CA Algebra I: Slope and Y-intercept

WEBVTT
Kind: captions
Language: en

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We're on problem 27.

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And the question is, which
equation best represents the

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graph above?

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So before even looking at the
choices, let's see what we can

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figure out about the graph.

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So what's its y-intercept?

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So if we said that this is the
equation of a line, we said y

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is equal to mx plus b where m is
the slope and y is-- and b

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is the y-intercept.

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It's late, I'm getting
slightly muddled.

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So what's the y-intercept?

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Well when x is equal to
0, y is equal to 0.

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So this is going to
be equal to 0.

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The y-intercept is 0.

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When x is equal to 0,
why is equal to 0.

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So the y intercept is 0.

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So we know that this has a form
y is equal to mx where m

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is a slope.

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Let's figure out the slope.

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Slope is equal to change in y
for a given change in x or

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change in y over change in x.

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So when we increase x by 1, how
much are we increase-- or

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decrease y by?

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Well then y increases by 2.

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So we could say that equals y
changes by 2, by plus 2 when x

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changes by plus 1.

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So we get the slope is equal to
2, so the equation of this

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line is y is equal to 2x.

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Which is choice B.

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Next problem.

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Which point lies on the line
defined by 3x plus

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6y is equal to 2?

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Well the best thing to do is
just probably to substitute

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these numbers in for x and y
and see which one works.

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So here x is 0, y is 2.

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So let's see.

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3 times 0 plus 6 times 2
is equal to 0 plus 12.

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Doesn't equal 2, it equals 12.

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This one doesn't work.

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I'm just taking 3 times
x plus 6 times y and

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seeing what it equals.

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In this case, we have 3 times
0 plus 6 times y.

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Plus 6 times 6.

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Well that's 0 plus 36.

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That doesn't equal 2.

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Can't be that choice.

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This one, we have 3,
this 3, times 1.

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Plus 6 times this y.

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6 times minus 1/6.

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So let's see.

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That's 3.

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That's equal to 3.

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And then, 6 times 1/6
is 1, but we have a

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negative out there.

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So it's minus 1.

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That is equal to 2.

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So that works.

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3 times 1 plus 6 times negative
1/6 is equal to 2.

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So our answer is C.

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Problem 29.

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Let me see if I need to cut
and paste this one in.

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All right.

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So yeah, I think it's
a good idea.

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Let me see.

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Copy and paste it.

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Let me copy and paste the next
several problems. We could

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just move-- be streamlined
about it.

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OK, they all fit.

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Good enough.

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What is the equation of the line
that has slope of 4 and

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passes through the point
3 comma minus 10?

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So if the slope is 4, we know
that the line-- so just

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rewrite it in slope y-intercept
form again.

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mx plus b.

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They're telling us that
the slope is 4.

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So we know the equation of the
line is y is equal to 4x plus

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the y-intercept.

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And then we could figure
out the y-intercept by

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substituting in this point that
they say it goes through.

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So it goes through the point
3 comma negative 10.

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So y is equal to minus 10
when x is equal to 3.

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So 4 times x.

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x is equal to 3.

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Plus b.

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So what is that?

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That's minus 10 is equal
to 12 plus b.

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We could subtract 12 from both
sides of this equation, and we

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get minus 22.

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Minus 10 minus 12 is minus 22.

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This 12 obviously goes away.

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Is equal to b.

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So the equation of our line is y
is equal to 4x plus b, which

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we just figured out
is minus 22.

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4x minus 22.

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That's choice A.

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Problem 30.

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The date in the table show the
cost of renting a bicycle by

00:04:28.270 --> 00:04:29.800
the hour, including a deposit.

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If hours, h, were graphed on the
horizontal axis-- let me

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see if I can draw that.

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So that is on the horizontal
axis I have hours, h.

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So this is h.

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And costs were graphed on the
vertical axis, so let me draw

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the vertical axis.

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So cost is on the
vertical axis.

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What would be the equation of
the line that fits the data?

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OK, so they just want to know
cost as a function of hours.

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So let's see.

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It's a linear relationship.

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We could treat this just
like any other line.

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So if we write it as
y equals mx plus b.

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I didn't even have to do this.

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So y is equal to mx plus b.

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So m, or the slope, is equal to
change in y over-- or let

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me write it differently.

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Because we want it in terms of--
so we could write c is

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equal to the slope times
the hours plus b.

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Instead of x, we have hours, and
instead of y, we have the

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cost.

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So what's the slope?

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Slope is equal to change in the
dependent variable, change

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in cost, divided by change in
the independent variable,

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divided by change in hours.

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So let's see.

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If when our hours increased
by 3, how much did our

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change in c go by?

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When we go from 2 to 5, so if
we say 5 minus 2, that's how

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much we changed in hours.

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That's delta h.

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Then how much did we
change in cost?

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Well it's 30 minus 15.

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So it equals 15 over 3.

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which is equal to 5.

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So we figured out the slope.

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Slope is equal to 5.

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Now we just have to figure
out the y-intercept.

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So this equation, we could
rewrite it as cost is equal to

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the slope, which is 5, times
the hours plus whatever our

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y-intercept is.

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And then we just have to
substitute one of the points

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in to solve for b, just like we
did in the previous video.

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So let's substitute the point
when h is equal to 2, c is

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equal to 15.

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So the point 2 comma 15.

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So when h is equal to
2, c is equal to 15.

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c is equal to 15 when
h is equal to 2.

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And now we can just
solve for b.

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So you get 15 is equal
to 10 plus b.

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Subtract 10 from both sides,
you get b is equal to what?

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5.

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So the equation of this line is
cost is equal to 5 times h

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plus our y-intercept, or plus
our c-intercept we could say.

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So plus 5.

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5h plus 5, and that
is choice C.

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And if you try it out
with these points,

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it should all work.

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OK, next problem.

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Let me see.

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I think this cutting and
pasting works out well.

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I think I have time for
at least two more.

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OK, let me clear this.

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Clear this one right here.

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Let me paste it in.

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OK, it says, some ordered pairs
for a linear function of

00:08:01.500 --> 00:08:03.790
x are given in the
table below.

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It's kind of like
the last one.

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Which of the following equations
was used to generate

00:08:07.400 --> 00:08:08.850
the table above?

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Well we could do the same thing
we did the last time.

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We could say OK, this is going
to have some relation.

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You know y is equal
to mx plus b.

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It's going to be a line
and this is the

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slope y-intercept form.

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And they tell us it's
the linear function.

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So that's why we can immediately
say that's going

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to be a line.

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There's no x squared or
anything like that.

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So a good place to start
is just with the slope.

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So slope is equal to change
in y over change in x.

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So let's see.

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When x goes from 1 to 3,
what does y go for.

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It goes from 1 to 7.

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So change in x is equal to
3 minus 1 or just 2.

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And what's change in y?

00:08:50.980 --> 00:08:54.230
7 minus 1.

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So that is equal to
6 over-- oh, no.

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Sorry.

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Change in x was 3 minus 1.

00:09:01.560 --> 00:09:03.140
Sorry about that.

00:09:03.140 --> 00:09:04.020
3 minus 1.

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So it's 6 over 2.

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When x changes by 2-- that's
this 2, change in x-- y

00:09:13.940 --> 00:09:15.090
changes by 6.

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That's change in y.

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So the slope is 3.

00:09:19.260 --> 00:09:23.660
So the equation of our line is
going to be y is equal to 3x

00:09:23.660 --> 00:09:25.900
plus some y-intercept.

00:09:25.900 --> 00:09:27.660
And now we could substitute
some points in.

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Let's substitute this one.

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This 1, 1 seems easy.

00:09:29.670 --> 00:09:33.530
So y is equal to 1 when
x is equal to 1.

00:09:36.500 --> 00:09:40.010
So we get 1 is equal
to 3 plus b.

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Subtract 3 from both sides of
this equation and we get what?

00:09:44.930 --> 00:09:48.180
1 minus 3 is minus 2 is equal
to-- obviously, this

00:09:48.180 --> 00:09:50.280
disappears, 3 minus 3.

00:09:50.280 --> 00:09:52.450
So b is equal to minus 2.

00:09:52.450 --> 00:09:54.760
So the equation of our line,
we have a slope of 3 and a

00:09:54.760 --> 00:09:56.080
y-intercept of minus 2.

00:09:56.080 --> 00:10:01.840
The equation is going to be
y is equal to 3x plus b.

00:10:01.840 --> 00:10:02.960
Well b is minus 2.

00:10:02.960 --> 00:10:05.130
So minus 2.

00:10:05.130 --> 00:10:06.940
And that is choice C.

00:10:13.310 --> 00:10:14.730
32.

00:10:14.730 --> 00:10:18.870
The equation of line l is
6x plus 5y equals 3.

00:10:18.870 --> 00:10:24.570
And equation of line q is 5x
minus 6y is equal to 0.

00:10:24.570 --> 00:10:27.070
Which of the statements about
the two lines are true?

00:10:27.070 --> 00:10:27.820
OK, let's see.

00:10:27.820 --> 00:10:29.360
They have the same
y intercept.

00:10:29.360 --> 00:10:30.220
They're parallel.

00:10:30.220 --> 00:10:33.050
They have the same
x-intercept.

00:10:33.050 --> 00:10:36.510
Lines l and q are
perpendicular.

00:10:36.510 --> 00:10:38.970
Well I'll tell you how you can
identify perpendicular lines.

00:10:38.970 --> 00:10:41.470
They have the negative inverse
of each other's slope.

00:10:41.470 --> 00:10:43.710
But I don't know if that's going
to come into play yet.

00:10:43.710 --> 00:10:47.960
I always find it easier to look
at these lines in kind of

00:10:47.960 --> 00:10:49.310
the slope y-intercept form.

00:10:49.310 --> 00:10:49.940
So let's do that.

00:10:49.940 --> 00:10:57.540
So line l can be 6x plus
5y is equal to 3.

00:10:57.540 --> 00:11:01.160
So if we were to subtract 6x
from both sides you get 5y is

00:11:01.160 --> 00:11:04.560
equal to minus 6x plus 3.

00:11:04.560 --> 00:11:08.300
And then you divide both sides
by 5 and you get y is equal to

00:11:08.300 --> 00:11:14.160
minus 6/5 x plus 3/5.

00:11:14.160 --> 00:11:18.380
So this is line l right here.

00:11:18.380 --> 00:11:20.100
I'll do line q in a
different color.

00:11:20.100 --> 00:11:27.180
So q is 5x minus 6y
is equal to 0.

00:11:27.180 --> 00:11:29.580
So let's subtract 5x
from both sides.

00:11:29.580 --> 00:11:35.080
You get minus 6y is
equal to minus 5x.

00:11:35.080 --> 00:11:39.760
If you divide both sides by
negative 6, you get y is equal

00:11:39.760 --> 00:11:42.000
to-- minus 5 divided
by negative 6.

00:11:42.000 --> 00:11:46.010
That's 5/6 x.

00:11:46.010 --> 00:11:48.470
Well, it actually looks like
D comes into play.

00:11:48.470 --> 00:11:52.060
Because the slope of this
one is minus 6/5.

00:11:52.060 --> 00:11:54.110
That's the slope of line l.

00:11:54.110 --> 00:11:59.620
The slope of line q is the
inverse of that made negative.

00:11:59.620 --> 00:12:01.390
At this point it's a good
thing to memorize.

00:12:01.390 --> 00:12:04.580
I haven't proven
it to you yet.

00:12:04.580 --> 00:12:07.670
If you want a line that has a
slope perpendicular to this

00:12:07.670 --> 00:12:10.590
one, or a line that is
perpendicular to this one will

00:12:10.590 --> 00:12:12.470
have a negative inverse slope.

00:12:12.470 --> 00:12:16.630
So the inverse of this is minus
5/6 and the negative

00:12:16.630 --> 00:12:19.030
inverse is plus 5/6.

00:12:19.030 --> 00:12:21.650
So this is the negative
inverse of that.

00:12:21.650 --> 00:12:24.520
So these lines are actually
perpendicular.

00:12:24.520 --> 00:12:27.740
So that is D.

00:12:27.740 --> 00:12:29.560
See you in the next video.


CA Algebra I: Graphing Inequalities

WEBVTT
Kind: captions
Language: en

00:00:00.790 --> 00:00:02.720
We're on problem 21.

00:00:02.720 --> 00:00:06.190
Stan's solution to an equation
is shown below.

00:00:06.190 --> 00:00:08.440
All right, and I haven't looked
at it yet, but let's

00:00:08.440 --> 00:00:09.540
just see what they ask us.

00:00:09.540 --> 00:00:12.440
Which statement about Stan's
solution is true?

00:00:12.440 --> 00:00:14.550
So let's just work through it
and see if he got it right or

00:00:14.550 --> 00:00:15.920
if he made a mistake.

00:00:15.920 --> 00:00:17.220
So let's see.

00:00:17.220 --> 00:00:18.490
Let's think about it
how we would do it.

00:00:18.490 --> 00:00:25.980
So if we had n plus 8 times
n plus 20 is equal to 110.

00:00:25.980 --> 00:00:28.380
So the first thing you want to
do is you'd want to distribute

00:00:28.380 --> 00:00:31.330
this 8 times n plus 20.

00:00:31.330 --> 00:00:33.730
You don't want to multiply
this whole n plus 8.

00:00:33.730 --> 00:00:35.680
Remember, you're doing order
of operations, so it's

00:00:35.680 --> 00:00:37.480
multiplication first.
Let's do that.

00:00:37.480 --> 00:00:40.330
So you get the n plus and
you distribute this 8.

00:00:40.330 --> 00:00:47.520
So it's 8n plus 160
is equal to 110.

00:00:47.520 --> 00:00:51.080
8 times n plus 8 times
20 is 160.

00:00:51.080 --> 00:00:52.660
And already I see a discrepancy

00:00:52.660 --> 00:00:53.670
between ours and his.

00:00:53.670 --> 00:00:55.560
We have n there.

00:00:55.560 --> 00:00:57.580
We have plus 8n there.

00:00:57.580 --> 00:01:00.410
But where he has a
20, we have 160.

00:01:00.410 --> 00:01:02.400
He didn't distribute the 8.

00:01:02.400 --> 00:01:05.810
8 times n plus 20
is 8n plus 160.

00:01:05.810 --> 00:01:06.470
So that's wrong.

00:01:06.470 --> 00:01:10.220
So he made a mistake
in step 1.

00:01:10.220 --> 00:01:12.560
Unfortunately, he got all that
other work that he had to do

00:01:12.560 --> 00:01:14.935
just because of his-- it's all
going to be wrong because of

00:01:14.935 --> 00:01:16.300
his mistake.

00:01:16.300 --> 00:01:18.060
Problem 22.

00:01:18.060 --> 00:01:20.380
When is the statement true?

00:01:20.380 --> 00:01:25.085
The opposite of a number is less
then the original number.

00:01:29.280 --> 00:01:32.910
Well, I mean, if I have a
negative number, the opposite

00:01:32.910 --> 00:01:34.670
of a negative is positive.

00:01:34.670 --> 00:01:35.720
So it's not going to be less.

00:01:35.720 --> 00:01:38.580
But if l positive number, then
the opposite of it is

00:01:38.580 --> 00:01:40.520
negative, which is
less than it.

00:01:40.520 --> 00:01:43.390
Because a negative is less
than a positive.

00:01:43.390 --> 00:01:52.280
Or if we say that x is greater
than 0, then minus x is going

00:01:52.280 --> 00:01:55.760
to be less than x.

00:01:55.760 --> 00:01:57.580
If we say x is positive,
then definitely.

00:01:57.580 --> 00:02:00.170
You know, minus 3 is less
than positive 3.

00:02:00.170 --> 00:02:03.130
So the statement is true
for positive numbers.

00:02:03.130 --> 00:02:04.380
So that's C.

00:02:06.700 --> 00:02:07.930
It's definitely not
never true.

00:02:07.930 --> 00:02:09.960
I mean we showed a case
where it's true.

00:02:09.960 --> 00:02:13.210
If I have positive 3, negative
3 is less than that.

00:02:13.210 --> 00:02:13.830
Statement B.

00:02:13.830 --> 00:02:14.550
The statement is always true.

00:02:14.550 --> 00:02:16.650
Well no, if we start with
negative 3, the opposite is

00:02:16.650 --> 00:02:17.620
going to be larger.

00:02:17.620 --> 00:02:19.840
The opposite's going
to be positive 3.

00:02:19.840 --> 00:02:20.290
And D.

00:02:20.290 --> 00:02:21.710
The statement is true for
negative numbers.

00:02:21.710 --> 00:02:22.910
No, it's not.

00:02:22.910 --> 00:02:24.550
Negative 3, the opposite is 3.

00:02:24.550 --> 00:02:27.730
And that's larger
than negative 3.

00:02:27.730 --> 00:02:29.570
Problem 23.

00:02:29.570 --> 00:02:32.230
What is the y-intercept
of the graph?

00:02:32.230 --> 00:02:33.520
OK, let me see.

00:02:33.520 --> 00:02:35.060
So 23.

00:02:35.060 --> 00:02:40.330
They say 4x plus 2y
is equal to 12.

00:02:40.330 --> 00:02:43.960
So if you were to graph this
line, not that I'm going to--

00:02:43.960 --> 00:02:47.210
the y-intercept is when it
intersects the y-axis.

00:02:47.210 --> 00:02:49.120
So it's that point there.

00:02:49.120 --> 00:02:50.350
It's what's this y-coordinate.

00:02:50.350 --> 00:02:53.660
Well the x-coordinate, it's
when x is equal to 0.

00:02:53.660 --> 00:02:57.940
So it's going to be 0 comma
some y-intercept.

00:02:57.940 --> 00:02:59.690
So the easiest thing to do
is say when x is equal

00:02:59.690 --> 00:03:01.550
to 0, what is y?

00:03:01.550 --> 00:03:05.990
So when x is equal to 0, this
4 times 0 is equal to 0.

00:03:05.990 --> 00:03:08.030
So that just becomes 0.

00:03:08.030 --> 00:03:10.950
So you get 2y is equal to 12.

00:03:10.950 --> 00:03:16.000
y is equal to 6, and
that's choice C.

00:03:16.000 --> 00:03:19.020
All they want to know is, of
this line, when x is equal to

00:03:19.020 --> 00:03:19.790
0, what's y?

00:03:19.790 --> 00:03:22.110
That's what the y-intercept
is.

00:03:22.110 --> 00:03:23.120
Next problem.

00:03:23.120 --> 00:03:26.090
OK, they've drawn us a graph.

00:03:26.090 --> 00:03:28.750
They say-- let me copy
and paste this one.

00:03:32.740 --> 00:03:33.990
OK.

00:03:35.980 --> 00:03:37.230
Copied it.

00:03:40.490 --> 00:03:44.460
OK, which inequality is shown
on the graph below?

00:03:44.460 --> 00:03:46.160
So first of all, let's figure
out the equation of

00:03:46.160 --> 00:03:47.690
this line right now.

00:03:47.690 --> 00:03:50.250
So what's its y-intercept,
first of all?

00:03:50.250 --> 00:03:57.300
So when x is equal to 0,
y is equal to minus 1.

00:03:57.300 --> 00:04:00.320
If we put it in slope y
intercept form, the equation

00:04:00.320 --> 00:04:04.300
of really, any line is y
is equal to mx plus b.

00:04:04.300 --> 00:04:07.970
Where m is the slope and
b is the y-intercept.

00:04:07.970 --> 00:04:10.560
Here, when x is 0,
y is negative 1.

00:04:10.560 --> 00:04:12.740
So the y-intercept is minus 1.

00:04:12.740 --> 00:04:14.920
So b is minus 1.

00:04:14.920 --> 00:04:19.630
So we know that y is equal
to mx minus 1.

00:04:19.630 --> 00:04:21.470
And now we just have to
figure out the slope.

00:04:21.470 --> 00:04:25.720
And the slope is just change in
y for a given change in x.

00:04:25.720 --> 00:04:31.920
So if we say change in y for a
given change in x is equal

00:04:31.920 --> 00:04:32.420
to-- let's see.

00:04:32.420 --> 00:04:39.930
When I increase x by 2,
I increase y by 1.

00:04:39.930 --> 00:04:43.280
So change in y is 1 when
the change in x is 2.

00:04:43.280 --> 00:04:48.030
And you could say when you
increase x by 4-- I'm going

00:04:48.030 --> 00:04:50.290
four spots-- I increase
y by 2.

00:04:50.290 --> 00:04:51.110
So you could have done that.

00:04:51.110 --> 00:04:53.490
You could have said change in
y over change in x is 2/4,

00:04:53.490 --> 00:04:55.030
which is also equal to 1/2.

00:04:55.030 --> 00:04:57.000
But either way, we now
know the slope.

00:04:57.000 --> 00:05:00.320
The slope is change in
y over change in x.

00:05:00.320 --> 00:05:07.570
So the equation of the line is
y is equal to 1/2 x minus 1.

00:05:07.570 --> 00:05:08.930
That's the equation
of this line.

00:05:08.930 --> 00:05:11.820
Now they want to know what the
equation of the inequality is.

00:05:11.820 --> 00:05:14.520
So first of all, they're
including-- they have this

00:05:14.520 --> 00:05:16.720
grey area and they put
this line in bold.

00:05:16.720 --> 00:05:18.800
So that means that we're
including the line.

00:05:18.800 --> 00:05:20.940
If they had drawn a dotted line
here that means we're not

00:05:20.940 --> 00:05:21.540
including the line.

00:05:21.540 --> 00:05:23.130
But since they're including
the line, it's going to be

00:05:23.130 --> 00:05:25.600
either greater than or less
than or equal to.

00:05:25.600 --> 00:05:28.560
That equal to is because we're
including the line.

00:05:28.560 --> 00:05:30.310
But just think about it.

00:05:30.310 --> 00:05:34.830
So our choices are y is greater
than or equal to 1/2 x

00:05:34.830 --> 00:05:40.900
minus 1 or y is less than or
equal to 1/2 x minus 1.

00:05:40.900 --> 00:05:44.230
Now think about for
any given x.

00:05:44.230 --> 00:05:47.550
Let's say when x
is equal to 2.

00:05:47.550 --> 00:05:52.600
When x is equal to 2, if you put
it into this equation you

00:05:52.600 --> 00:05:55.440
get y is equal to 0 and that's
this point on the line.

00:05:55.440 --> 00:05:59.890
Now is the grey area all y
values greater than 0 or all y

00:05:59.890 --> 00:06:01.140
values less than 0?

00:06:01.140 --> 00:06:04.440
Well clearly, it's all y
values greater than 0.

00:06:04.440 --> 00:06:06.760
It's all the y values
above the y value

00:06:06.760 --> 00:06:08.010
dictated by this equation.

00:06:10.650 --> 00:06:14.810
The equation of this inequality
or this area is y

00:06:14.810 --> 00:06:18.040
is greater than 1/2 x minus 1.

00:06:18.040 --> 00:06:20.390
And just a very easy way to
eyeball it is OK, they're

00:06:20.390 --> 00:06:22.280
including the line, so I'm
going to have this equal

00:06:22.280 --> 00:06:24.900
then-- it's going to either be
greater than or equal to or

00:06:24.900 --> 00:06:27.930
less than or equal to, and since
it's the area above the

00:06:27.930 --> 00:06:32.020
line it's going to be greater
than or equal to.

00:06:32.020 --> 00:06:34.280
And that is choice D.

00:06:36.870 --> 00:06:41.220
Next problem, problem 25.

00:06:41.220 --> 00:06:44.720
This is another one where I
think it makes sense to copy

00:06:44.720 --> 00:06:45.970
and paste it.

00:06:48.800 --> 00:06:50.630
It's a big one.

00:06:50.630 --> 00:06:51.880
OK.

00:06:55.950 --> 00:06:58.560
Which best represents
the graph of y id

00:06:58.560 --> 00:07:01.180
equal to 2x minus 2.

00:07:01.180 --> 00:07:02.870
So the slope is 2.

00:07:02.870 --> 00:07:06.940
Its positive 2 and its
y-intercept is minus 2.

00:07:06.940 --> 00:07:10.290
So the y-intercept is
easy to eyeball.

00:07:10.290 --> 00:07:12.280
Here the y-intercept
is minus 2.

00:07:12.280 --> 00:07:13.130
That's right.

00:07:13.130 --> 00:07:16.150
Here it's minus 2 as well.

00:07:16.150 --> 00:07:18.260
In both of these, the
y-intercept is plus 2.

00:07:18.260 --> 00:07:19.850
So neither of these are
going to, right?

00:07:19.850 --> 00:07:22.430
It intersects the y-axis
at positive 2.

00:07:22.430 --> 00:07:25.180
But we know that the y-intercept
is minus 2.

00:07:25.180 --> 00:07:27.540
So we know-- whoops.

00:07:27.540 --> 00:07:29.580
Where'd I go?

00:07:29.580 --> 00:07:30.620
There we go.

00:07:30.620 --> 00:07:33.010
So we know our choices are
one of the top two.

00:07:33.010 --> 00:07:36.170
It's either A or C.

00:07:36.170 --> 00:07:38.790
And which of these has
a positive slope,

00:07:38.790 --> 00:07:39.980
positive slope of 2?

00:07:39.980 --> 00:07:48.680
Well in choice A when I go to
the right by 2, I go up by 4.

00:07:48.680 --> 00:07:52.590
So change in y over
change in x.

00:07:52.590 --> 00:07:57.420
I go up by 4, when I
go up in x by 2.

00:07:57.420 --> 00:08:01.760
I go up in y by 4 when I go up
in x by 2, so the slope is 2.

00:08:01.760 --> 00:08:05.020
So this has a positive slope
of 2, so that's choice A.

00:08:05.020 --> 00:08:08.000
But just for seeing why choice C
doesn't work or figuring out

00:08:08.000 --> 00:08:10.940
the slope, think about
it this way.

00:08:10.940 --> 00:08:12.380
Let me start at a random
point here.

00:08:15.450 --> 00:08:18.220
If I increase x value
by 2, what's

00:08:18.220 --> 00:08:19.540
happening to my y value?

00:08:19.540 --> 00:08:22.040
Is it increasing by 2 or
is it decreasing by 2?

00:08:22.040 --> 00:08:22.330
Well it's decreasing.

00:08:22.330 --> 00:08:24.420
Well it's decreasing by 4.

00:08:24.420 --> 00:08:26.740
It's decreasing at
twice the rate.

00:08:26.740 --> 00:08:28.000
It's decreasing by 4.

00:08:28.000 --> 00:08:32.299
So in this case, change
in y over change in x.

00:08:32.299 --> 00:08:36.400
Whenever x is positive 2,
we increased by 2 here.

00:08:36.400 --> 00:08:37.559
What happened to y.

00:08:37.559 --> 00:08:39.480
It went down 4.

00:08:39.480 --> 00:08:41.669
So here the slope is minus 2.

00:08:41.669 --> 00:08:42.600
And you can look at that.

00:08:42.600 --> 00:08:44.270
You can kind of eyeball it
because it's going from the

00:08:44.270 --> 00:08:45.750
top left to the bottom right,
so it's going to

00:08:45.750 --> 00:08:46.680
be a negative number.

00:08:46.680 --> 00:08:48.790
But the easiest way I always do
is to draw these arrows and

00:08:48.790 --> 00:08:51.390
say, OK, when I increase
x, y is decreasing.

00:08:51.390 --> 00:08:54.700
So that's going to be
a negative slope.

00:08:54.700 --> 00:08:55.240
Let's do it.

00:08:55.240 --> 00:08:57.560
So the answer was A.

00:08:57.560 --> 00:08:58.810
Next problem.

00:09:02.740 --> 00:09:05.150
OK, so they're giving us another
one of these where

00:09:05.150 --> 00:09:07.520
they shaded a graph.

00:09:07.520 --> 00:09:08.350
These are good.

00:09:08.350 --> 00:09:10.110
Thank God I have the cut
and paste feature.

00:09:18.980 --> 00:09:20.860
So they ask us, which inequality
does the shaded

00:09:20.860 --> 00:09:23.160
region of the graph represent?

00:09:23.160 --> 00:09:27.550
So once again, it'll be good
just to figure out what the

00:09:27.550 --> 00:09:28.780
equation of this line is.

00:09:28.780 --> 00:09:31.230
We could immediately say, OK,
the y-intercept is 2.

00:09:31.230 --> 00:09:32.910
And what's its slope?

00:09:32.910 --> 00:09:40.930
When x increases by 1,
what happens to y?

00:09:40.930 --> 00:09:42.685
y is going down by 3.

00:09:45.400 --> 00:09:47.830
So the y-intercept
is equal to 2.

00:09:47.830 --> 00:09:49.850
We figured that out.
b is equal to 2.

00:09:49.850 --> 00:09:54.710
The slope is equal to change
in y over change in x.

00:09:54.710 --> 00:09:59.500
When x increases by 1,
the slope-- sorry.

00:09:59.500 --> 00:10:01.330
The change in y is minus 3.

00:10:01.330 --> 00:10:03.960
It goes down by 3.

00:10:03.960 --> 00:10:06.490
So the slope is equal
to minus 3.

00:10:06.490 --> 00:10:09.820
So the equation of this
line is y is equal to

00:10:09.820 --> 00:10:13.300
minus 3x plus 2.

00:10:13.300 --> 00:10:15.540
mx plus b.

00:10:15.540 --> 00:10:19.050
And if I look at the choices,
already it looks a little bit

00:10:19.050 --> 00:10:20.400
different than that, but
we'll get to that.

00:10:20.400 --> 00:10:22.410
But let's see what the
inequality is.

00:10:22.410 --> 00:10:24.700
So first of all, it's going to
be either greater than or

00:10:24.700 --> 00:10:26.430
equal or less than or equal.

00:10:26.430 --> 00:10:27.900
And we say the or equal
because they

00:10:27.900 --> 00:10:29.610
filled in the line.

00:10:29.610 --> 00:10:31.540
So let's think about it.

00:10:31.540 --> 00:10:37.870
If we pick any x, say when x is
equal to 2, then according

00:10:37.870 --> 00:10:40.060
to this equation, according
to this line, y would be

00:10:40.060 --> 00:10:41.880
equal to minus 4.

00:10:41.880 --> 00:10:44.610
Now, are we including the y's
that are greater than that

00:10:44.610 --> 00:10:47.110
value as well or are we
including the y's that are

00:10:47.110 --> 00:10:48.165
less than that value?

00:10:48.165 --> 00:10:50.980
Well, we're including the ones
that are less than that point

00:10:50.980 --> 00:10:52.230
for a given x.

00:10:52.230 --> 00:10:57.460
So the shaded region is y
is less than or equal to

00:10:57.460 --> 00:10:59.770
minus 3x plus 2.

00:10:59.770 --> 00:11:00.710
And you can eyeball this.

00:11:00.710 --> 00:11:04.060
You can kind of say, oh, this
is below the graph, so it's

00:11:04.060 --> 00:11:05.260
less than or equal.

00:11:05.260 --> 00:11:07.720
And that equal was because
they shaded in the line.

00:11:07.720 --> 00:11:10.350
If it was a dotted line it
would just be less than.

00:11:10.350 --> 00:11:12.820
Once again, this isn't
the exact format

00:11:12.820 --> 00:11:13.610
that they have here.

00:11:13.610 --> 00:11:16.040
They put all the x's and the y's
on the same side, and we

00:11:16.040 --> 00:11:17.110
can do that.

00:11:17.110 --> 00:11:20.200
Let's just add 3x to both sides
of this equation and you

00:11:20.200 --> 00:11:24.800
get 3x plus y is less
than or equal to.

00:11:24.800 --> 00:11:26.245
When you add 3x on the
right-hand side that

00:11:26.245 --> 00:11:27.690
just becomes 0.

00:11:27.690 --> 00:11:29.690
So less than or equal to 2.

00:11:29.690 --> 00:11:34.800
And that is 3x plus y is less
than or equal to positive 2.

00:11:34.800 --> 00:11:38.760
And that is choice A.

00:11:38.760 --> 00:11:40.700
And I'm all out of time, see
you in the next video.


CA Algebra I: Simple Logical Arguments

WEBVTT
Kind: captions
Language: en

00:00:00.660 --> 00:00:03.860
We're on problem 14.

00:00:03.860 --> 00:00:08.109
And it asks us what is the
solution to the inequality x

00:00:08.109 --> 00:00:12.710
minus 5 is greater than 14?

00:00:12.710 --> 00:00:14.530
Well, to do this, this
is just like solving

00:00:14.530 --> 00:00:17.820
any equality or equation.

00:00:17.820 --> 00:00:20.000
What we do to one side, we have
to do the other side.

00:00:20.000 --> 00:00:22.060
And we want to get rid of this
minus 5, and the best way to

00:00:22.060 --> 00:00:25.206
get rid of a minus 5 is to add
5, so lets add 5 to both sides

00:00:25.206 --> 00:00:26.610
of this equation.

00:00:26.610 --> 00:00:29.910
So 5 plus and then a plus 5.

00:00:29.910 --> 00:00:32.340
And then a 5 plus a
minus 5, that's 0.

00:00:32.340 --> 00:00:34.530
That's why we added the
5 in the first place.

00:00:34.530 --> 00:00:36.240
So we're just left with
an x on that side.

00:00:36.240 --> 00:00:40.610
We get x is greater than
14 plus 5, which is 19.

00:00:40.610 --> 00:00:42.650
That's choice B.

00:00:42.650 --> 00:00:44.980
Problem 15.

00:00:44.980 --> 00:00:47.195
The lengths of the sides
of a triangle are--

00:00:47.195 --> 00:00:49.300
so we have a triangle.

00:00:49.300 --> 00:00:54.090
They tell us the lengths of the
sides are y, y plus 1, and

00:00:54.090 --> 00:00:56.180
7 centimeters.

00:00:56.180 --> 00:00:58.530
They also tell us the perimeter
is 56 centimeters.

00:00:58.530 --> 00:01:01.660
The perimeter is equal
to 56 centimeters.

00:01:01.660 --> 00:01:03.670
What is the value of y?

00:01:03.670 --> 00:01:08.590
So the perimeter of any shape is
just the sum of the sides.

00:01:08.590 --> 00:01:14.240
So y plus y plus 1 plus 7.

00:01:14.240 --> 00:01:16.820
That's the distance around
the triangle.

00:01:16.820 --> 00:01:18.200
And that is equal to
the perimeter,

00:01:18.200 --> 00:01:19.600
which is equal to 56.

00:01:19.600 --> 00:01:27.110
Let's see, we get y plus y is
2y, plus 1 plus 7 is 8, is

00:01:27.110 --> 00:01:31.560
equal to 56, so you get 2y
is equal to what is that?

00:01:31.560 --> 00:01:32.600
56.

00:01:32.600 --> 00:01:36.870
So if we subtract 8 from both
sides of this equation, on the

00:01:36.870 --> 00:01:38.240
left-hand side, we
just get 2y.

00:01:38.240 --> 00:01:42.460
On the right-hand side,
56 minus 8, that's 48.

00:01:42.460 --> 00:01:49.360
Divide both sides by 2 and
you get y is equal to 24.

00:01:49.360 --> 00:01:53.100
And that is choice A.

00:01:53.100 --> 00:01:57.480
Problem 16.

00:01:57.480 --> 00:02:00.690
Now what do they
want us to do?

00:02:00.690 --> 00:02:02.620
All right, I think this is one
I should copy and paste.

00:02:08.910 --> 00:02:12.700
OK, now they're telling us--
let me pick a good color.

00:02:12.700 --> 00:02:15.970
Which number serves as a
counterexample to the

00:02:15.970 --> 00:02:16.410
statement below?

00:02:16.410 --> 00:02:19.370
So, a counterexample, an example
that shows that this

00:02:19.370 --> 00:02:20.440
isn't always true.

00:02:20.440 --> 00:02:22.430
So the statement is all
positive integers are

00:02:22.430 --> 00:02:24.480
divisible by 2 or 3.

00:02:24.480 --> 00:02:27.770
So we just have to find a
positive integer that is not

00:02:27.770 --> 00:02:31.950
divisible by 2 or 3,
by neither 2 nor 3.

00:02:31.950 --> 00:02:34.810
Well, 100 is divisible
by 2, right?

00:02:34.810 --> 00:02:38.230
So that just verifies or it's
just another example of a

00:02:38.230 --> 00:02:40.350
positive integer that's
divisible by one of these two,

00:02:40.350 --> 00:02:41.790
so it's not choice A.

00:02:41.790 --> 00:02:43.640
It's not a counterexample.

00:02:43.640 --> 00:02:48.670
57, It's not divisible by 2,
but it is divisible by 3.

00:02:48.670 --> 00:02:51.550
19 times 3 is 57.

00:02:51.550 --> 00:02:53.440
So it's not choice B.

00:02:53.440 --> 00:02:56.870
57 is another positive integer
that's divisible by 2 or 3.

00:02:56.870 --> 00:02:58.670
It's divisible by 3,
so it's not that.

00:02:58.670 --> 00:03:00.390
30's divisible by both, so
that's definitely not a

00:03:00.390 --> 00:03:02.010
counterexample.

00:03:02.010 --> 00:03:03.360
But here we have 25.

00:03:03.360 --> 00:03:08.205
It's a positive integer and it
is neither divisible by 2 nor

00:03:08.205 --> 00:03:10.600
3, so it disproves
the statement.

00:03:10.600 --> 00:03:12.160
So it is a counterexample.

00:03:12.160 --> 00:03:15.230
And so the answer is D.

00:03:15.230 --> 00:03:19.260
Problem 17.

00:03:19.260 --> 00:03:20.620
Let me copy and paste
this one.

00:03:23.470 --> 00:03:26.410
Copied it.

00:03:26.410 --> 00:03:27.210
Pasted it.

00:03:27.210 --> 00:03:28.460
All right.

00:03:28.460 --> 00:03:33.550
What is the conclusion of the
statement in the box below?

00:03:33.550 --> 00:03:37.330
OK, they say if x squared
is equal to 4.

00:03:37.330 --> 00:03:42.470
So if we know that x squared is
equal to 4, then we know,

00:03:42.470 --> 00:03:44.440
and we know this from algebra,
we could have solved it, that

00:03:44.440 --> 00:03:49.710
x is equal to minus 2 or 2.

00:03:49.710 --> 00:03:53.020
All right, so what is the
conclusion of the statement in

00:03:53.020 --> 00:03:54.420
the box below?

00:03:54.420 --> 00:03:55.380
Oh, OK.

00:03:55.380 --> 00:03:58.360
I think I'm reading
this too much.

00:03:58.360 --> 00:04:01.840
This is the condition and
this is the conclusion.

00:04:01.840 --> 00:04:03.370
They're saying if this
has happened,

00:04:03.370 --> 00:04:04.670
then we conclude that.

00:04:04.670 --> 00:04:07.910
So they're actually just
asking us to label it.

00:04:07.910 --> 00:04:09.450
This is the conclusion.

00:04:09.450 --> 00:04:13.060
Then x is equal to
minus 2 or 2.

00:04:13.060 --> 00:04:15.810
I don't like that question
so much.

00:04:15.810 --> 00:04:20.480
My reaction was, OK, if this is
the whole statement, what

00:04:20.480 --> 00:04:22.920
can I conclude from it?

00:04:22.920 --> 00:04:25.210
And there's not a lot that I
could conclude from it unless

00:04:25.210 --> 00:04:27.090
they told me that this
indeed was true.

00:04:27.090 --> 00:04:28.810
But anyway, I don't want
to get too complicated.

00:04:28.810 --> 00:04:32.110
They're just saying that if
this is true, then we can

00:04:32.110 --> 00:04:33.010
conclude this.

00:04:33.010 --> 00:04:34.970
And they're actually just saying
what is the conclusion?

00:04:34.970 --> 00:04:37.350
So what part of this statement
can we label as the

00:04:37.350 --> 00:04:38.370
conclusion?

00:04:38.370 --> 00:04:42.160
Well, that's the conclusion,
so that's choice D.

00:04:42.160 --> 00:04:43.410
I didn't like that.

00:04:45.330 --> 00:04:48.740
That was more meaning
of words than math.

00:04:48.740 --> 00:04:52.990
But anyway, problem 18.

00:04:52.990 --> 00:04:55.400
All right, let me copy and
paste this one, too.

00:05:00.830 --> 00:05:02.490
Which of the following
is a valid

00:05:02.490 --> 00:05:05.800
conclusion to the statement?

00:05:05.800 --> 00:05:06.730
OK.

00:05:06.730 --> 00:05:10.100
You see, now if a student is a
high school band member, then

00:05:10.100 --> 00:05:12.280
the student is a
good musician.

00:05:12.280 --> 00:05:16.130
So if you're in the high school
band, high school band

00:05:16.130 --> 00:05:19.450
member, then you're going
to be a good musician.

00:05:19.450 --> 00:05:20.510
That's what this says.

00:05:20.510 --> 00:05:22.370
And if you compare this
to the last question,

00:05:22.370 --> 00:05:24.020
that could be confusing.

00:05:24.020 --> 00:05:28.030
Because a valid conclusion--
so once again, here they're

00:05:28.030 --> 00:05:29.690
giving us a statement and they
want us to come to some

00:05:29.690 --> 00:05:30.490
conclusion.

00:05:30.490 --> 00:05:31.310
They might have said
what is the

00:05:31.310 --> 00:05:33.410
conclusion of this statement?

00:05:33.410 --> 00:05:34.580
And then you would have
said, oh, the

00:05:34.580 --> 00:05:35.730
student is a good musician.

00:05:35.730 --> 00:05:37.180
But that's not what
they're asking.

00:05:37.180 --> 00:05:40.370
They're saying, OK, if this
whole thing is a statement, if

00:05:40.370 --> 00:05:44.260
we know that this whole thing
is true, if they're in the

00:05:44.260 --> 00:05:47.630
high school band, then they're
a good musician, which I kind

00:05:47.630 --> 00:05:49.110
of wrote in shorthand here.

00:05:49.110 --> 00:05:51.630
They're asking, OK, if we know
that to be true, which of

00:05:51.630 --> 00:05:54.440
these statements can we
conclude from that?

00:05:54.440 --> 00:05:56.870
Statement A says, all
good musicians are

00:05:56.870 --> 00:05:58.510
high school band members.

00:05:58.510 --> 00:05:59.810
Well, that doesn't
tell us that.

00:05:59.810 --> 00:06:04.460
It doesn't say that all good
musicians-- the good musicians

00:06:04.460 --> 00:06:06.530
could be this circle and the
high school band members would

00:06:06.530 --> 00:06:08.580
be a subset of it.

00:06:08.580 --> 00:06:11.680
High school band, and this
could be good musicians.

00:06:11.680 --> 00:06:14.090
So right, all high school band
members are good musicians,

00:06:14.090 --> 00:06:16.410
but there could be people out
here who are good musicians

00:06:16.410 --> 00:06:17.890
who aren't high school
band members.

00:06:17.890 --> 00:06:20.170
So it's not choice A.

00:06:20.170 --> 00:06:23.520
A student is a high school
band member.

00:06:23.520 --> 00:06:25.600
Well, no.

00:06:25.600 --> 00:06:28.060
High school band members, they
didn't tell us that all of the

00:06:28.060 --> 00:06:30.380
students are high school band
members or the high school

00:06:30.380 --> 00:06:32.870
band is comprised of all
of the students.

00:06:32.870 --> 00:06:36.010
The student body might
be like this circle.

00:06:36.010 --> 00:06:37.680
Some of them might be in
the high school band.

00:06:37.680 --> 00:06:39.640
Some of them aren't and
are good musicians.

00:06:39.640 --> 00:06:41.410
Some of them aren't and
aren't good musicians.

00:06:41.410 --> 00:06:44.300
So, I mean, you don't know
whether someone is a high

00:06:44.300 --> 00:06:47.480
school band member just
by being a student.

00:06:47.480 --> 00:06:48.950
All students are
good musicians.

00:06:48.950 --> 00:06:50.840
Once again, you know, this
could be an example.

00:06:50.840 --> 00:06:55.410
There could be people out here
who are students, the yellow

00:06:55.410 --> 00:06:57.170
could be students.

00:06:57.170 --> 00:06:58.330
There could be people
who are students

00:06:58.330 --> 00:07:00.110
who aren't good musicians.

00:07:00.110 --> 00:07:02.600
This statement doesn't in
any way constrain that.

00:07:02.600 --> 00:07:04.150
So let's get rid of that.

00:07:04.150 --> 00:07:06.600
All high school band members
are good musicians.

00:07:06.600 --> 00:07:08.140
Well, let's look at
this Venn diagram.

00:07:08.140 --> 00:07:10.680
I mean, that's almost like
a restating of what they

00:07:10.680 --> 00:07:11.790
already told us.

00:07:11.790 --> 00:07:15.070
All high school band members
have to be good musicians

00:07:15.070 --> 00:07:17.260
because they told us, if you're
in the high school

00:07:17.260 --> 00:07:20.190
band, then you're
a good musician.

00:07:20.190 --> 00:07:22.430
So it's almost like repeating
the same statement twice, but

00:07:22.430 --> 00:07:24.940
the answer is D.

00:07:24.940 --> 00:07:30.050
Problem 19.

00:07:30.050 --> 00:07:31.910
Let me erase this one.

00:07:31.910 --> 00:07:33.200
This looks like another
one that I'm going to

00:07:33.200 --> 00:07:37.400
have to copy and paste.

00:07:37.400 --> 00:07:40.520
OK, copied it.

00:07:43.730 --> 00:07:47.270
OK, let me do a dark color.

00:07:47.270 --> 00:07:49.930
The chart below shows an
expression evaluated for four

00:07:49.930 --> 00:07:52.500
different values of x.

00:07:52.500 --> 00:07:55.000
When x is equal to 1, x squared
plus x plus 5, right.

00:07:55.000 --> 00:07:56.550
1 plus 1 plus 5 is 7.

00:07:56.550 --> 00:07:59.810
When x is equal to 2, then 2
squared plus 2 plus 5 is 11.

00:07:59.810 --> 00:08:01.060
Fair enough.

00:08:01.060 --> 00:08:03.740
Josiah or Hosiah, I don't know
how to pronounce that,

00:08:03.740 --> 00:08:07.170
concluded that for all positive
values of x, x

00:08:07.170 --> 00:08:14.220
squared plus x plus 5 produces
a prime number.

00:08:14.220 --> 00:08:21.440
Which value of x serves as
a counterexample to prove

00:08:21.440 --> 00:08:24.180
Josiah's conclusion false?

00:08:24.180 --> 00:08:27.580
So a counterexamples says-- he
says that whenever I put any

00:08:27.580 --> 00:08:30.840
positive number here, I get
a prime number here.

00:08:30.840 --> 00:08:33.650
We have to say which one
proves him false?

00:08:33.650 --> 00:08:41.605
If you put 5 there, what is-- 5
squared is 25 plus 5 plus 5,

00:08:41.605 --> 00:08:43.159
and what is that equal to?

00:08:43.159 --> 00:08:45.930
That's equal to 35.

00:08:45.930 --> 00:08:47.960
So that's a counterexample
right there.

00:08:47.960 --> 00:08:52.930
If x is equal to 5, I produce
35, which is not a prime

00:08:52.930 --> 00:08:55.510
number, not prime.

00:08:58.170 --> 00:09:02.130
So clearly, his statement, his
conclusion, was incorrect,

00:09:02.130 --> 00:09:04.480
that when you put a positive
number here, it doesn't always

00:09:04.480 --> 00:09:05.540
produce a prime number.

00:09:05.540 --> 00:09:06.500
That was his conclusion.

00:09:06.500 --> 00:09:07.740
This one isn't prime.

00:09:07.740 --> 00:09:11.040
So statement A is a
counterexample, or the number

00:09:11.040 --> 00:09:13.740
5 is a counterexample here.

00:09:13.740 --> 00:09:21.490
Next problem, problem 20.

00:09:21.490 --> 00:09:24.870
OK, this is one of those where
we have to pinpoint the

00:09:24.870 --> 00:09:26.120
incorrect step.

00:09:28.960 --> 00:09:32.250
John's solution to an equation
is given below.

00:09:32.250 --> 00:09:34.950
For which step of real numbers
did-- oh, which property of

00:09:34.950 --> 00:09:37.240
real numbers did John
use for step 2?

00:09:37.240 --> 00:09:39.430
OK, so we don't even have
to look at step 1.

00:09:39.430 --> 00:09:40.650
We just have to say,
OK, how did he get

00:09:40.650 --> 00:09:42.290
from there to there?

00:09:44.970 --> 00:09:47.810
So when you look at this, when
you say x plus 2 times x plus

00:09:47.810 --> 00:09:52.940
3 is equal to 0, you're saying
some number x plus 2 times

00:09:52.940 --> 00:09:55.640
some other number x plus
3 is equal to 0.

00:09:55.640 --> 00:09:58.060
So that's like saying some
number times some other number

00:09:58.060 --> 00:10:00.100
is equal to 0.

00:10:00.100 --> 00:10:03.250
So that means that one of those
numbers or both of them

00:10:03.250 --> 00:10:04.440
have to be equal
to zero, right?

00:10:04.440 --> 00:10:06.610
Because the only way to get
zero is if one of these or

00:10:06.610 --> 00:10:08.590
both of these have to
be equal to zero.

00:10:08.590 --> 00:10:11.250
And that's where he gets this
conclusion that either x plus

00:10:11.250 --> 00:10:13.830
2 has to be 0 or
x plus 3 is 0.

00:10:13.830 --> 00:10:15.090
They're both zero.

00:10:15.090 --> 00:10:16.560
So let's see how they--
what do they call it?

00:10:16.560 --> 00:10:19.770
Multiplication property
of equality.

00:10:19.770 --> 00:10:22.020
I don't even know
what that means.

00:10:22.020 --> 00:10:23.930
Zero property of
multiplication?

00:10:23.930 --> 00:10:26.690
This seems like the closest
one for me so far.

00:10:26.690 --> 00:10:30.570
That anything times zero is zero
Or for two things to be

00:10:30.570 --> 00:10:32.580
multiplied to equal zero,
at least one of

00:10:32.580 --> 00:10:33.750
them has to be zero.

00:10:33.750 --> 00:10:35.640
And that's what this
is right here.

00:10:35.640 --> 00:10:37.200
Commutative property
of multiplication?

00:10:37.200 --> 00:10:37.890
No, that's not it.

00:10:37.890 --> 00:10:40.160
Distributive property of
multiplication over addition?

00:10:40.160 --> 00:10:42.840
No, we're not doing anything
like that.

00:10:42.840 --> 00:10:45.010
If we went from this step
to that step where we're

00:10:45.010 --> 00:10:46.730
multiplying it, that might
have been, because you're

00:10:46.730 --> 00:10:48.256
really just doing the
distributive property, but I

00:10:48.256 --> 00:10:49.020
don't want to confuse you.

00:10:49.020 --> 00:10:51.070
Yeah, we're just saying that
if two numbers when you

00:10:51.070 --> 00:10:53.100
multiply them equal zero,
one of them have

00:10:53.100 --> 00:10:53.850
to be equal to zero.

00:10:53.850 --> 00:10:59.690
And I guess the label for that
is the zero product property

00:10:59.690 --> 00:11:00.970
of multiplication.

00:11:00.970 --> 00:11:03.120
Anyway, see you in
the next video.


CA Algebra I: Simplifying Expressions

WEBVTT
Kind: captions
Language: en

00:00:00.670 --> 00:00:02.560
All right, we're on
problem number 8.

00:00:02.560 --> 00:00:07.400
And they ask us which equation
is equivalent to 5x minus 2

00:00:07.400 --> 00:00:10.700
times 7x plus 1 is
equal to 14x?

00:00:10.700 --> 00:00:12.820
So I'm guessing they just want
to simplify this a little bit

00:00:12.820 --> 00:00:14.530
and see if we get to one
of these choices.

00:00:14.530 --> 00:00:15.190
So let's do that.

00:00:15.190 --> 00:00:20.950
So we start with 5x minus
2 times 7x plus

00:00:20.950 --> 00:00:25.910
1 is equal to 14x.

00:00:25.910 --> 00:00:28.280
This is I think the most obvious
thing to simplify.

00:00:28.280 --> 00:00:33.550
This 2 times the 7x plus 1, or
we can even say minus 2 times

00:00:33.550 --> 00:00:38.576
7x plus 1, so then this becomes
5x plus-- I'm just

00:00:38.576 --> 00:00:40.940
going to distribute the minus
2 times all of this.

00:00:40.940 --> 00:00:49.470
Plus minus 2 times
17x is minus 14x.

00:00:49.470 --> 00:00:56.020
And then minus 2 times 1 is
minus 2 is equal to 14x.

00:00:56.020 --> 00:00:58.690
And let's see, on all of these
choices they have 14x on the

00:00:58.690 --> 00:00:59.240
right-hand side.

00:00:59.240 --> 00:01:00.380
So they just want us
to simplify this.

00:01:00.380 --> 00:01:06.940
So this simplifies to 5x, plus
minus 14, so that's minus 14x

00:01:06.940 --> 00:01:10.080
minus 2 is equal to 14x.

00:01:10.080 --> 00:01:11.785
So we have 5x minus 14x.

00:01:11.785 --> 00:01:14.170
So what's 5 minus 14?

00:01:14.170 --> 00:01:14.410
It's what?

00:01:14.410 --> 00:01:16.600
Minus 9, right?

00:01:16.600 --> 00:01:23.270
Minus 9x minus 2 is
equal to 14x.

00:01:23.270 --> 00:01:26.350
That is choice A.

00:01:26.350 --> 00:01:28.730
Now, one thing that maybe
I realized, I kind

00:01:28.730 --> 00:01:31.490
of skipped a step.

00:01:31.490 --> 00:01:33.960
Let me just do this, just so
that you understand this step

00:01:33.960 --> 00:01:34.840
that I did here.

00:01:34.840 --> 00:01:37.290
We could have just said minus
and then distribute the 2.

00:01:37.290 --> 00:01:42.540
So we could have right there
said 5x minus positive 2 times

00:01:42.540 --> 00:01:48.810
7x, 14x, plus 2 times
1, plus 2.

00:01:48.810 --> 00:01:49.950
I'm just doing the
left-hand side.

00:01:49.950 --> 00:01:52.150
And then we could have
distributed the minus sign.

00:01:52.150 --> 00:01:56.970
So you get 5x minus
14x minus 2.

00:01:56.970 --> 00:01:59.610
Either way, it would have gotten
us to this point and

00:01:59.610 --> 00:02:03.090
then we could have simplified
here and gotten choice A.

00:02:03.090 --> 00:02:05.160
Next problem.

00:02:05.160 --> 00:02:09.090
Which equation is equivalent
to-- OK, so another case where

00:02:09.090 --> 00:02:13.772
I'm guessing they just want us
to simplify these equations.

00:02:13.772 --> 00:02:15.800
So let me write that down.

00:02:15.800 --> 00:02:26.810
4 times 2 minus 5x is equal to
6 minus 3 times 1 minus 3x.

00:02:26.810 --> 00:02:29.830
All right, let's distribute
this 4 first. So you get 4

00:02:29.830 --> 00:02:32.090
times 2 is 8.

00:02:32.090 --> 00:02:36.470
4 times minus 5x is minus 20x.

00:02:36.470 --> 00:02:42.190
Minus 20x is equal to 6 minus--
maybe you might have

00:02:42.190 --> 00:02:44.550
been tempted to say, oh, 6 minus
3, that's 3, and then

00:02:44.550 --> 00:02:44.760
distribute it.

00:02:44.760 --> 00:02:46.140
Remember, order of operations.

00:02:46.140 --> 00:02:50.150
Multiplication comes first. So
we have to multiply the 3

00:02:50.150 --> 00:02:53.810
times the 1 minus 3x before
we can deal with the 6.

00:02:53.810 --> 00:02:55.650
Well, on the other side there
was just the multiplication

00:02:55.650 --> 00:02:57.660
first. All right.

00:02:57.660 --> 00:03:03.770
So here, 3 times 1 minus
3x, so it'll be 3.

00:03:03.770 --> 00:03:05.760
I'm just going to do
minus a positive 3

00:03:05.760 --> 00:03:06.320
times each of these.

00:03:06.320 --> 00:03:11.020
So 3 times a negative
3 is minus 9x.

00:03:11.020 --> 00:03:13.090
And now I can distribute
this minus sign.

00:03:13.090 --> 00:03:15.600
So let's see, on the left-hand
side I have-- let me do it in

00:03:15.600 --> 00:03:16.940
a brighter color.

00:03:16.940 --> 00:03:22.930
The left-hand side I have 8
minus 20x is equal to 6, and

00:03:22.930 --> 00:03:24.450
then I'm going to subtract
this whole thing.

00:03:24.450 --> 00:03:27.090
So you can view it as
distributing a negative 1

00:03:27.090 --> 00:03:28.000
times both of these.

00:03:28.000 --> 00:03:34.070
So 6 plus minus 3, or just 6
minus 3, and then a minus

00:03:34.070 --> 00:03:35.770
times a minus plus 9x.

00:03:39.120 --> 00:03:39.610
All right.

00:03:39.610 --> 00:03:42.620
Now let's see what we could
do to help simplify this.

00:03:42.620 --> 00:03:45.445
What happens if we add
20x to both sides?

00:03:45.445 --> 00:03:48.190
Then on the left-hand side, it
goes away, so we're just left

00:03:48.190 --> 00:03:53.130
with 8 is equal to-- well,
let me just do that.

00:03:53.130 --> 00:03:55.020
6 minus 3.

00:03:55.020 --> 00:03:58.720
I added 20x, so plus 29x.

00:03:58.720 --> 00:04:00.940
I really should have simplified
this 6 minus 3, but

00:04:00.940 --> 00:04:02.050
I didn't want to do too
much in one step.

00:04:02.050 --> 00:04:08.910
So 8 is equal to 3 plus 29x, and
then if we subtract 3 from

00:04:08.910 --> 00:04:13.120
both sides here, we get
8 minus 3 is 5.

00:04:13.120 --> 00:04:17.500
That 3 goes away and you get
29x, and that is choice C.

00:04:17.500 --> 00:04:21.890
29x is equal to 5.

00:04:21.890 --> 00:04:26.230
Next question, problem
number 10.

00:04:26.230 --> 00:04:28.340
And I should be careful that
you don't think that all of

00:04:28.340 --> 00:04:30.320
this stuff is for
this problem.

00:04:30.320 --> 00:04:33.810
The total cost, c, in dollars
of renting a sailboat for n

00:04:33.810 --> 00:04:37.930
days is given by
this equation.

00:04:37.930 --> 00:04:42.050
If the total cost was $360,
for how many days was the

00:04:42.050 --> 00:04:42.830
sailboat rented?

00:04:42.830 --> 00:04:45.920
So they're giving
us c. c is $360.

00:04:45.920 --> 00:04:50.080
So they're telling us that $360,
that's the total cost,

00:04:50.080 --> 00:04:54.720
and that's going to be equal to
120 plus 60 times with the

00:04:54.720 --> 00:04:57.230
total days rented, times 60n.

00:04:57.230 --> 00:04:59.090
And that's what they want
us to figure out, n.

00:04:59.090 --> 00:05:02.630
How many days was the
sailboat rented?

00:05:02.630 --> 00:05:05.460
Let's see, so if you should
subtract 120 from both sides

00:05:05.460 --> 00:05:12.140
of this equation, you get
240 is equal to 60n.

00:05:12.140 --> 00:05:13.290
I'll go here.

00:05:13.290 --> 00:05:16.510
And if you divide both sides of
this by 60, 240 divided by

00:05:16.510 --> 00:05:21.300
60, that's the same thing as 24
divided by 6, so that's 4.

00:05:21.300 --> 00:05:25.430
And then 60n divided by 60 is,
of course, 1, so that is n.

00:05:25.430 --> 00:05:29.930
So the sailboat was
rented for 4 days.

00:05:29.930 --> 00:05:35.650
All right, problem number 11.

00:05:35.650 --> 00:05:39.250
And I cut and pasted part of
it, but they want us to

00:05:39.250 --> 00:05:43.020
determine, what was the first
incorrect step that they did

00:05:43.020 --> 00:05:43.740
in this problem?

00:05:43.740 --> 00:05:45.060
So let's see if we can
determine that.

00:05:45.060 --> 00:05:45.970
So let's see.

00:05:45.970 --> 00:05:49.420
Solve 3 times x plus 5 is
equal to 2x plus 35.

00:05:49.420 --> 00:05:50.300
Fair enough.

00:05:50.300 --> 00:05:52.960
So in this first step they
distributed 3 times

00:05:52.960 --> 00:05:54.780
x and 3 times 5.

00:05:54.780 --> 00:05:59.190
Right, 3x plus 15 is equal to
2x plus 35, so step 1 looks

00:05:59.190 --> 00:06:00.600
good to me.

00:06:00.600 --> 00:06:02.340
Let me do that in
a darker color.

00:06:02.340 --> 00:06:03.490
Step 1 looks good.

00:06:03.490 --> 00:06:05.400
They distributed the 3.

00:06:05.400 --> 00:06:08.370
Now let's see what they
did in step 2.

00:06:08.370 --> 00:06:11.490
Let's see, if I wanted to solve
this, I would subtract

00:06:11.490 --> 00:06:13.780
2x from both sides, and it looks
like that's what they

00:06:13.780 --> 00:06:17.830
attempted to do because they
got rid of this 2x, right?

00:06:17.830 --> 00:06:21.020
On the right-hand side, they
went from 2x plus 35 to just

00:06:21.020 --> 00:06:24.550
35, so somehow they got rid of
this 2x, and the only way to

00:06:24.550 --> 00:06:27.220
get rid of it is to subtract
2x from both sides.

00:06:31.170 --> 00:06:33.620
Now, if I subtract 2x from both
sides, sure enough, the

00:06:33.620 --> 00:06:36.860
right-hand side will just
be left with the 35.

00:06:36.860 --> 00:06:40.220
The left-hand side will have
the 15, of course.

00:06:40.220 --> 00:06:44.140
And then I have minus
2x plus 3x.

00:06:44.140 --> 00:06:46.440
Well, minus 2x plus
3x, that's x.

00:06:46.440 --> 00:06:48.350
So this 5x, that shouldn't
be there.

00:06:48.350 --> 00:06:51.860
It should be x plus
15 is equal to 35.

00:06:51.860 --> 00:06:54.980
So even though they were trying
to get rid of this 2x

00:06:54.980 --> 00:06:57.170
and they should have subtracted
2x from both sides,

00:06:57.170 --> 00:07:00.180
they inadvertently added 2x
here, so they said 2x plus 3x

00:07:00.180 --> 00:07:01.280
is equal to 5x.

00:07:01.280 --> 00:07:01.760
Well, that was wrong.

00:07:01.760 --> 00:07:02.375
You have to subtract.

00:07:02.375 --> 00:07:05.610
And you're just left with x
plus 15 is equal to 35.

00:07:05.610 --> 00:07:09.730
So step 2 is when they made
their first error.

00:07:09.730 --> 00:07:16.870
Problem 120.

00:07:16.870 --> 00:07:21.220
A 120-foot-long rope is
cut into 3 pieces.

00:07:21.220 --> 00:07:23.160
The first piece of the
rope-- let me draw.

00:07:23.160 --> 00:07:25.145
This is begging for a diagram.

00:07:27.950 --> 00:07:31.510
All right, so they say the first
piece of rope is twice

00:07:31.510 --> 00:07:35.360
as long as the second
piece of rope.

00:07:35.360 --> 00:07:37.330
And then they say the third
piece of rope is three times

00:07:37.330 --> 00:07:39.600
as long as the second
piece of rope.

00:07:39.600 --> 00:07:42.000
Everything is a multiple of
the second piece of rope.

00:07:44.770 --> 00:07:47.100
So the second one was the
shortest one, just reading it.

00:07:47.100 --> 00:07:49.470
The first is twice as
long as the second.

00:07:49.470 --> 00:07:52.420
The third is three
times as long.

00:07:52.420 --> 00:07:55.750
So if I just drew it.

00:07:55.750 --> 00:07:58.680
So if this is the second
length of rope,

00:07:58.680 --> 00:08:00.950
let's call that x.

00:08:00.950 --> 00:08:02.810
That's the second
length of rope.

00:08:02.810 --> 00:08:09.980
And they told us the first piece
of rope is twice as long

00:08:09.980 --> 00:08:12.050
as the second piece of rope.

00:08:12.050 --> 00:08:14.230
OK, so if this is the first
piece of rope right here, it's

00:08:14.230 --> 00:08:17.000
going to be twice as long as
the second piece of rope.

00:08:17.000 --> 00:08:18.910
So that's 2x.

00:08:18.910 --> 00:08:19.490
Fair enough.

00:08:19.490 --> 00:08:22.530
And then they tell us-- I'll
do this in magenta.

00:08:22.530 --> 00:08:26.700
The third piece of rope is three
times as long as the

00:08:26.700 --> 00:08:28.500
second piece of rope.

00:08:28.500 --> 00:08:30.760
So this is the third
piece of rope here.

00:08:30.760 --> 00:08:33.549
It's three times as long as
the second piece of rope.

00:08:33.549 --> 00:08:35.280
So it's 3x.

00:08:35.280 --> 00:08:38.179
And they say, what is
the length of the

00:08:38.179 --> 00:08:40.539
longest piece of rope?

00:08:40.539 --> 00:08:42.440
All right, so let's
think about it.

00:08:42.440 --> 00:08:45.650
If I add up all of these
segments, what

00:08:45.650 --> 00:08:46.550
do they add up to?

00:08:46.550 --> 00:08:49.510
Well, they tell us that it's
120-foot-long rope.

00:08:49.510 --> 00:08:51.120
So let me add that up.

00:08:51.120 --> 00:08:57.140
So if I do 2x plus x plus
3x, that's going

00:08:57.140 --> 00:08:59.470
to equal 120 feet.

00:08:59.470 --> 00:09:01.570
That is equal to 120 feet.

00:09:01.570 --> 00:09:04.790
That whole distance is 120.

00:09:04.790 --> 00:09:11.660
So 2x plus x is 3x plus 3x, you
get 6x is equal to 120.

00:09:11.660 --> 00:09:15.810
You divide both sides by 6, You
get x is-- let's see, 120

00:09:15.810 --> 00:09:18.040
divided by 6 is 20.

00:09:18.040 --> 00:09:18.990
All right, that's what x is.

00:09:18.990 --> 00:09:22.570
That's the second piece of rope
or the shortest piece of

00:09:22.570 --> 00:09:24.380
rope because they
tell us that.

00:09:24.380 --> 00:09:26.510
The other ones are some multiple
of that length.

00:09:26.510 --> 00:09:28.590
But they want to know, what is
the length of the longest

00:09:28.590 --> 00:09:29.840
piece of rope?

00:09:32.040 --> 00:09:35.140
Well, the longest piece of
rope is clearly piece 3.

00:09:35.140 --> 00:09:37.030
It's three times the
second piece.

00:09:37.030 --> 00:09:39.000
Piece 1 was only two times.

00:09:39.000 --> 00:09:40.080
So this is what they
want to know.

00:09:40.080 --> 00:09:42.520
They want to know what
3x is equal to.

00:09:42.520 --> 00:09:48.740
So 3x is equal to three times
20, which is equal to 60 feet.

00:09:48.740 --> 00:09:52.150
And that's choice C.

00:09:52.150 --> 00:09:53.815
Next problem.

00:09:53.815 --> 00:09:55.320
Let's see if I have time.

00:09:55.320 --> 00:09:57.616
Yeah, I think I have time.

00:09:57.616 --> 00:10:00.170
I might as well attempt
to fit it in.

00:10:04.130 --> 00:10:05.380
All right.

00:10:08.700 --> 00:10:11.350
Don't want to rush
them, though.

00:10:11.350 --> 00:10:13.230
Let's see.

00:10:13.230 --> 00:10:16.850
The cost to rent a construction
crane is $750 per

00:10:16.850 --> 00:10:21.440
day plus $250 per hour of use.

00:10:21.440 --> 00:10:24.490
What is the maximum number of
hours the crane can be used

00:10:24.490 --> 00:10:33.140
each day if the rental cost is
not to exceed $2,500 per day?

00:10:33.140 --> 00:10:36.610
OK, so how much is
the cost per day?

00:10:36.610 --> 00:10:41.030
So the cost per day
is equal to $750.

00:10:41.030 --> 00:10:43.480
So no matter how little or much
we use it, we have to pay

00:10:43.480 --> 00:10:45.730
that $750 per day.

00:10:45.730 --> 00:10:55.120
So it's equal to $750 plus $250
per hour, so plus $250

00:10:55.120 --> 00:10:56.670
times the number of hours.

00:10:56.670 --> 00:10:59.620
This is what our total
cost is per day.

00:10:59.620 --> 00:11:03.870
And they want to say, what is
the maximum number of hours

00:11:03.870 --> 00:11:07.260
the crane can be used each day
if the rental cost is not to

00:11:07.260 --> 00:11:09.680
exceed $2,500 per day?

00:11:09.680 --> 00:11:13.900
So this whole thing has to be
less than or equal to $2,500.

00:11:13.900 --> 00:11:18.080
The total cost has to be less
than or equal to $2,500.

00:11:18.080 --> 00:11:21.530
So now let's just solve this
equation right there.

00:11:21.530 --> 00:11:22.330
And I'll do it up here.

00:11:22.330 --> 00:11:28.480
So we know that $750 plus
$250-- let me do

00:11:28.480 --> 00:11:30.170
it down here actually.

00:11:30.170 --> 00:11:31.480
More space.

00:11:31.480 --> 00:11:32.880
So what's the first thing
we could do here?

00:11:32.880 --> 00:11:36.520
We could subtract 750 from both
sides of this equation.

00:11:36.520 --> 00:11:45.130
So we get 250 times h is less
than or equal to-- 2,500 minus

00:11:45.130 --> 00:11:46.490
750, that's what?

00:11:46.490 --> 00:11:48.510
1,750?

00:11:48.510 --> 00:11:52.210
Yeah, because 1,500
minus 750 is 750.

00:11:52.210 --> 00:11:53.420
1,750.

00:11:53.420 --> 00:11:57.210
I just subtracted 750 from both
sides of this equation.

00:11:57.210 --> 00:11:59.630
Let me write that down.

00:11:59.630 --> 00:12:02.910
So I said minus 750 plus that.

00:12:02.910 --> 00:12:05.280
And then I said minus 750.

00:12:05.280 --> 00:12:09.400
And, of course, minus 750 plus
750, that goes to 0.

00:12:09.400 --> 00:12:11.420
That's why you don't see
anything over here.

00:12:11.420 --> 00:12:15.870
And on the right-hand side,
2,500 minus 750 is 1,750.

00:12:15.870 --> 00:12:19.050
All right, now let's just divide
both sides by 250.

00:12:19.050 --> 00:12:21.180
We don't have to do anything to
the inequality sign since

00:12:21.180 --> 00:12:23.320
250 is a positive number.

00:12:23.320 --> 00:12:26.610
So the hours, the maximum number
of hours, the hours

00:12:26.610 --> 00:12:33.730
have to be less than or equal to
1,750/250, which is-- let's

00:12:33.730 --> 00:12:38.570
see, 1,750 divided by 250.

00:12:38.570 --> 00:12:41.860
That's the same thing as
175 divided by 25.

00:12:41.860 --> 00:12:45.235
25 goes into 175,
what is that?

00:12:45.235 --> 00:12:48.760
I want to say seven times?

00:12:48.760 --> 00:12:51.030
7 times 5 is 35.

00:12:51.030 --> 00:12:53.510
7 times 2 is 14 plus 3 is 17.

00:12:53.510 --> 00:12:54.330
Right.

00:12:54.330 --> 00:12:59.070
So h has to be less than
or equal to 7 hours.

00:12:59.070 --> 00:13:01.500
And that is choice C.

00:13:01.500 --> 00:13:03.410
See you in the next video.


CA Algebra I: Number properties and absolute value

WEBVTT
Kind: captions
Language: en

00:00:00.990 --> 00:00:04.280
We're now going to take the
California Standards Test,

00:00:04.280 --> 00:00:05.960
Algebra I released questions.

00:00:05.960 --> 00:00:08.480
In the last series, I had
done the Algebra II.

00:00:08.480 --> 00:00:09.920
I guess I'm going in
reverse order.

00:00:09.920 --> 00:00:13.110
Let me copy and paste this first
question because I think

00:00:13.110 --> 00:00:14.905
it's good to see the
whole thing.

00:00:14.905 --> 00:00:20.860
So let me see, I
have copied it.

00:00:20.860 --> 00:00:25.800
Let me move this pointer
all the way up, and

00:00:25.800 --> 00:00:27.370
then, there we go.

00:00:27.370 --> 00:00:28.880
All right.

00:00:28.880 --> 00:00:34.060
And they're asking us is the
equation 3 times 2x minus 4

00:00:34.060 --> 00:00:38.960
equal to minus 18 equivalent to
6x minus 12 is equal to 18?

00:00:38.960 --> 00:00:39.770
So let's think about this.

00:00:39.770 --> 00:00:41.490
If we just distribute this
3, what do we get?

00:00:41.490 --> 00:00:44.820
3 times 2x is 6x.

00:00:44.820 --> 00:00:47.770
3 times minus 4 is minus 12.

00:00:47.770 --> 00:00:50.480
And that, of course, is
equal to minus 18.

00:00:50.480 --> 00:00:51.890
So sure, they're
the same thing.

00:00:51.890 --> 00:00:55.290
If you just distribute the 3
over the 2x minus 4, you get

00:00:55.290 --> 00:00:56.950
6x minus 12.

00:00:56.950 --> 00:00:58.270
So the answer is
definitely yes.

00:00:58.270 --> 00:01:00.140
It's not the no one down here.

00:01:00.140 --> 00:01:03.540
And it says yes, the equations
are equivalent by the

00:01:03.540 --> 00:01:04.080
associative?

00:01:04.080 --> 00:01:04.340
No.

00:01:04.340 --> 00:01:04.959
Communicative?

00:01:04.959 --> 00:01:05.249
No.

00:01:05.249 --> 00:01:09.033
The equations are equivalent by
the distributive property?

00:01:09.033 --> 00:01:09.800
[FIRE TRUCK SIREN]

00:01:09.800 --> 00:01:13.480
There's some type of a fire
truck going on outside.

00:01:13.480 --> 00:01:14.260
Let's see.

00:01:14.260 --> 00:01:15.540
Where was I?

00:01:15.540 --> 00:01:16.770
Oh, yes.

00:01:16.770 --> 00:01:19.050
Yes, the equations are
equivalent by the distributive

00:01:19.050 --> 00:01:20.760
property of multiplication
over addition.

00:01:20.760 --> 00:01:21.470
Right, that's that.

00:01:21.470 --> 00:01:24.400
We distributed this 3
over the 2x minus 4.

00:01:24.400 --> 00:01:26.950
And they say over addition
because you could view this as

00:01:26.950 --> 00:01:28.460
a plus minus 4.

00:01:28.460 --> 00:01:30.540
Addition and subtraction is
really the same thing when you

00:01:30.540 --> 00:01:32.180
think of the distributive
property.

00:01:32.180 --> 00:01:35.740
Anyway, let's do the
next problem.

00:01:35.740 --> 00:01:39.650
The next problem I can
just write out.

00:01:39.650 --> 00:01:41.770
This is problem number 2.

00:01:41.770 --> 00:01:46.550
They say the square root of
16 plus the cube root

00:01:46.550 --> 00:01:48.120
of 8 is equal to?

00:01:48.120 --> 00:01:49.975
Well, what's the square
root of 16?

00:01:49.975 --> 00:01:51.680
And when you just have a square
root there, you might

00:01:51.680 --> 00:01:53.830
say, maybe it's plus or minus 4,
but when they write it this

00:01:53.830 --> 00:01:57.680
way is means the principal root,
so it's just plus 4.

00:01:57.680 --> 00:01:59.590
They would write a plus or minus
out front if they wanted

00:01:59.590 --> 00:02:01.520
you to get the negative
square root.

00:02:01.520 --> 00:02:06.430
So it's 4 plus-- now, what to
the third power is equal to 8?

00:02:06.430 --> 00:02:09.539
Well, 2 to the third power
is equal to 8, right?

00:02:09.539 --> 00:02:12.950
So we could write 2 to the
third is equal to 8.

00:02:12.950 --> 00:02:17.510
That's the same thing as saying
the cube root of 8 is

00:02:17.510 --> 00:02:19.170
equal to 2.

00:02:19.170 --> 00:02:22.570
You could also view this
as 8 to the 1/3 power.

00:02:22.570 --> 00:02:26.490
Anyway, the cube root of 8 is
then 2, so 4 plus 2 is equal

00:02:26.490 --> 00:02:30.140
to 6, and that is choice B.

00:02:30.140 --> 00:02:31.390
Problem 3.

00:02:35.900 --> 00:02:38.590
Let me scroll down
a little bit.

00:02:38.590 --> 00:02:41.070
OK, and they want to know--
I could copy and

00:02:41.070 --> 00:02:42.510
paste the whole thing.

00:02:48.200 --> 00:02:49.910
There we go.

00:02:49.910 --> 00:02:52.010
And they want to know which
expression is equivalent to x

00:02:52.010 --> 00:02:54.790
to the sixth times x squared?

00:02:54.790 --> 00:02:58.400
So x to the sixth times
x squared, they

00:02:58.400 --> 00:02:59.390
have the same base.

00:02:59.390 --> 00:03:01.470
When you're multiplying both of
these expressions, we can

00:03:01.470 --> 00:03:02.390
add the exponents.

00:03:02.390 --> 00:03:06.750
So that is equal to x to
the-- 6 plus 2 is 8.

00:03:06.750 --> 00:03:08.640
That's not one of the choices
here, so we have to say, which

00:03:08.640 --> 00:03:11.740
of these also are the same
thing as x to the eighth?

00:03:11.740 --> 00:03:14.570
And so which two exponents when
I add them is equal to 8?

00:03:14.570 --> 00:03:15.895
4 plus 3 is equal to 7.

00:03:15.895 --> 00:03:20.330
5 plus 3, this is equal to
x to the eighth as well.

00:03:20.330 --> 00:03:23.320
So that is choice B.

00:03:23.320 --> 00:03:28.130
Next problem, problem 4.

00:03:28.130 --> 00:03:30.740
All right, let me-- this is
another one where I'll copy

00:03:30.740 --> 00:03:31.990
and paste it.

00:03:36.520 --> 00:03:38.470
All right.

00:03:38.470 --> 00:03:41.940
They want to know which number
does not have a reciprocal?

00:03:41.940 --> 00:03:44.900
So the reciprocal of negative 1,
that's just 1 over negative

00:03:44.900 --> 00:03:46.930
1, which is equal
to negative 1.

00:03:46.930 --> 00:03:49.220
The reciprocal of
0, that's what?

00:03:49.220 --> 00:03:52.700
1/0, which is not defined.

00:03:52.700 --> 00:03:54.660
So the choice is B.

00:03:54.660 --> 00:03:55.100
0.

00:03:55.100 --> 00:03:56.500
We don't know what 1/0 is.

00:03:56.500 --> 00:03:58.040
Maybe that's a project
for you to think of

00:03:58.040 --> 00:03:59.290
what it should mean.

00:03:59.290 --> 00:04:01.000
And, of course, these
have reciprocals.

00:04:01.000 --> 00:04:07.180
1 over 1/1,000 is just equal to
1 times 1,000 over 1, which

00:04:07.180 --> 00:04:09.590
is equal to 1,000.

00:04:09.590 --> 00:04:13.080
And the reciprocal of 3
is, of course, 1/3.

00:04:13.080 --> 00:04:14.330
Next problem.

00:04:18.480 --> 00:04:21.140
They say-- so there's a lot of
terminology here, but I guess

00:04:21.140 --> 00:04:23.040
that's good.

00:04:23.040 --> 00:04:24.870
So they want to know--
let me just copy.

00:04:24.870 --> 00:04:26.210
Maybe I'll do the
next one, too.

00:04:28.870 --> 00:04:31.260
OK.

00:04:31.260 --> 00:04:33.080
I could probably just
do it up here.

00:04:37.810 --> 00:04:38.410
All right.

00:04:38.410 --> 00:04:41.510
They want to know, what is the
multiplicative inverse of 1/2?

00:04:41.510 --> 00:04:44.840
So essentially, what can
I multiply 1/2 by

00:04:44.840 --> 00:04:46.325
and then get 1?

00:04:49.480 --> 00:04:51.630
It's the same thing as saying
what's the inverse of 1/2.

00:04:51.630 --> 00:04:54.630
So if I multiply by 1/2 by--
well, the inverse of 1/2, I'd

00:04:54.630 --> 00:04:56.060
say 1 over 1/2.

00:04:56.060 --> 00:04:58.760
That's the same thing
as 1 times 2/1,

00:04:58.760 --> 00:05:00.230
which is equal to 2.

00:05:00.230 --> 00:05:05.140
Or another way to think of it is
2 times 1/2 is equal to 1.

00:05:05.140 --> 00:05:10.080
So the multiplicative inverse
of 1/2 is just 2.

00:05:10.080 --> 00:05:13.460
That's choice D.

00:05:13.460 --> 00:05:14.420
Problem 6.

00:05:14.420 --> 00:05:17.160
What is the solution
for this equation?

00:05:17.160 --> 00:05:21.070
All right, sometimes these
absolute value signs can

00:05:21.070 --> 00:05:22.360
appear daunting, but
you just have to

00:05:22.360 --> 00:05:24.650
think it through logically.

00:05:24.650 --> 00:05:27.880
If the absolute value of 2x
minus 3 is equal to 5, that

00:05:27.880 --> 00:05:28.730
tells us what?

00:05:28.730 --> 00:05:34.210
That means that 2x minus
3 is equal to 5, right?

00:05:34.210 --> 00:05:37.370
Because inside the absolute
value is equal to 5, then the

00:05:37.370 --> 00:05:38.920
absolute value of
5 is equal to 5.

00:05:38.920 --> 00:05:40.110
So that's fair enough.

00:05:40.110 --> 00:05:43.190
But what could 2x minus
3 also be equal to?

00:05:43.190 --> 00:05:47.680
What happens if 2x minus 3
within the absolute value sign

00:05:47.680 --> 00:05:49.600
is equal to minus 5?

00:05:49.600 --> 00:05:51.470
Well, then you would take the
absolute value of that and you

00:05:51.470 --> 00:05:52.490
would get 5, right?

00:05:52.490 --> 00:05:58.800
So 2x minus 3 could also
be equal to minus 5.

00:05:58.800 --> 00:06:00.480
When you see this absolute
value sign, you say, OK,

00:06:00.480 --> 00:06:04.150
whatever's inside the absolute
value is either 5 or negative

00:06:04.150 --> 00:06:06.960
5 because we're taking the
absolute value of it to get 5.

00:06:06.960 --> 00:06:09.680
So we just solve both
of these equations.

00:06:09.680 --> 00:06:11.390
If you add 3 to both sides
of this one, you get

00:06:11.390 --> 00:06:14.200
2x is equal to 8.

00:06:14.200 --> 00:06:17.570
x is equal to 4.

00:06:17.570 --> 00:06:20.030
On the second one, you
add 3 to both sides.

00:06:20.030 --> 00:06:25.306
You get 2x is equal to-- minus
5 plus 3 is minus 2.

00:06:25.306 --> 00:06:30.645
x is equal to minus 2 divided
by 2 is negative 1.

00:06:30.645 --> 00:06:35.450
So x could be equal to 4 or x
could be equal to negative 1.

00:06:35.450 --> 00:06:42.160
And that is choice C, x is
negative 1 or x is equal to 4.

00:06:42.160 --> 00:06:45.060
Next problem.

00:06:45.060 --> 00:06:49.240
The Algebra I ones go faster
than the Algebra II problems.

00:06:49.240 --> 00:06:51.110
Those tend to be hairier.

00:06:51.110 --> 00:06:52.360
Let me clear all of this.

00:06:56.790 --> 00:06:57.880
I'll just write this one down.

00:06:57.880 --> 00:07:04.100
They say what is the solution
set for the inequality 5 minus

00:07:04.100 --> 00:07:08.530
the absolute value of x
plus 4 is less than or

00:07:08.530 --> 00:07:11.660
equal to minus 3?

00:07:11.660 --> 00:07:13.730
So at first, this is
really daunting.

00:07:13.730 --> 00:07:16.210
I can't even do that logic that
I did last time because I

00:07:16.210 --> 00:07:17.040
have that 5 out there.

00:07:17.040 --> 00:07:17.520
But let's think of
it this way.

00:07:17.520 --> 00:07:20.240
Let's try to simplify it, so
we have just the absolute

00:07:20.240 --> 00:07:22.410
value of something is
less than or equal

00:07:22.410 --> 00:07:23.200
to something else.

00:07:23.200 --> 00:07:25.540
So one thing we can do is, if we
want to get rid of this 5,

00:07:25.540 --> 00:07:27.770
remember, what we do to both
sides of an equation or

00:07:27.770 --> 00:07:31.340
inequality-- whatever we do to
one side of an equation or an

00:07:31.340 --> 00:07:33.540
inequality, we do
to both sides.

00:07:33.540 --> 00:07:37.690
So let's subtract 5 from both
sides of this equation.

00:07:37.690 --> 00:07:41.940
If you subtract 5 from the left
side, this 5 disappears.

00:07:41.940 --> 00:07:45.400
I'm just going to do minus--
let me write that out.

00:07:45.400 --> 00:07:48.750
Minus 5 plus, and I'm going
to do a minus 5 there.

00:07:52.690 --> 00:07:53.240
That's a plus.

00:07:53.240 --> 00:07:57.020
So minus 5 plus 5 is 0, so I'm
just left with minus absolute

00:07:57.020 --> 00:08:02.690
value of x plus 4 is less than
or equal to-- now, what's

00:08:02.690 --> 00:08:05.140
minus 3 minus 5?

00:08:05.140 --> 00:08:07.850
That's minus 8.

00:08:07.850 --> 00:08:10.290
All right, now this next step,
this is something-- maybe it

00:08:10.290 --> 00:08:13.510
wasn't obvious for you and
putting the inequality there--

00:08:13.510 --> 00:08:16.050
you know, if this was an
inequality, you would just

00:08:16.050 --> 00:08:17.970
say, OK, I'm going to multiply
or divide both sides by

00:08:17.970 --> 00:08:20.340
negative 1 to get rid of
the negative signs.

00:08:20.340 --> 00:08:23.260
But one thing you have to
remember, whenever you

00:08:23.260 --> 00:08:27.800
multiply or divide both sides of
an inequality by a negative

00:08:27.800 --> 00:08:30.910
number, you have to switch
the inequality.

00:08:30.910 --> 00:08:34.850
So if this is true, then if I'm
multiplying both sides of

00:08:34.850 --> 00:08:38.950
this by negative 1, so negative
1 times negative x

00:08:38.950 --> 00:08:43.840
plus 4, I'm going to switch
the inequality, so that's

00:08:43.840 --> 00:08:47.210
going to become greater than
or equal to negative 8.

00:08:47.210 --> 00:08:49.100
And I did the negative 1 on
this side, so I have to

00:08:49.100 --> 00:08:51.480
multiply it times negative
1 on that side.

00:08:51.480 --> 00:08:53.800
And so this negative cancels
out that negative, so we're

00:08:53.800 --> 00:08:59.140
just left with x plus 4 is
greater than or equal to--

00:08:59.140 --> 00:09:02.150
negative 8 times a negative
1 is equal to 8.

00:09:02.150 --> 00:09:04.330
Now we can just use the
logic that we had

00:09:04.330 --> 00:09:06.520
from that last problem.

00:09:06.520 --> 00:09:08.530
This tells us what?

00:09:08.530 --> 00:09:12.510
This tells us that the magnitude
of x plus 4 is

00:09:12.510 --> 00:09:16.320
greater than or equal to 8.

00:09:16.320 --> 00:09:18.110
Let me draw a number line here
because I really want you to

00:09:18.110 --> 00:09:21.350
get the intuition of what
magnitude means.

00:09:21.350 --> 00:09:27.680
So if that's the number line and
you can view magnitude as

00:09:27.680 --> 00:09:30.150
kind of the distance from, or
the absolute value, you can

00:09:30.150 --> 00:09:33.930
kind of view it as the distance
from 0, right?

00:09:33.930 --> 00:09:40.500
So if this is 0 right here and
this is positive 8 and this is

00:09:40.500 --> 00:09:44.540
minus 8, the absolute value of
whatever this quantity was is

00:09:44.540 --> 00:09:45.780
greater than 8.

00:09:45.780 --> 00:09:49.820
That means its distance from
0 has to be greater than 8.

00:09:49.820 --> 00:09:53.490
You can just say distance from
0 of this number has to be

00:09:53.490 --> 00:09:57.570
greater than 8, greater
than or equal to 8.

00:09:57.570 --> 00:09:59.970
That means that this number is
definitely going to be greater

00:09:59.970 --> 00:10:02.070
than or equal to positive 8.

00:10:02.070 --> 00:10:03.880
On the number line,
it would be all of

00:10:03.880 --> 00:10:06.340
those numbers, right?

00:10:06.340 --> 00:10:08.620
Or, remember, we're saying the
magnitude, so we don't care

00:10:08.620 --> 00:10:09.850
about the direction.

00:10:09.850 --> 00:10:14.360
The magnitude has to be greater
than positive 8, so it

00:10:14.360 --> 00:10:18.150
also includes the negative
numbers less than negative 8.

00:10:18.150 --> 00:10:19.100
And why does that make sense?

00:10:19.100 --> 00:10:20.160
Well, take negative 9.

00:10:20.160 --> 00:10:22.580
What's the absolute value
of negative 9?

00:10:22.580 --> 00:10:29.440
The absolute value of negative
9 is greater than 8 because 9

00:10:29.440 --> 00:10:32.340
is greater than 8, so any number
to the left of negative

00:10:32.340 --> 00:10:34.380
8 or to the right
of positive 8.

00:10:34.380 --> 00:10:36.540
So what does that tell us
about this equation?

00:10:36.540 --> 00:10:39.890
So that means that-- well, the
easy one is x plus 4 could be

00:10:39.890 --> 00:10:41.235
greater than or equal to 8.

00:10:41.235 --> 00:10:44.140
So let's write that down.

00:10:44.140 --> 00:10:45.710
Let me write it here.

00:10:45.710 --> 00:10:48.775
x plus 4 greater than
or equal to 8.

00:10:48.775 --> 00:10:51.010
And that takes into
consideration that the

00:10:51.010 --> 00:10:53.670
magnitude is greater than
or equal to 8 there.

00:10:53.670 --> 00:11:00.160
Or x plus 4 less than
or equal to minus 8.

00:11:00.160 --> 00:11:02.060
That's the magnitude
to the left of this

00:11:02.060 --> 00:11:04.610
negative 8 right there.

00:11:04.610 --> 00:11:05.570
And now we solve it.

00:11:05.570 --> 00:11:08.050
And it's very important to think
of absolute value in

00:11:08.050 --> 00:11:10.450
these terms. Otherwise, it can
become very confusing and you

00:11:10.450 --> 00:11:11.250
start testing numbers.

00:11:11.250 --> 00:11:13.630
But if you really just visualize
the number line and

00:11:13.630 --> 00:11:18.000
you think of absolute value as
distance from 0, magnitude of

00:11:18.000 --> 00:11:20.150
the distance from 0, you say,
oh, the distance from 0 has to

00:11:20.150 --> 00:11:23.330
be greater than or equal to 8,
so that means my number has to

00:11:23.330 --> 00:11:27.310
be-- this thing has to be less
than or equal to minus 8 or it

00:11:27.310 --> 00:11:31.150
has to be greater than or
equal to positive 8.

00:11:31.150 --> 00:11:32.420
So let's solve.

00:11:32.420 --> 00:11:35.000
x plus 4 is greater than
or equal to 8.

00:11:35.000 --> 00:11:38.250
Subtract 4 from both sides, so
you get x is greater than or

00:11:38.250 --> 00:11:39.480
equal to 4.

00:11:39.480 --> 00:11:41.440
I just subtracted 4
from both sides.

00:11:41.440 --> 00:11:44.920
Subtract 4 from both sides here,
you get x is less than

00:11:44.920 --> 00:11:47.570
or equal to minus 12.

00:11:47.570 --> 00:11:51.700
So the solution here is x is
greater than or equal to 4 or

00:11:51.700 --> 00:11:55.510
x is less than or equal
to minus 12, and

00:11:55.510 --> 00:11:57.860
that is choice D.

00:11:57.860 --> 00:11:59.420
Anyway, I'll see you
in the next video.