Let's do some problems that deal with equations and
inequalities.
So let's start with this problem 2 here.
So it says define the variables and translate the
following expressions into inequalities.
Let's see, part a: A bus can seat 65 passengers or fewer.
So in Part a, let's let x be the number bus can seat.
So the number of people you can sit in the bus, we're
going to call that x.
And they're saying that the bus can fit
65 people or fewer.
So x has to be less than or equal to 65.
And we put that equal there, that less than or equal,
because it could be 65.
It doesn't say definitely less than 65.
It could be 65 passengers or fewer.
So that's why it's less than or equal to.
b: the sum of two consecutive integers is less than 54.
So if you say x is first integer, and then x
plus 1 is the next.
They're saying the sum of these two is less than 54.
So x plus 1 is less than 54.
And if you actually wanted to figure out what x is, what the
first integer is, you could just solve this inequality.
c: an amount of money is invested at 5% annual
interest. Let x be the amount of money.
It's invested at 5% annual interest. The interest earned
at the end of the year is greater than or equal to $250.
So what's the interest earned at the end of the year?
It's the amount of money you invest times your annual
interest rate, times 5% or, if you write this as a decimal ,
it's 0.05 So 0.05 or 5% times the amount of money invested,
that is going to be greater than or equal to-- this is the
amount of interest earned --that is going to be greater
than or equal to $250.
So that's what we did.
We set up the problem into an inequality.
And part d.
You buy hamburgers at a fast food restaurant.
A hamburger costs $0.49.
You have at most $3.00 to spend.
Write an inequality for the number of
hamburgers you can buy.
So let's change the letters.
Let h be equal to the number of hamburgers.
So the total I'm going to spend is the number of
hamburgers times the amount per hamburger, times 0.49.
That's the total amount I'm going to spend, and that can
at most be $3.00.
So it has to be less than or equal to $3.00.
So that is our inequality.
Let's do problem 4 here.
Check that the given number is a solution to the
corresponding inequality.
All right.
Part a here.
So the inequality is 2 times x plus 6 is less
than or equal to 8x.
And they're saying that x is equal to 12 is a solution.
Let's verify that.
So if you put a 12 here-- do it in a different color --if
this is a 12 and then this would be a 12.
So you get 2 times 12 plus 6.
So this is 2 times 18 is less than or equal to 8 times 12.
8 times 12 is 96.
2 times 18 is 36.
So 36 is definitely less than or equal to 96.
So this x is equal to 12 is a solution.
Part b.
We have 1.4 times z plus 5.2 is greater than 0.4z.
For the solution, if z is equal to minus 9-- negative 9
I should say.
I always say minus 9.
The correct wording is negative 9.
So let's put a negative 9 here.
So it's 1.4-- they're saying that that's one of the
solutions --times negative 9 plus 5.2 is greater than 0.4
times negative 9.
And I'm just going to use the calculator on this one just
for the sake of time.
You could do that in your head if you like.
So we have on the left-hand side, 1.4 times negative 9.
That's equal to minus 12.6.
So this is minus is 12.6 plus 5.2 is equal to negative 7.4.
So this left-hand side is negative 7.4.
Then they're claiming that that is greater than-- This I
can do in my head.
4 times 9 is 36.
I'm going to have to put a negative sign because it's a
positive times a negative.
And I have one number behind the decimal point.
So this is saying negative 7.4 is greater than negative 3.6.
This isn't true.
If you draw a number line right here.
So if this is 0, this is negative 3.6,
this is negative 7.4.
It's less than negative 3.6.
So this is not true.
I'll do it in a big red color.
That is not true.
z is equal to negative 9 is not a solution of this
inequality right there.
It does not satisfy that inequality.
Part c.
They have minus 5/2 y plus 1/2 is less than negative 18.
And they're claiming y is equal 40 is a solution.
Let's try that out.
So minus 5/2-- instead of y I can write 40 just to see if it
works --plus 1/2 is less than minus 18.
That's their claim.
So 5/2 times 40-- Divide the numerator and
denominator by 2.
So 1 becomes a 20.
So it becomes minus 5 times 20 is minus 100 plus 1/2 is less
than minus 18.
Well, you could view this as negative 99 and 1/2.
You could do that in your head if you like, or you could view
this as 99.5-- you're adding 0.5 to minus 100 --is less
than negative 18, which is definitely true.
This is more negative than this is.
So this is correct.
And then finally part d.
Let me scroll down or let me clear some space for myself.
Part d.
They're saying 80 is greater than or equal to 10
times 3t plus 2.
And they're claiming that t is equal to 0.4 as a solution.
Let's try that out.
80 is greater than or equal to 3 times 0.4 plus 2.
So that's saying that 80 is greater than or equal to 10
times-- 3 times 0.4 is 1.2 --plus 2, or 80 is greater
than or equal to 10 times-- This is 3.2.
Or that 80 is greater than or equal to 32, which is
absolutely right.
So d is also a valid solution.
Problem 5.
The cost of a Ford Focus is 27% of the
price of a Lexus GS450H.
If the price of a Ford is $15,000-- So the Ford is equal
to $15,000, what is the price of a Lexus?
So they tell us that the Ford is equal to 27% of the price
of the Lexus.
Is equal to 0.27, or 27%, times the price of a Lexus.
I could write F and L, but I'll just write
out the words there.
So the Ford we know is $15,000.
So we know that $15,000 is equal to 27%, 0.27, times the
price of a Lexus.
So to figure out the price of a Lexus we just divide both
sides by 0.27, and divide this side by 0.27.
That just becomes a 1.
And so you have the price of your Lexus is equal to $15,000
divided by 0.27.
See what we get.
So let me clear it.
We have 15-- 1, 2, 3, $15,000 divided by 0.27 is equal to
$55,555.55.
So we'll say $55,556 just to round up.
So this is equal to $55,556.
So you're spending a lot more on the Lexus than you would on
the Ford Focus.
It better be a much better car.
Finally, number 6.
On your new job you can be paid in one of two ways.
All right.
You can either be paid-- So this is option one.
And let's write option 2 here.
You can either be paid $1,000 per month plus 6% commission
of total sales.
So let's let s is equal to total sales.
So your first option you'll be paid $1,000 per month plus 6%
of total sales.
So on a monthly basis you'll be paid $1,000 plus 6% of
total sales.
So plus 6% times your total sales.
That's option 1.
That's right there.
You can either be paid $1,000 per month plus 6% commission
of total sales.
And then option 2 is $1,200 per month--
Let me switch colors.
Option 2-- I'll do it in yellow --paid $1,200 per month
plus 5% commission on sales over $2,000.
Let me scroll over a little bit.
Plus 5% of sales over $2,000.
So if my sales are s, my sales over $2,000 are going to be s
minus $2,000.
If my sales are, let's say they're $3,000 in a month.
The sales over $2,000 are going to be
$3,000 minus $2,000.
It's going to be $1,000 over $2,000.
So these are our two options.
Let me highlight that right over there.
And they say for what amount of sales is the first option
better than the second option?
And assume they're always sales over $2,000.
So they're essentially saying, OK, this is always going to be
non-0 right here.
So we want to know the situation where the first
option is better.
Better, probably meaning for me, meaning that
I'll get more money.
So when is option 1 going to be, we could either say
greater than or greater than or equal to, option 2.
And then we could solve this equation.
So let's see what we can do.
The first thing, let's subtract
$1,000 from both sides.
If you subtract $1,000 from the left-hand side you're just
left with 0.06s is greater than or equal to-- So subtract
$1,000 from the left-hand side.
I'll take the $1,000 from here.
You're left with 200 plus 0.05 times s minus 2,000.
And then let's see, I probably want to multiply that out.
This is going to be 200 plus 0.05s minus-- This 0.05,
that's the same 5%, that's the same thing as 1/20.
1/20 of 2,000, that's 100.
So this is minus 100.
That times that is 100.
I'm just distributing this 5% or this 0.05.
So that's the right-hand side of the equation.
On the left-hand side we have 0.06, and we just simplify the
right-hand side a little bit.
We have 200 minus 100, so that just becomes 200 minus 100, we
could just write as plus 100 right there.
So our equation is 0.06 is greater than or equal to
0.05-- Sorry, this is 0.06s is greater than or equal to-- I
don't want to lose that s over there --0.05s plus 100.
Now let's subtract 0.05s from both sides of the equation.
So 0.06 minus 0.05s is going to be 0.01s is greater than or
equal to-- I subtracted this from both sides, so it's not
going to be here anymore --is greater than or equal to 100.
And now I just divide both sides by 0.01.
So 0.01, 0.01.
That becomes a 1.
And then we are left with s is to going to be greater than or
equal to-- What's 100 divided by 0.01?
This is the same thing as 100 divided by 1 over 100, which
is the same thing as 100 times 100.
100 times 100 is 10,000.
So option 1 is definitely better for you if your total
sales for the month, if s is greater
than or equal to $10,000.
If it's less than that you're better with option 2.