Transcription
Welcome to the presentation on quadratic inequalities. Before we get to quadratic inequalities, let's just start graphing some functions and interpret them; then we'll slowly move to the inequalities.
Let's say I had f(x) = x² + x - 6. Well, if we wanted to figure out where this function intersects the x-axis, or the roots of it, we learned in our factoring quadratics that we could just set f(x) = 0, right? Because f(x) = 0 when you're intersecting the x-axis. So you would say x² + x - 6 = 0. And you just factor this quadratic: (x + 3)(x - 2) = 0. And you would learn that the roots of this quadratic function are x = -3, and x = 2.
How would we visualize this? Well, let's draw this quadratic function. Those are my very uneven lines. So the roots are x = -3. So this is, right here, x is at -3, y = 0 -- by definition, one of the roots is where f(x) = 0. So the y, or the f(x) axis here is 0. The coordinate is ( -3, 0). And this point here is (2, 0). Once again, this is the x-axis, and this is the f(x)-axis. We also know that the y-intercept is -6. This isn't the vertex; this is the y-intercept. And that the graph is going to look something like this -- not as bumpy as what I'm drawing, which I think you get the general idea if you've ever seen a clean parabola. It looks like that with x = -3 here, and x = 2 here.
Pretty straightforward. We figured out the roots; we figured out what it looks like. Now what if we, instead of wanting to know where f(x) = 0, which is these two points, what if we wanted to know where f(x) > 0? What x values make f(x) > 0? Or another way of saying it, what values make the statement true: x² + x - 6 > 0? Right, this is just f(x). Well, if we look at the graph, when is f(x) > 0? Well, this is the f(x) axis, and when are we in positive territory? Well, f(x) > 0 here -- let me draw that another color -- is greater than 0 here, right? Because it's above the x-axis. And f(x) > 0 here. So just visually looking at it, what x values make this true? Well, this is true whenever x < -3, right, or whenever x > 2. Because when x > 2, f(x) > 0, and when x < -3, f(x) > 0. So we would say the solution to this quadratic inequality, and we pretty much solved this visually, is x < -3, or x > 2. And you could test it out. You could try out the number -4, and you should get f(x) being greater than 0. You could try it out here. Or you could try the number 3 and make sure that this works. And you can just make sure that, you could, for example, try out the number 0 and make sure that 0 doesn't work, right, because 0 is between the two roots. It actually turns out that when x = 0, f(x) = -6, which is definitely less than 0. So I think this will give you a visual intuition of what this quadratic inequality means.
Now with that visual intuition in the back of your mind, let's do some more problems, and maybe we won't have to go through the exercise of drawing it, but maybe I will draw it just to make sure that the point hits home. Let me give you a slightly trickier problem. Let's say I had -x² - 3x + 28 > 0. Well, I want to get rid of this negative sign in front of the x². I just don't like it there because it makes it look more confusing to factor. I'm going to multiply everything by -1. Both sides. I get x² + 3x - 28, and when you multiply or divide by a negative, with any inequality you have to swap the sign. So this is now going to be < 0. And if we were to factor this, we get (x + 7)(x - 4) < 0. So if this was equal to 0, we would know that the two roots of this function -- let's define the function f(x) -- let's define the function as f(x) = -- well we can define it as this or this because they're the same thing. But for simplicity let's define it as (x + 7)(x - 4). That's f(x), right? Well, after factoring it, we know that the roots of this, the roots are x = -7, and x = 4. Now what we want to know is what x values make this inequality true? If this was any equality we'd be done. But we want to know what makes this inequality true. I'll give you a little bit of a trick; it's always going to be the numbers in between the two roots or outside of the two roots. So what I do whenever I'm doing this on a test or something, I just test numbers that are either between the roots or outside of the two roots. So let's pick a number that's between x = -7 and x = 4. Well, let's try x = 0. Well, f(0) = -- we could do it right here -- f(0) = (0 + 7)(0 - 4) = 7 * -4, which is -28. So f(0) = -28. Now is this -- this is the function we're working with -- is this < 0? Well, yeah, it is. So it actually turns that a number, an x value between the two roots works. So actually I immediately know that the answer here is all of the x's that are between the two roots. So we could say that the solution to this is -7 < x < 4. Because now the other way. You could have tried a number that's outside of the roots, either less than -7 or greater than 4 and have tried it out. Let's say if you had tried out 5. Try x = 5. Well then f(5) would be (12)(1), right, which is equal to 12. f(5) = 12. Is that < 0? No. So that wouldn't have worked. So once again, that gives us a confidence that we got the right interval. And if we wanted to think about this visually, because we got this answer, when you do it visually it actually makes, I think, a lot of sense, but maybe I'm biased. If you look at it visually it looks like this. If you draw visually and this is the parabola, this is f(x), the roots here are (-7, 0) and (4, 0), we're saying that for all x values between these two numbers, f(x) < 0. And that makes sense, because when is f(x) < 0? Well, this is the graph of f(x). And when is f(x) < 0? Right here. So what x values give us that? Well, the x values that give us that are right here. I hope I'm not confusing you too much with these visual graphs. And you're probably saying, well, how do I know I don't include 0? Well, you could try it out, but if you -- oh, well, how come I don't include the roots? Well, at the roots, f(x) = 0. So if this was this, if this was ≤ 0, then the answer would be -7 ≤ x ≤ 4. I hope that gives you a sense. You pretty much just have to try a number in between the roots, and try a number outside of the roots, and that tells you what interval will make the inequality true. I'll see you in the next presentation.