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.