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Welcome to the presentation on quadratic inequalities.

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Before we get to quadratic inequalities, let's just start

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graphing some functions and interpret them and then we'll

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slowly move to the inequalities.

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Let's say I had f of x is equal to x squared plus x minus 6.

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Well, if we wanted to figure out where this function

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intersects the x-axis or the roots of it, we learned in our

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factoring quadratics that we could just set f of x

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is equal to 0, right?

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Because f of x equals 0 when you're intersecting the x-axis.

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So you would say x squared plus x minus 6 is equal to 0.

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And you just factor this quadratic.

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x plus 3 times x minus 2 equals 0.

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And you would learn that the roots of this quadratic

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function are x is equal to minus 3, and x is equal to 2.

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How would we visualize this?

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Well let's draw this quadratic function.

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Those are my very uneven lines.

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So the roots are x is equal to negative 3.

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So this is, right here, x is at minus 3y0 -- by definition one

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of the roots is where f of x is equal to 0.

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So the y, or the f of x axis here is 0.

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The coordinate is 0.

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And this point here is 2 comma 0.

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Once again, this is the x-axis, and this is the f of x-axis.

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We also know that the y intercept is minus 6.

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This isn't the vertex, this is the y intercept.

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And that the graph is going to look something like this -- not

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as bumpy as what I'm drawing, which I think you get the

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general idea if you've ever seen a clean parabola.

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It looks like that with x minus 3 here, and x is 2 here.

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Pretty straightforward.

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We figured out the roots, we figured out what it looks like.

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Now what if we, instead of wanting to know where f of x is

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equal to 0, which is these two points, what if we wanted

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to know where f of x is greater than 0?

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What x values make f of x greater than 0?

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Or another way of saying it, what values make

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the statement true?

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x squared plus x minus 6 is greater than 0, Right,

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this is just f of x.

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Well if we look at the graph, when is f of x greater than 0?

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Well this is the f of x axis, and when are we

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in positive territory?

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Well f of x is greater than 0 here -- let me draw that

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another color -- is greater than 0 here, right?

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Because it's above the x-axis.

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And f of x is greater than 0 here.

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So just visually looking at it, what x values make this true?

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Well, this is true whenever x is less than minus 3, right, or

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whenever x is greater than 2.

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Because when x is greater than 2, f of x is greater than 0,

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and when x is less than negative 3, f of x

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is greater than 0.

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So we would say the solution to this quadratic inequality, and

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we pretty much solved this visually, is x is less than

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minus 3, or x is greater than 2.

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And you could test it out.

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You could try out the number minus 4, and you should get f

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of x being greater than 0.

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You could try it out here.

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Or you could try the number 3 and make sure that this works.

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And you can just make sure that, you could, for example,

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try out the number 0 and make sure that 0 doesn't work,

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right, because 0 is between the two roots.

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It actually turns out that when x is equal to 0, f

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of x is minus 6, which is definitely less than 0.

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So I think this will give you a visual intuition of what this

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quadratic inequality means.

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Now with that visual intuition in the back of your mind, let's

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do some more problems and maybe we won't have to go through the

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exercise of drawing it, but maybe I will draw it just to

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make sure that the point hits home.

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Let me give you a slightly trickier problem.

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Let's say I had minus x squared minus 3x plus 28, let me

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say, is greater than 0.

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Well I want to get rid of this negative sign in

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front of the x squared.

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I just don't like it there because it makes it look

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more confusing to factor.

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I'm going to multiply everything by negative 1.

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Both sides.

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I get x squared plus 3x minus 28, and when you multiply or

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divide by a negative, with any inequality you have

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to swap the sign.

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So this is now going to be less than 0.

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And if we were to factor this, we get x plus 7 times x

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minus 4 is less than 0.

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So if this was equal to 0, we would know that the two roots

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of this function -- let's define the function f of x --

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let's define the function as f of x is equal to -- well we can

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define it as this or this because they're the same thing.

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But for simplicity let's define it as x plus 7 times x minus 4.

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That's f of x, right?

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Well, after factoring it, we know that the roots of this,

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the roots are x is equal to minus 7, and x is equal to 4.

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Now what we want to know is what x values make

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this inequality true?

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If this was any equality we'd be done.

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But we want to know what makes this inequality true.

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I'll give you a little bit of a trick, it's always going to be

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the numbers in between the two roots or outside

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of the two roots.

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So what I do whenever I'm doing this on a test or something, I

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just test numbers that are either between the roots or

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outside of the two roots.

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So let's pick a number that's between x equals minus

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7 and x equals 4.

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Well let's try x equals 0.

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Well, f of 0 is equal to -- we could do it right here -- f of

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0 is 0 plus 7 times 0 minus 4 is just 7 times minus

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4, which is minus 28.

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So f of 0 is minus 28.

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Now is this -- this is the function we're working with

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-- is this less than 0?

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Well yeah, it is.

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So it actually turns that a number, an x value between

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the two roots works.

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So actually I immediately know that the answer here

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is all of the x's that are between the two roots.

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So we could say that the solution to this is

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minus 7 is less than x which is less than 4.

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Because now the other way.

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You could have tried a number that's outside of the roots,

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either less than minus 7 or greater than 4 and

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have tried it out.

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Let's say if you had tried out 5.

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Try x equals 5.

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Well then f of 5 would be 12 times 1, right,

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which is equal to 12.

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f of 5 is 12.

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Is that less than 0?

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No.

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So that wouldn't have worked.

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So once again, that gives us a confidence that we

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got the right interval.

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And if we wanted to think about this visually, because we got

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this answer, when you do it visually it actually makes, I

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think, a lot of sense, but maybe I'm biased.

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If you look at it visually it looks like this.

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If you drive visually and this is the parabola, this is f of

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x, the roots here are minus 7, 0 and 4, 0, we're saying that

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for all x values between these two numbers, f of

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x is less than 0.

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And that makes sense, because when is f of x less than 0?

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Well this is the graph of f of x.

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And when is f of x less than 0?

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Right here.

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So what x values give us that?

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Well the x values that give us that are right here.

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I hope I'm not confusing you too much with

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these visual graphs.

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And you're probably saying, well how do I know

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I don't include 0?

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Well you could try it out, but if you -- oh, well how come

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I don't include the roots?

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Well at the roots, f of x is equal to 0.

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So if this was this, if this was less than or equal to 0,

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then the answer would be negative 7 is less than

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or equal to x is less than or equal to 4.

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I hope that gives you a sense.

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You pretty much just have to try number in between the

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roots, and try number outside of the roots, and that tells

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you what interval will make the inequality true.

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I'll see you in the next presentation.

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