Tip:
Highlight text to annotate it
X
All right, we're on problem number 8.
They ask us
which equation is equivalent 5x-2(7x+1)=14x
So I'm guessing they just want to simplify this a little bit
and see if we get to one of these choices.
So let's do that.
So we start with 5x-2(7x+1)=14x
This is I think the most obvious thing to simplify.
This 2(7x+1) or we can even say -2(7x+1)
so then this becomes 5x plus
I'm just going to distribute the -2 times all of this.
Plus -2 times 17x is minus 14x.
minus 14x
And then -2 times 1 is minus 2 is equal to 14x.
And let's see, on all of these choices
they have 14x on the right hand side.
So they just want us to simplify this.
So this simplifies to 5x, plus -14, so that's -14x
minus 2 is equal to 14x.
So we have 5x-14x.
What's 5-14?
It's what? -9, right?
-9x-2=14x
That is choice A.
Now, one thing that maybe I realized,
I kind of skipped a step.
We could've just...
Let me just do this
just so that you understand this step that I did here.
We could have just said minus
and then distribute the 2
So we could have right there said
5x minus positive 2 times 7x, 14x, plus 2 times 1, plus 2.
I'm just doing the left-hand side.
And then we could have distributed the minus sign.
So you get 5x-14x-2.
Either way, it would have gotten us to this point and then
we could have simplified here and gotten choice A.
Next problem.
Which equation is equivalent to
OK, so another case where
I'm guessing they just want us
to simplify these equations
So let me write that down.
4(2-5x)=6-3(1-3x)
All right, let's distribute this 4 first.
So you get 4 times 2 is 8.
4 times -5x is -20x.
-20x is equal to 6 minus--
maybe you might have been tempted to say,
oh, 6 minus 3, that's 3, and then distribute it.
Remember, order of operations.
Multiplication comes first.
So we have to multiply the 3(1-3x)
before we can deal with the 6.
Well, on the other side
there was just the multiplication first.
All right.
So here, 3(1-3x), so it'll be 3.
I'm just going to do
minus a positive 3 times each of these.
So 3 times a -3 is -9x.
And now I can distribute this minus sign.
So let's see, on the left-hand side I have
let me do it in a brighter color.
The left-hand side I have 8-20x= 6,
and then I'm going to subtract this whole thing.
So you can view it as
distributing a -1 times both of these.
So 6 plus -3, or just 6-3,
and then a minus times a minus plus 9x.
All right.
Now let's see what we could do to help simplify this.
What happens if we add 20x to both sides?
Then on the left-hand side, it goes away
so we're just left with 8 is equal to
well,let me just do that.
6 minus 3.
I added 20x, so plus 29x.
I really should have simplified this 6 minus 3
but I didn't want to do too much in one step.
So 8=3+29x,
and then if we subtract 3 from both sides here
we get 8 minus 3 is 5.
That 3 goes away and you get 29x
and that is choice C. 29x=5.
Next question, problem number 10.
And I should be careful that you don't think that
all of this stuff is for this problem.
The total cost, c, in dollars
of renting a sailboat for n days
is given by this equation.
If the total cost was $360
for how many days was the sailboat rented?
So they're giving us c. c is $360.
So they're telling us that $360, that's the total cost,
and that's going to be equal to
120 plus 60 times the total days rented, times 60n.
And that's what they want us to figure out, n.
How many days was the sailboat rented?
Let's see, so if you should subtract 120
from both sides of this equation
you get 240 is equal to 60n.
I'll go here.
And if you divide both sides of this by 60
240 divided by 60
that's the same thing as 24 divided by 6
so that's 4.
And then 60n divided by 60 is, of course, 1
so that is n.
So the sailboat was rented for 4 days.
All right, problem number 11.
And I cut and pasted part of it
but they want us to determine
what was the first incorrect step
that they did in this problem?
So let's see if we can determine that.
So let's see.
Solve 3(x+5)=2x+35
Fair enough.
So in this first step they distributed 3*x 3x5
Right, 3x+15=2x+35
so step 1 looks good to me.
Let me do that in a darker color.
Step 1 looks good.
They distributed the 3.
Now let's see what they did in step 2.
Let's see, if I wanted to solve this
I would subtract 2x from both sides,
and it looks like that's what they attempted to do
because they got rid of this 2x, right?
On the right-hand side
they went from 2x+35 to just 35
so somehow they got rid of this 2x
and the only way to get rid of it
is to subtract 2x from both sides.
Right? Subtract 2x from both sides
Now, if I subtract 2x from both sides, sure enough
the right-hand side will just be left with the 35.
The left-hand side will have the 15, of course.
And then I have -2x+3x.
Well, -2x+3x, that's x.
So this 5x, that shouldn't be there.
It should be x+15=35.
So even though they were trying to get rid of this 2x
and they should have subtracted 2x from both sides
they inadvertently added 2x here
so they said 2x+3x=5x.
Well, that was wrong.
You have to subtract.
And you're just left with x+15 =35.
So step 2 is when they made their first error.
Problem 120.
120
A 120-foot-long rope is cut into 3 pieces.
The first piece of the rope let me draw.
This is begging for a diagram.
All right, so they say the first piece of rope
is twice as long as the second piece of rope.
Ok, and then they say the third piece of rope
is three times as long as the second piece of rope.
Everything is a multiple of the second piece of rope.
So the second one was the shortest one
just reading it, right?
The first is twice as long as the second.
The third is three times as long.
So if I just drew it.
So if this is the second length of rope,
let's call that x.
That's the second length of rope.
And they told us the first piece of rope
the first piece of rope is twice as long
as the second piece of rope.
OK, so if this is the first piece of rope right here
it's going to be twice as long as the second piece of rope.
So that's 2x.
Fair enough.
And then they tell us
I'll do this in magenta
The third piece of rope is three times as long
as the second piece of rope.
So this is the third piece of rope here.
It's three times as long as the second piece of rope.
So it's 3x.
And they say, what is the length
of the longest piece of rope?
All right, so let's think about it.
If I add up all of these segments, right
what do they add up to?
Well, they tell us that it's 120-foot-long rope.
So let me add that up.
So if I do 2x plus x plus 3x
that's going to equal 120 feet.
That is equal to 120 feet, right
That whole distance is 120.
So 2x plus x is 3x plus 3x
you get 6x is equal to 120
You divide both sides by 6
You get x is 120 divided by 6 is 20.
All right, that's what x is.
That's the second piece of rope
or the shortest piece of rope
because they tell us that.
The other ones are some multiples of that length.
But they want to know
what is the length of the longest piece of rope?
the longest piece of rope
Well, the longest piece of rope is clearly piece 3.
Right? It's three times the second piece.
Piece 1 was only two times.
So this is what they want to know.
They want to know what 3x is equal to.
So 3x is equal to three times 20
which is equal to 60 feet.
And that's choice C.
Next problem.
Let's see if I have time.
Yeah, I think I have time.
I might as well attempt to fit it in.
All right.
Don't want to rush them, though.
Let's see.
The cost to rent a construction crane is $750 per day
plus $250 per hour of use.
What is the maximum number of hours
the crane can be used each day
if the rental cost is not to exceed $2,500 per day?
OK, so how much is the cost per day?
So the cost per day is equal to $750.
So no matter how little or much we use it
we have to pay that $750 per day.
So it's equal to $750 plus $250 per hour
so plus $250 times the number of hours.
This is what our total cost is per day.
And they want to say
what is the maximum number of hours
the crane can be used each day
if the rental cost is not to exceed $2,500 per day?
So this whole thing has to be
less than or equal to $2,500.
The total cost has to be less than or equal to $2,500.
So now let's just solve this equation right there.
And I'll do it up here.
So we know that $750 plus $250
let me do it down here actually.
More space.
So what's the first thing we could do here?
We could subtract 750s from both sides of this equation.
So we get 250 times h is less than or equal to
2,500 minus 750, that's what?
1,750?
Yeah, because 1,500 minus 750 is 750.
1,750.
I just subtracted 750 from both sides of this equation.
Let me write that down.
So I said minus 750, plus that.
And then I said minus 750.
And, of course, minus 750 plus 750, that goes to 0.
That's why you don't see anything over here.
And on the right-hand side, 2,500 minus 750 is 1,750.
All right, now let's just divide both sides by 250.
We don't have to do anything to the inequality sign
since 250 is a positive number.
So the hours, the maximum number of hours
the hours have to be less than or equal to 1,750 over 250
which is-- let's see, 1,750 divided by 250.
That's the same thing as 175 divided by 25.
25 goes into 175, what is that?
I want to say seven times?
7 times 5 is 35.
7 times 2 is 14 plus 3 is 17, right.
So h has to be less than or equal to 7 hours.
And that is choice C.
See you in the next video.