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# CAHSEE practice: Problems 47-51

Video transcript

Problem 47. Which graph represents
the system of equations shown below? So each of these equations
represents a line. So this is going to be a line,
well, both of them have y-intercepts of plus 3. And this line has a
slope of minus 1. So let's see. We need a downward sloping
thing that intersects at positive 3. Actually, they're both going
to intersect at positive 3. So choice A, neither of them. Both of their y-intercepts
are minus 3. We need to see a plus 3 here. So it's not choice A. In choice C, one of the lines
intersects at positive 3, but the other line intersects
at minus 3. And they also have the same
slope, and these guys obviously don't have
the same slope. So it's not going
to be choice C. Choice B. So I have two lines. Both of them have a
y-intercept of 3. This looks good so far. Both of them have y-intercepts
of 3. And then I have one line that
has a positive slope of 1. For every 1 I go over,
I go up 1. And I have one that has
a negative slope of 1. When I move over
1, I go down 1. This line right here is y is
equal to x, because I have a slope of 1. 1x plus 3. 3 is its y-intercept. And this line right here
is y is equal to-- its slope is minus 1. You move to the right
1, you go down 1. So minus 1x, or minus x, and
its y-intercept is plus 3. So our answer is B. These are the two equations
that we have up there. You can't see it yet. That we have right here. Our answer is B. And D, both of these guys have
the same slope, so you can't have-- clearly it doesn't
apply to the two equations up there. And they also have two different
y-intercepts, so it's definitely not choice D. Problem 48. Yoshi has exactly $1 in
dimes, which is $0.10, and nickels, $0.05. If Yoshi has twice as many dimes
as nickels, how many nickels does she have? So she has $1. So let n equal the number of
nickels, and let d is equal to the number of the dimes. So it says Yoshi has twice
as many dimes as nickels. So the dimes are twice as many
as the number of nickels. That's what this tells us. Yoshi has twice as many
dimes as nickels. That translates into the dimes
are 2 times the nickels. And then we also know
that she has $1. So how do you get $1? Well, it's $0.10, so 0.1 times
the number of dimes. Plus 0.05 times the number of
nickels is equal to $1. Or maybe just so we can avoid
decimals, I could rewrite this statement right here,
exactly $1. Instead of writing in dollars,
I could write in cents. I could write 10 cents times
each dime, plus 5 cents times each nickel is equal
to 100 cents. So if you compare this equation
and this equation, I just multiplied both
sides by 100. Here I'm dealing with cents--
$1 is 100 cents. Here I'm dealing with dollars. A dime is 1/10 of a dollar. A dime is 10 cents. This one right here is probably
a little bit easier to deal with. So what are they asking? How many nickels
does she have? So we want to solve for n, we
want to solve for nickels. We have this piece of
information right here. The dimes are equal to 2 times
the number of nickels. And we have this right here. So what we can do is
substitute for d. We can replace this d, and we
know that d is equal to 2n. So we could put 2n in
the place of that. So let me write it over here. So I have 10d plus 5n
is equal to 100. 10 cents times my number of
dimes plus 5 cents times my number of nickels is
equal to 100 cents. And then my dimes
is equal to 2n. So 10 times 2n plus 5n
is equal to 100. 10 times 2n is just 20n, plus
5n, is equal to 100. And then we get 25n
is equal to 100. Divide both sides of this
equation by 25, and you get n is equal to 100 divided
by 25 is 4. Yoshi has 4 nickels. And if you want to know
how many dimes, you multiply it by 2. She has 8 dimes. So she has $0.80 in dimes
and she has $0.20 in nickels, 4 times 5. Problem 49. What are all the possible values
of x such that 10 times the absolute value of
x is equal to 2.5? So we can simplify this. So we have 10 times the
absolute value of x is equal to 2.5. Or if we divide both sides of
this equation by 10, we get-- let me do it in blue. We get the absolute value of
x is equal to-- what's 2.5 divided by 10? You might be able to just
do it in your head. You say, oh, you know, there's
4 quarters in $1, or 25 times 4 is 100. But this is 25 divided by 10. You multiply 2.5 times 4,
you're going to get 10. So you can say this is 1/4. If that wasn't obvious
to you, you could actually do the division. You can divide 10 into 2.5. Maybe throw some 0's there. Put the decimal there. 10 goes into 2 zero times. 0 times 10 is 0. Bring down the 2. 2 minus 0 is 2. Bring down the 5. 10 goes into 25 how
many times? 10 goes into 25 two times. 2 times 10 is 20. 25 minus 20 is 5. Bring down another 0. 10 goes into 50 exactly
five times. 5 times 10 is 50. Subtract, you get 0. So 10 goes into 2.5, 0.25
times, which is the same thing as 1/4. The absolute value of
x is equal to 0.25. Now that tells us that x, what
are the two values for x? x could definitely be equal to
0.25, because the absolute value of 0.25 is 0.25. Or what other number,
when I take its absolute value, is 0.25? Well, it could be minus 0.25. I take the absolute value of
minus 0.25, I also get 0.25. So my two choices are 0.25 or
minus 0.25, and that is choice A right there. And when we did this division,
I did it kind of a long and slow way. And if you didn't recognize
this as 1/4. The other way to say it is when
you divide by 10, that's equivalent to moving your
decimal 1 slot to the left. So you could have just said, oh,
2.5 divided by 10 is equal to-- let's just move the decimal
over to the left-- is equal to 0.25. All of those would have been
equivalent ways of doing it. Problem 50. Which of the following is
equivalent to 1 minus 2x is greater than 3 times
x minus 2? So let's solve this. So we have 1 minus 2x
is greater than 3 times x minus 2. Or we get 1 minus 2x is
greater than-- let's distribute our 3-- 3x minus 6. 3 times minus 2 is minus 6. Now we can subtract
1 from both sides. So we have a minus 1 plus
here, and we're going to subtract a 1 from that side. The whole reason why I'm doing
that is to cancel out that 1 right there. So this side, those cancel out,
so I'm just left with minus 2x is greater than
3x, and then minus 6 minus 1 is minus 7. And now here's the vaguely
tricky part. No, actually, we're not at the
vaguely tricky part yet. Let's subtract 3x
from both sides. So we have minus 3x on
that side, and then minus 3x on that side. And the whole reason why
I'm doing that is to cancel out this 3x. So that these two guys
obviously cancel out. Now minus 3x plus minus 2x,
which is minus 5x, is greater than minus 7. And now here's the vaguely
tricky part. We want to divide both
sides by minus 5. But when you have an inequality,
and you're dividing or multiplying both
sides by a negative number, you need to switch
the inequality. So if we divide both sides by
minus 5, so I could multiply by a fraction, minus
1/5 times minus 5x. And then I'm going to have a
minus 7 times minus 1/5. I'm going to switch
the inequality. So it goes from greater
than to less than. And the whole reason why it's
switched is because I'm multiplying both sides, or
depending on how you do it, dividing both sides by
a negative number. So then this becomes-- I picked
minus 1/5 so this cancels out. So this just becomes x is less
than, and then minus 7 times minus 1/5, the negatives cancel
out, or a negative times a negative
is a positive. So you're left with 7/5. So x is less than 7/5. And it shows you that maybe you
don't want to go work out the whole problem. Because right here, they didn't
expect me to figure out the whole problem, so I just
wasted a little bit of both of our time. They just wanted us to go a
couple of steps into it. So what did they just do? Well, they just wanted that
first step, where I distributed the 3. 3x minus 6. So that's all they did here. Because they didn't change the
left-hand side anywhere, they didn't change the inequality
anywhere. So all I have to do is pick
choice C right there. But hopefully you learned
a little bit about inequalities as well. I hate to waste your time like
that, but this was actually a fairly straightforward
problem. But actually, the very
first step we did got us to the answer. So that teaches us both a
lesson: look at the choices before you proceed
with the problem. 51. Which equation represents the
line on the graph below? So we can look at this, we can
just kind of eyeball it. But sometime it's hard
to eyeball it. So what we want to do is look
for some point, my first intuition is to say, OK, y
is equal to mx plus b. Let's look for the y-intercept
and the slope. But here the y-intercept,
it's not completely obvious where it is. Although it looks like
it's at 1.5. And we can actually verify that
it's at 1.5, because its slope, you can see right here. If we start at 1, and if we
go 2 to the right, we go down by exactly 1. So its slope, m, is equal to
change in y over change in x. When your change in x is
positive 2, your change in y is minus 1. So your slope is minus 1/2. So if you go back 1, you're
going to go up 1/2. Just like that. So our y-intercept is 1.5. It's right there. It's equal to 1.5. Just like that. I can verify that
from the slope. If I start at 2, and I go 1 to
the right, that means I'm going to go 1/2 down. Because my slope is minus 1/2. So we go 1/2 down, and that'll
get me to 1.5. So the equation of this line is
y is equal to minus 1/2 x, that's my slope, plus 1.5. And that doesn't look like any
of these choices here, so we need to simplify this
a little bit. Well, the first thing we might
want to do is multiply both sides of this equation by 2. So you get 2y is equal to 2
times minus 1/2 x, is just minus x, and then 2
times 1.5 is 3. Plus 3. And then if we add x to both
sides of this equation, and I'm doing that to get rid of
this minus x on the right-hand side, we get x plus
2y is equal to 3. And that is choice A right
there, which was completely equivalent to what
we did up here.