 --> 00:07:47.830
Actually there's
an easier way.

00:07:47.830 --> 00:07:49.660
Let's subtract 1 from
both sides.

00:07:49.660 --> 00:07:52.340
You get 1 is equal to 2/y.

00:07:52.340 --> 00:07:56.920
Multiply both sides by y and
you get y is equal to 2.

00:07:56.920 --> 00:07:59.305
And that is choice D.

00:07:59.305 --> 00:08:01.400
And you could've actually
eyeballed it, because, say, 1

00:08:01.400 --> 00:08:04.470
plus 1 is equal to 2,
so this has to be 1.

00:08:04.470 --> 00:08:05.590
So how am I going to
make that a 1?

00:08:05.590 --> 00:08:07.390
That equals 2, but sometimes
it's easier just to

00:08:07.390 --> 00:08:09.130
mechanically go through the
algebra, because sometimes,

00:08:09.130 --> 00:08:10.470
you can't eyeball it,
and it's good to

00:08:10.470 --> 00:08:14.610
just keep moving forward.

00:08:14.610 --> 00:08:19.120
Problem 236.

00:08:19.120 --> 00:08:27.100
If a, b, and c are consecutive
positive integers, and a is

00:08:27.100 --> 00:08:31.210
less than b, which is less than
c, which of the following

00:08:31.210 --> 00:08:32.539
must be true?

00:08:32.539 --> 00:08:38.520
Consecutive positive, so, you
know, we could say that b

00:08:38.520 --> 00:08:42.539
would be equal to a plus 1, and
c would be equal to a plus

00:08:42.539 --> 00:08:46.340
2 or b plus 1.

00:08:46.340 --> 00:08:47.740
And, of course, a
is equal to a.

00:08:47.740 --> 00:08:49.110
I don't know if that helps,
I just felt like

00:08:49.110 --> 00:08:51.220
writing that down.

00:08:51.220 --> 00:08:54.720
So in statement 1, they say
c minus a is equal to 2.

00:08:54.720 --> 00:08:56.050
Sure.

00:08:56.050 --> 00:08:58.460
They're consecutive positive
integers, so c is going to be

00:08:58.460 --> 00:08:59.680
2 more than a.

00:08:59.680 --> 00:09:01.040
So this is right.

00:09:01.040 --> 00:09:04.005
Statement 2, they tell
us abc is even.

00:09:07.680 --> 00:09:09.640
Well, instead of going into some
algebraic thing where I

00:09:09.640 --> 00:09:11.390
represent even and odd integers
and all of that,

00:09:11.390 --> 00:09:14.590
let's just try it with two
cases, one where a is odd and

00:09:14.590 --> 00:09:15.820
one where a is even.

00:09:15.820 --> 00:09:17.240
So let's say a is even.

00:09:17.240 --> 00:09:22.170
So the numbers would be 2--
sorry, would be 2 times 3

00:09:22.170 --> 00:09:24.070
times 4, right?

00:09:24.070 --> 00:09:26.470
So then we would get 8.

00:09:26.470 --> 00:09:27.260
Well, you know what?

00:09:27.260 --> 00:09:30.510
This has to be even because one
of these numbers are going

00:09:30.510 --> 00:09:32.300
to be even, right?

00:09:32.300 --> 00:09:34.500
And so if you're divisible, and
by definition, if one of

00:09:34.500 --> 00:09:36.490
these numbers are even,
it's divisible by 2.

00:09:36.490 --> 00:09:38.840
So the whole thing, when you
multiply it together, it's got

00:09:38.840 --> 00:09:40.530
to be divisible by 2, right?

00:09:40.530 --> 00:09:42.830
Either a and c are even, in
which case this whole thing

00:09:42.830 --> 00:09:46.180
has to be even, or b is even, in
which case this whole thing

00:09:46.180 --> 00:09:46.750
is even, right?

00:09:46.750 --> 00:09:50.370
If you multiply a string of
numbers and at least one of

00:09:50.370 --> 00:09:53.120
them is even, then that whole
number is going to be even,

00:09:53.120 --> 00:09:55.270
because that number that's even
is a factor of that whole

00:09:55.270 --> 00:09:56.340
string of numbers.

00:09:56.340 --> 00:09:58.690
But anyway, you could try it
with 2, 3, 4, or you could try

00:09:58.690 --> 00:10:00.560
it with 3, 4, 5.

00:10:00.560 --> 00:10:02.470
Either way, you're going
to get an even number.

00:10:02.470 --> 00:10:04.840
So this is also right.

00:10:04.840 --> 00:10:06.840
Statement 3.

00:10:06.840 --> 00:10:13.730
a plus b plus c over
3 is an integer.

00:10:16.800 --> 00:10:19.440
Well, let's think about it.

00:10:19.440 --> 00:10:19.830
I don't know.

00:10:19.830 --> 00:10:25.500
Let's say if we were to
take a plus b plus c.

00:10:25.500 --> 00:10:27.940
I don't know, even if we were to
take-- let's see, 3 plus 4

00:10:27.940 --> 00:10:29.500
plus-- no, that works.

00:10:29.500 --> 00:10:31.840
2 plus 3-- no, that works.

00:10:31.840 --> 00:10:36.540
Let's write it as-- so this
is a plus a plus 1, right?

00:10:36.540 --> 00:10:42.700
B is a plus 1, plus a plus
2 over 3, right?

00:10:42.700 --> 00:10:50.735
And so this is equal to 3a plus
3 over 3. b is just a

00:10:50.735 --> 00:10:52.390
plus 1, c is a plus 2.

00:10:52.390 --> 00:10:53.430
You add them all together.

00:10:53.430 --> 00:10:55.360
Sure enough, you know, you could
divide the top and the

00:10:55.360 --> 00:11:01.630
bottom by 3 and you
get a plus 1.

00:11:01.630 --> 00:11:04.510
So yes, statement 3
is also correct.

00:11:04.510 --> 00:11:07.050
This is always going to evaluate
to a plus 1, which is

00:11:07.050 --> 00:11:07.890
an integer.

00:11:07.890 --> 00:11:11.690
So all three are true,
so that is choice E.

00:11:11.690 --> 00:11:13.400
And I'm all out of time.

00:11:13.400 --> 00:11:15.420
See you in the next video.