Quadratic Inequalities Solving quadratic inequalities using factoring
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- 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?
- 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.
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