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Let's do some problems that deal with equations and

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inequalities.

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So let's start with this problem 2 here.

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So it says define the variables and translate the

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following expressions into inequalities.

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Let's see, part a: A bus can seat 65 passengers or fewer.

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So in Part a, let's let x be the number bus can seat.

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So the number of people you can sit in the bus, we're

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going to call that x.

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And they're saying that the bus can fit

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65 people or fewer.

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So x has to be less than or equal to 65.

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And we put that equal there, that less than or equal,

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because it could be 65.

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It doesn't say definitely less than 65.

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It could be 65 passengers or fewer.

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So that's why it's less than or equal to.

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b: the sum of two consecutive integers is less than 54.

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So if you say x is first integer, and then x

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plus 1 is the next.

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They're saying the sum of these two is less than 54.

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So x plus 1 is less than 54.

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And if you actually wanted to figure out what x is, what the

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first integer is, you could just solve this inequality.

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c: an amount of money is invested at 5% annual

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interest. Let x be the amount of money.

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It's invested at 5% annual interest. The interest earned

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at the end of the year is greater than or equal to $250.

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So what's the interest earned at the end of the year?

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It's the amount of money you invest times your annual

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interest rate, times 5% or, if you write this as a decimal ,

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it's 0.05 So 0.05 or 5% times the amount of money invested,

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that is going to be greater than or equal to-- this is the

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amount of interest earned --that is going to be greater

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than or equal to $250.

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So that's what we did.

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We set up the problem into an inequality.

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And part d.

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You buy hamburgers at a fast food restaurant.

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A hamburger costs $0.49.

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You have at most $3.00 to spend.

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Write an inequality for the number of

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hamburgers you can buy.

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So let's change the letters.

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Let h be equal to the number of hamburgers.

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So the total I'm going to spend is the number of

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hamburgers times the amount per hamburger, times 0.49.

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That's the total amount I'm going to spend, and that can

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at most be $3.00.

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So it has to be less than or equal to $3.00.

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So that is our inequality.

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Let's do problem 4 here.

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Check that the given number is a solution to the

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corresponding inequality.

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All right.

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Part a here.

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So the inequality is 2 times x plus 6 is less

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than or equal to 8x.

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And they're saying that x is equal to 12 is a solution.

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Let's verify that.

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So if you put a 12 here-- do it in a different color --if

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this is a 12 and then this would be a 12.

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So you get 2 times 12 plus 6.

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So this is 2 times 18 is less than or equal to 8 times 12.

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8 times 12 is 96.

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2 times 18 is 36.

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So 36 is definitely less than or equal to 96.

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So this x is equal to 12 is a solution.

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Part b.

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We have 1.4 times z plus 5.2 is greater than 0.4z.

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For the solution, if z is equal to minus 9-- negative 9

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I should say.

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I always say minus 9.

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The correct wording is negative 9.

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So let's put a negative 9 here.

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So it's 1.4-- they're saying that that's one of the

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solutions --times negative 9 plus 5.2 is greater than 0.4

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times negative 9.

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And I'm just going to use the calculator on this one just

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for the sake of time.

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You could do that in your head if you like.

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So we have on the left-hand side, 1.4 times negative 9.

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That's equal to minus 12.6.

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So this is minus is 12.6 plus 5.2 is equal to negative 7.4.

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So this left-hand side is negative 7.4.

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Then they're claiming that that is greater than-- This I

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can do in my head.

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4 times 9 is 36.

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I'm going to have to put a negative sign because it's a

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positive times a negative.

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And I have one number behind the decimal point.

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So this is saying negative 7.4 is greater than negative 3.6.

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This isn't true.

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If you draw a number line right here.

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So if this is 0, this is negative 3.6,

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this is negative 7.4.

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It's less than negative 3.6.

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So this is not true.

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I'll do it in a big red color.

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That is not true.

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z is equal to negative 9 is not a solution of this

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inequality right there.

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It does not satisfy that inequality.

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Part c.

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They have minus 5/2 y plus 1/2 is less than negative 18.

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And they're claiming y is equal 40 is a solution.

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Let's try that out.

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So minus 5/2-- instead of y I can write 40 just to see if it

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works --plus 1/2 is less than minus 18.

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That's their claim.

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So 5/2 times 40-- Divide the numerator and

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denominator by 2.

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So 1 becomes a 20.

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So it becomes minus 5 times 20 is minus 100 plus 1/2 is less

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than minus 18.

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Well, you could view this as negative 99 and 1/2.

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You could do that in your head if you like, or you could view

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this as 99.5-- you're adding 0.5 to minus 100 --is less

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than negative 18, which is definitely true.

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This is more negative than this is.

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So this is correct.

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And then finally part d.

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Let me scroll down or let me clear some space for myself.

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Part d.

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They're saying 80 is greater than or equal to 10

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times 3t plus 2.

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And they're claiming that t is equal to 0.4 as a solution.

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Let's try that out.

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80 is greater than or equal to 3 times 0.4 plus 2.

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So that's saying that 80 is greater than or equal to 10

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times-- 3 times 0.4 is 1.2 --plus 2, or 80 is greater

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than or equal to 10 times-- This is 3.2.

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Or that 80 is greater than or equal to 32, which is

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absolutely right.

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So d is also a valid solution.

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Problem 5.

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The cost of a Ford Focus is 27% of the

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price of a Lexus GS450H.

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If the price of a Ford is $15,000-- So the Ford is equal

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to $15,000, what is the price of a Lexus?

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So they tell us that the Ford is equal to 27% of the price

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of the Lexus.

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Is equal to 0.27, or 27%, times the price of a Lexus.

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I could write F and L, but I'll just write

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out the words there.

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So the Ford we know is $15,000.

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So we know that $15,000 is equal to 27%, 0.27, times the

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price of a Lexus.

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So to figure out the price of a Lexus we just divide both

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sides by 0.27, and divide this side by 0.27.

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That just becomes a 1.

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And so you have the price of your Lexus is equal to $15,000

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divided by 0.27.

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See what we get.

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So let me clear it.

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We have 15-- 1, 2, 3, $15,000 divided by 0.27 is equal to

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$55,555.55.

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So we'll say $55,556 just to round up.

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So this is equal to $55,556.

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So you're spending a lot more on the Lexus than you would on

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the Ford Focus.

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It better be a much better car.

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Finally, number 6.

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On your new job you can be paid in one of two ways.

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All right.

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You can either be paid-- So this is option one.

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And let's write option 2 here.

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You can either be paid $1,000 per month plus 6% commission

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of total sales.

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So let's let s is equal to total sales.

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So your first option you'll be paid $1,000 per month plus 6%

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of total sales.

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So on a monthly basis you'll be paid $1,000 plus 6% of

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total sales.

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So plus 6% times your total sales.

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That's option 1.

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That's right there.

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You can either be paid $1,000 per month plus 6% commission

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of total sales.

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And then option 2 is $1,200 per month--

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Let me switch colors.

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Option 2-- I'll do it in yellow --paid $1,200 per month

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plus 5% commission on sales over $2,000.

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Let me scroll over a little bit.

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Plus 5% of sales over $2,000.

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So if my sales are s, my sales over $2,000 are going to be s

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minus $2,000.

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If my sales are, let's say they're $3,000 in a month.

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The sales over $2,000 are going to be

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$3,000 minus $2,000.

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It's going to be $1,000 over $2,000.

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So these are our two options.

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Let me highlight that right over there.

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And they say for what amount of sales is the first option

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better than the second option?

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And assume they're always sales over $2,000.

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So they're essentially saying, OK, this is always going to be

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non-0 right here.

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So we want to know the situation where the first

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option is better.

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Better, probably meaning for me, meaning that

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I'll get more money.

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So when is option 1 going to be, we could either say

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greater than or greater than or equal to, option 2.

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And then we could solve this equation.

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So let's see what we can do.

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The first thing, let's subtract

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$1,000 from both sides.

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If you subtract $1,000 from the left-hand side you're just

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left with 0.06s is greater than or equal to-- So subtract

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$1,000 from the left-hand side.

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I'll take the $1,000 from here.

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You're left with 200 plus 0.05 times s minus 2,000.

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And then let's see, I probably want to multiply that out.

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This is going to be 200 plus 0.05s minus-- This 0.05,

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that's the same 5%, that's the same thing as 1/20.

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1/20 of 2,000, that's 100.

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So this is minus 100.

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That times that is 100.

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I'm just distributing this 5% or this 0.05.

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So that's the right-hand side of the equation.

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On the left-hand side we have 0.06, and we just simplify the

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right-hand side a little bit.

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We have 200 minus 100, so that just becomes 200 minus 100, we

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could just write as plus 100 right there.

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So our equation is 0.06 is greater than or equal to

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0.05-- Sorry, this is 0.06s is greater than or equal to-- I

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don't want to lose that s over there --0.05s plus 100.

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Now let's subtract 0.05s from both sides of the equation.

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So 0.06 minus 0.05s is going to be 0.01s is greater than or

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equal to-- I subtracted this from both sides, so it's not

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going to be here anymore --is greater than or equal to 100.

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And now I just divide both sides by 0.01.

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So 0.01, 0.01.

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That becomes a 1.

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And then we are left with s is to going to be greater than or

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equal to-- What's 100 divided by 0.01?

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This is the same thing as 100 divided by 1 over 100, which

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is the same thing as 100 times 100.

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100 times 100 is 10,000.

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So option 1 is definitely better for you if your total

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sales for the month, if s is greater

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than or equal to $10,000.

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If it's less than that you're better with option 2.

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