Welcome to the presentation on quadratic inequalities.
I know that sounds very complicated, but hopefully
you'll see it's actually not that difficult.
Or at least, maybe the problems we're going to work on
aren't that difficult.
Well, let's get started with some problems and hopefully
you'll see where this is kind of slightly different
than solving regular quadratic equations.
So let's say I had the inequality x squared plus
3x is greater than 10.
And remember, whenever you solve a quadratic or I guess
you would call it a second degree equation-- I guess
this is an inequality.
I shouldn't use the word equation.
It's tempting to sometimes do it the same way you'd do a
linear equation, kind of getting all the x terms
on one side and all the constants on the other.
But it never works because you actually have an x term and
then you have an x squared term.
So you actually want to get it in kind of what I would call
the-- I don't know if it's actually called this-- the
standard form where you actually have all of the terms
on one side and then a 0 on the other side.
And then you can either factor it or use the
quadratic equation.
So let's do that.
Well this is pretty easy.
We just have to subtract 10 from both sides and we get x
squared plus 3x minus 10 is greater than 0.
Now let's see if we can factor it.
Are there two numbers that when you multiply it become negative
10 and when you add it become positive 3?
Well, yeah.
Positive 5 and negative 2.
And once again, at this point I think you already know how to
do factoring, so this should be hopefully, obvious to you.
So it's x plus 5 times x minus 2 is greater than 0.
Now this is the part where it's going to become a little bit
more difficult than just your traditional factoring problem.
We have two numbers, I guess you could view it.
We have x plus 5.
I view that as one number.
Or I guess we have two expressions.
We have x plus 5 and we have x minus 2.
And when we're multiplying them we're getting
something greater than 0.
Now let's think about what happens when you
multiply numbers.
If they're both positive and you multiply them, then
you get a positive number.
And if they're both negative and you multiply them, then you
also get a positive number.
So we know that either both of these expressions are the same
sign, that they're both greater than 0, they're both positive.
Or we know that they're both negative.
And I know this might be a little confusing, but just
think of it as-- if I told you that-- I'll do something
slightly separate out here.
If I told you that a times b is greater than 0 we know that
either a is greater than 0 and b is greater than 0.
Which just means that they're both positive.
Or a is less than 0 and b is less than 0.
Which means that they're both negative.
All we know is that they both have to be the same sign in
order for their product to be greater than 0.
Now we just do the same thing here.
So we know that either both of these are positive, so x plus
5 is greater than 0 and x minus 2 is greater than 0.
Or-- now this is a little confusing, but if you work
through these problems it actually makes a lot of sense.
Or they're both negative.
Or x plus 5 is less than 0 and x minus 2 is less than 0.
I know that's confusing, but just think of it in terms of we
have two expressions: they're either both positive or
they're either both negative.
Because when you multiple them you get something
larger than 0.
Well, let's solve this side.
So this says that x is greater than negative 5
and x is greater than 2.
We just 2 both sides of this equation.
Or, and if we solve this side-- x is less than negative
5 and x is less than 2.
I just solved both of these inequalities right here.
Now we can actually simplify this because here we say that
x is greater than negative 5 and x is greater than 2.
So in order for x ti be greater than negative 5 and for x to be
greater than 2, this just simplifies as saying, well,
x is just greater than 2.
Because if x is greater than 2, it's definitely
greater than negative 5.
So it just simplifies to this.
And we'd say or-- and here we said x is less than negative
5 or x is less than 2.
Well, we know if x is less than negative 5, then x is
definitely less than 2.
So we could just simplify it to or x is less than negative 5.
So the solutions to this problem is x could be greater
than 2 or x could be less than negative 5.
And so let's just think about how that looks
on the number line.
So if 2 is here, x could be greater than 2.
So it's all of these numbers.
And if this is negative 5-- I shouldn't have done it
so close to the bottom.
x is less the negative 5.
So these are the numbers that satisfy this equation.
And I'll leave it up to you to try out to see that
they actually work.
Let's try another one and hopefully, I can
confuse you even more.
Let's say I have minus x times 2x minus 14 is greater
than or equal to 24.
Well, the first thing we want to do is just manipulate this
so it looks in the standard form.
So we get negative 2x squared plus 14x-- I'm just
distributing the minus x-- is greater than or equal to 24.
I don't like any coefficient it front of my x squared term,
so let's divide both sides of this equation by negative 2.
So we get x squared-- we divided by negative
2-- minus 7x.
And remember, when you divide by a negative number you switch
the sign on the inequality, or you switch the direction
of the inequality.
So we're dividing by negative 2, so we switched it.
We went from greater than or equal to, to less
than or equal to.
And then 24 divided by negative 2 is minus 12.
And now we can just bring this minus 12 onto the left-hand
side of the equation.
Add 12 to both sides.
We get x squared minus 7x plus 12 is less than or equal to 0.
And then we can just factor that and we get, what is that?
It's x minus 3 times x minus 4 is less than or equal to 0.
So now we know that when we multiply these two terms
we get a negative number.
So that means that these expressions have to be
of different signs.
Does that make sense?
If I tell you I have two number and I multiply them,
I get a negative number.
You know that they have to be of different signs.
So we know that either x minus 3 is less than or equal to 0
and x minus 4 is greater than or equal to 0.
So that's one case.
And the other case is x minus 3 is greater than or equal to 0,
which means x minus 3 is positive.
And x minus 4 is less than or equal to 0--
oh, I went to the edge.
So let's solve this and hopefully it'll simplify more.
So this just says that x is less than or equal to 3.
And this says x is greater than or equal to 4.
So both of these things have to be true. x has to be less than
or equal to 3 and x has to be greater than or equal to 4.
Well, let me ask you a question.
Can something be both less than or equal to 3 and greater
than or equal to 4?
Well, no.
So we know that this situation can't happen.
There's no numbers that's less than or equal to 3 and
greater than or equal to 4.
So let's look at this situation.
This says x is greater than or equal to 3 and x is
less than or equal to 4.
Can this happen?
Sure.
That just means that x is some number between 3 and 4.
If we were to draw this on the number line, we would get--
if this is 3, this is 4.
And it's greater than or equal to so we fill it in.
And less than or equal to so we'd fill it in.
And it would be any number between 3 and 4 would
satisfy this equation.
And I'll leave it up to you to try it out.
I know this is confusing at first, and this is actually
something that they normally don't teach really well, I
think, in most high schools until 10th or 11th grade.
But just think about you're multiplying two expressions.
If the answer is negative then they must be
of different signs.
If the answer is positive they must be the same sign.
And then you just work through the logic.
And you say, well, no number can be less than 3 and greater
than 4, so this doesn't apply.
And then you do this side and you're like, oh, this
situation does work.
It's any number between 3 and 4.
Hopefully that gives you a sense of how to do
these type of problems.
I'll let you do the exercises now.
Have fun.