Topic D: Expanding, factoring, and distributing expressions
- [Voiceover] What I wanna do in this video is think about how does the value of an expression change as a variable that makes up that expression changes? So, let's start with a pretty simple expression. Let's say we have the expression 100 minus x. We wanna know how does this expression change as x increases? As x increases and like always, try to work it out for yourself. Pause the video before I try to have a go at it. Well, there's a couple of ways you could think about it. You say, "Okay, I have 100 and I'm subtracting x." So, as x increases, as x increases over here, I'm going to be subtracting larger and larger values. So, if I'm subtracting larger and larger values, I'm going to get smaller and smaller values. So, this whole thing, this whole thing is going to decrease. This whole thing is going to decrease and if you wanna make it a little bit more concrete, you can actually try out some values there. So, you can make a little table here. X and then, what is 100 - x going to be? I'm just gonna pick some values of x. I'm gonna make them increase. Say, x is zero, x is 50, x is 100. Well, when an x is zero, 100 - x is 100 - zero. So, it's 100. When x is 50, it's going be 100 - 50, so it's going to be 50. When x is 100, it's 100 - 100, so it's zero. So, it's pretty clear here that as x is increasing, as x is increasing, I'll just write incr. for increasing, we see that 100 - x is decreasing. We see that that is decreasing. Let's do this with a couple more expressions that have different forms. So, let's say that I have the expression. Let's say that I have the expression five over x plus five and x is decreasing, x is decreasing, but we also know that it is positive and even while it's decreasing, it's staying above zero. So, we're saying x is staying greater than zero. So, in this case, so ya know, this could be a situation where x is decreasing from 10 to nine or a million to 100,000, but it's staying positive while it's decreasing. Let's think about that. We are going to be dividing by smaller and smaller positive values. So, as you have smaller and smaller positive values of the denominator, you're dividing by smaller and smaller positive values. So, if you're dividing by smaller positive values, you're going to know that this thing is going to get larger. This entire expression is going to get larger as you divide by smaller and smaller positive values. So, if that expression gets larger, then you're just adding five to it. The whole thing is going to increase. The whole thing is going to increase as, the whole thing is going to increase as x decreases while staying positive. And once again, we can make a little table to take a look at that. So, this is x, this is five over x plus five. Now, let's see. I'll just try out some. Let's see, I'll go from x 100 to x is five, to x is one. So, this is clearly x is decreasing. X is decreasing. When x is 100, you're gonna have five divided by 100, which would be five hundredths plus five, so it would be 5.05. When x is five, you're gonna have five divided by five, which is one plus five, which is six. When x is one, you're gonna have five divided by one, which is five plus five. You just, it's 10. So notice, when x is staying positive, but decreasing, the whole expression, five over x plus five, this thing right over here is increasing. Let's do one more of these. So, let's say that we have the expression and we'll change up the variable here. 3y over 2y and I'm curious what happens to the value of this expression as y increases, as y increases. Well, as you have larger y's here on the numerator, you're also going to have larger y's here in the denominator and one way you could think about this, this is the same thing as three halves times y over y. And no matter what y you have here, as long as it's not equal to zero and actually, let's just assume that it doesn't cross zero because at that point, at that point, this thing would become undefined. Let's just say that y increases and y is greater than zero just for simplicity. So, y is a positive value here and actually, for any non-zero value, you take y, you divide by y, you're just gonna get one. So, it doesn't matter what y is. Y could be a million over a million. This thing is just gonna be one. Y could be five over five, it's still just going to be one. The value of the expression is not going to change. It's just going to be three halves. So, the expression is just going to be the same. You're not going to have any change as y increases, as y is positive and it increases.