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We're now going to take the California Standards Test,
Algebra I released questions.
In the last series, I had done the Algebra II.
I guess I'm going in reverse order.
Let me copy and paste this first question because I think
it's good to see the whole thing.
So let me see, I have copied it.
Let me move this pointer all the way up, and
then, there we go.
All right.
And they're asking us is the equation 3 times 2x minus 4
equal to minus 18 equivalent to 6x minus 12 is equal to 18?
So let's think about this.
If we just distribute this 3, what do we get?
3 times 2x is 6x.
3 times minus 4 is minus 12.
And that, of course, is equal to minus 18.
So sure, they're the same thing.
If you just distribute the 3 over the 2x minus 4, you get
6x minus 12.
So the answer is definitely yes.
It's not the no one down here.
And it says yes, the equations are equivalent by the
associative?
No.
Communicative?
No.
The equations are equivalent by the distributive property?
[FIRE TRUCK SIREN]
There's some type of a fire truck going on outside.
Let's see.
Where was I?
Oh, yes.
Yes, the equations are equivalent by the distributive
property of multiplication over addition.
Right, that's that.
We distributed this 3 over the 2x minus 4.
And they say over addition because you could view this as
a plus minus 4.
Addition and subtraction is really the same thing when you
think of the distributive property.
Anyway, let's do the next problem.
The next problem I can just write out.
This is problem number 2.
They say the square root of 16 plus the cube root
of 8 is equal to?
Well, what's the square root of 16?
And when you just have a square root there, you might
say, maybe it's plus or minus 4, but when they write it this
way is means the principal root, so it's just plus 4.
They would write a plus or minus out front if they wanted
you to get the negative square root.
So it's 4 plus-- now, what to the third power is equal to 8?
Well, 2 to the third power is equal to 8, right?
So we could write 2 to the third is equal to 8.
That's the same thing as saying the cube root of 8 is
equal to 2.
You could also view this as 8 to the 1/3 power.
Anyway, the cube root of 8 is then 2, so 4 plus 2 is equal
to 6, and that is choice B.
Problem 3.
Let me scroll down a little bit.
OK, and they want to know-- I could copy and
paste the whole thing.
There we go.
And they want to know which expression is equivalent to x
to the sixth times x squared?
So x to the sixth times x squared, they
have the same base.
When you're multiplying both of these expressions, we can
add the exponents.
So that is equal to x to the-- 6 plus 2 is 8.
That's not one of the choices here, so we have to say, which
of these also are the same thing as x to the eighth?
And so which two exponents when I add them is equal to 8?
4 plus 3 is equal to 7.
5 plus 3, this is equal to x to the eighth as well.
So that is choice B.
Next problem, problem 4.
All right, let me-- this is another one where I'll copy
and paste it.
All right.
They want to know which number does not have a reciprocal?
So the reciprocal of negative 1, that's just 1 over negative
1, which is equal to negative 1.
The reciprocal of 0, that's what?
1/0, which is not defined.
So the choice is B.
0.
We don't know what 1/0 is.
Maybe that's a project for you to think of
what it should mean.
And, of course, these have reciprocals.
1 over 1/1,000 is just equal to 1 times 1,000 over 1, which
is equal to 1,000.
And the reciprocal of 3 is, of course, 1/3.
Next problem.
They say-- so there's a lot of terminology here, but I guess
that's good.
So they want to know-- let me just copy.
Maybe I'll do the next one, too.
OK.
I could probably just do it up here.
All right.
They want to know, what is the multiplicative inverse of 1/2?
So essentially, what can I multiply 1/2 by
and then get 1?
It's the same thing as saying what's the inverse of 1/2.
So if I multiply by 1/2 by-- well, the inverse of 1/2, I'd
say 1 over 1/2.
That's the same thing as 1 times 2/1,
which is equal to 2.
Or another way to think of it is 2 times 1/2 is equal to 1.
So the multiplicative inverse of 1/2 is just 2.
That's choice D.
Problem 6.
What is the solution for this equation?
All right, sometimes these absolute value signs can
appear daunting, but you just have to
think it through logically.
If the absolute value of 2x minus 3 is equal to 5, that
tells us what?
That means that 2x minus 3 is equal to 5, right?
Because inside the absolute value is equal to 5, then the
absolute value of 5 is equal to 5.
So that's fair enough.
But what could 2x minus 3 also be equal to?
What happens if 2x minus 3 within the absolute value sign
is equal to minus 5?
Well, then you would take the absolute value of that and you
would get 5, right?
So 2x minus 3 could also be equal to minus 5.
When you see this absolute value sign, you say, OK,
whatever's inside the absolute value is either 5 or negative
5 because we're taking the absolute value of it to get 5.
So we just solve both of these equations.
If you add 3 to both sides of this one, you get
2x is equal to 8.
x is equal to 4.
On the second one, you add 3 to both sides.
You get 2x is equal to-- minus 5 plus 3 is minus 2.
x is equal to minus 2 divided by 2 is negative 1.
So x could be equal to 4 or x could be equal to negative 1.
And that is choice C, x is negative 1 or x is equal to 4.
Next problem.
The Algebra I ones go faster than the Algebra II problems.
Those tend to be hairier.
Let me clear all of this.
I'll just write this one down.
They say what is the solution set for the inequality 5 minus
the absolute value of x plus 4 is less than or
equal to minus 3?
So at first, this is really daunting.
I can't even do that logic that I did last time because I
have that 5 out there.
But let's think of it this way.
Let's try to simplify it, so we have just the absolute
value of something is less than or equal
to something else.
So one thing we can do is, if we want to get rid of this 5,
remember, what we do to both sides of an equation or
inequality-- whatever we do to one side of an equation or an
inequality, we do to both sides.
So let's subtract 5 from both sides of this equation.
If you subtract 5 from the left side, this 5 disappears.
I'm just going to do minus-- let me write that out.
Minus 5 plus, and I'm going to do a minus 5 there.
That's a plus.
So minus 5 plus 5 is 0, so I'm just left with minus absolute
value of x plus 4 is less than or equal to-- now, what's
minus 3 minus 5?
That's minus 8.
All right, now this next step, this is something-- maybe it
wasn't obvious for you and putting the inequality there--
you know, if this was an inequality, you would just
say, OK, I'm going to multiply or divide both sides by
negative 1 to get rid of the negative signs.
But one thing you have to remember, whenever you
multiply or divide both sides of an inequality by a negative
number, you have to switch the inequality.
So if this is true, then if I'm multiplying both sides of
this by negative 1, so negative 1 times negative x
plus 4, I'm going to switch the inequality, so that's
going to become greater than or equal to negative 8.
And I did the negative 1 on this side, so I have to
multiply it times negative 1 on that side.
And so this negative cancels out that negative, so we're
just left with x plus 4 is greater than or equal to--
negative 8 times a negative 1 is equal to 8.
Now we can just use the logic that we had
from that last problem.
This tells us what?
This tells us that the magnitude of x plus 4 is
greater than or equal to 8.
Let me draw a number line here because I really want you to
get the intuition of what magnitude means.
So if that's the number line and you can view magnitude as
kind of the distance from, or the absolute value, you can
kind of view it as the distance from 0, right?
So if this is 0 right here and this is positive 8 and this is
minus 8, the absolute value of whatever this quantity was is
greater than 8.
That means its distance from 0 has to be greater than 8.
You can just say distance from 0 of this number has to be
greater than 8, greater than or equal to 8.
That means that this number is definitely going to be greater
than or equal to positive 8.
On the number line, it would be all of
those numbers, right?
Or, remember, we're saying the magnitude, so we don't care
about the direction.
The magnitude has to be greater than positive 8, so it
also includes the negative numbers less than negative 8.
And why does that make sense?
Well, take negative 9.
What's the absolute value of negative 9?
The absolute value of negative 9 is greater than 8 because 9
is greater than 8, so any number to the left of negative
8 or to the right of positive 8.
So what does that tell us about this equation?
So that means that-- well, the easy one is x plus 4 could be
greater than or equal to 8.
So let's write that down.
Let me write it here.
x plus 4 greater than or equal to 8.
And that takes into consideration that the
magnitude is greater than or equal to 8 there.
Or x plus 4 less than or equal to minus 8.
That's the magnitude to the left of this
negative 8 right there.
And now we solve it.
And it's very important to think of absolute value in
these terms. Otherwise, it can become very confusing and you
start testing numbers.
But if you really just visualize the number line and
you think of absolute value as distance from 0, magnitude of
the distance from 0, you say, oh, the distance from 0 has to
be greater than or equal to 8, so that means my number has to
be-- this thing has to be less than or equal to minus 8 or it
has to be greater than or equal to positive 8.
So let's solve.
x plus 4 is greater than or equal to 8.
Subtract 4 from both sides, so you get x is greater than or
equal to 4.
I just subtracted 4 from both sides.
Subtract 4 from both sides here, you get x is less than
or equal to minus 12.
So the solution here is x is greater than or equal to 4 or
x is less than or equal to minus 12, and
that is choice D.
Anyway, I'll see you in the next video.