Basic inequalities
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Inequalities on a number line
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Inequalities on a number line
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Interpreting Inequalities
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Inequalities Using Addition and Subtraction
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Inequalities Using Multiplication and Division
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Inequalities
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One-Step Inequalities
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One-Step Inequalities 2
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Solving Inequalities
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One step inequalities
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Inequality examples
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Multi-Step Inequalities
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Multi-Step Inequalities 2
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Multi-Step Inequalities 3
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Multi-step linear inequalities
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Writing and using inequalities 2
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Writing and using inequalities 3
Inequality examples
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- 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.
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At 5:31, how is the moon large enough to block the sun? Isn't the sun way larger?
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