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Studying for a test? Prepare with these 3 lessons on Module 2: Rational numbers.
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- [Voiceover] Let's give ourselves some practice substituting positive and negative values for variables. So we're told to evaluate X, we're told to evaluate X minus negative Y, where X is equal to negative two and Y is equal to five. So everywhere we see an X, we can replace with a negative two. Everywhere we see a Y, we replace with a five. So this is a Y right over here, and then of course, and then of course this is an X. Let's do that. So instead of that X, let's write negative two. So we have negative two minus and then we have in the parentheses a negative Y. Y is equal to, Y is equal to five. Now what is this going to evaluate to? Well, this is the same thing as negative two, now subtracting a negative five, so all of this business here, subtracting a negative five, that's the same thing as adding a five. So it's going to be negative two plus five, which is equal to three. This is going to be equal to three. And there's several ways to think about this. my brain thinks okay, I'm starting at negative two. If I add two I get to zero and then I would have to add another, and then I have to add another three so that gets me to three. You can even view this as, you can even view this, my brain kind of says negative two, well let's see, I have to add, i have to add two to get back to zero and then I have to add another three if I want to add a total of five. So this is going to be zero plus three gets me to three. Another way to think about it, negative two plus five is the same thing as five minus two. Five minus two, which is of course equal to three. Or you could of course draw it on a number line. And you would say if you start at negative two and you take five steps to the right, you get to positive three. Let's do another one of these. This one's a little bit more, a little bit more complex. So let's see. We're told to evaluate three minus negative six plus negative H plus negative four, where H is equal to negative seven. Well there's two ways you could do it. I could just take the negative seven and replace it, the H with that negative seven, or I could actually try to simplify this expression first and then do the substitution for H. Let's actually do that. My brain feels like doing that. So this expression, I have the three, but instead of subtracting a negative six, instead of subtracting a negative six, that's gonna be the same thing as just adding six. So three plus six. And adding a negative H, adding a negative H, adding a negative H, that's the same thing as just subtracting H. So three plus six minus H, and then adding a negative four, adding a negative four, that's the same thing as subtracting four. That's the same thing as subtracting four. And now of course we can do this in any, you know, addition and subtraction we have we can change the order in which we do it. So let's do that just to kind of simplify all of this. So, actually first of all, I can figure out what three plus six is. Three plus six is equal to nine. And then I have this minus four here. So I could say nine minus four, actually I want to be careful not to skip any steps. So three plus six, three plus six, in a color that you can see, so three plus six is nine. So that's nine minus H minus four, minus four. Now I could change the order in which I do this addition or subtraction, so this is going to be the equivalent of nine minus four. Nine minus four minus H, minus H. And I just did that so I can simplify and figure out what nine minus four is. Nine minus four, of course, let me do this in blue, navy blue. Nine minus four is five. So this whole thing simplified to five minus H before I even did the substitution. And now I can substitute H with negative seven. So this is going to be equal to, this is going to be equal to, when I do the substitution, I'll write it up here, it's going to be five minus, I'll do the minus in that magenta color, minus and now where I see an H, I'm gonna replace it with negative seven. Five minus negative seven. You want to be very careful there, you might be tempted to say, oh I have a negative here, negative here, let me just replace H with a seven. Remember, H is negative seven, so you're subtracting H. You're gonna subtract negative seven. So this is five minus negative seven, which is the same thing, which is the same thing as five plus seven. Five plus seven, which we all know is equal to 12. And we're done. Let's do a few more of these. You can't really get enough practice here, this is some important foundational skills for the rest of your mathematical lives. (laughing) Alright. So consider, I don't make you too stressed about it. Consider the following number line. Alright, so we've got a number line here and let's see, they didn't mark off all the numbers here. This is negative four, E is at this point, this is then we go to two. So it looks like we're counting by twos here, that this is negative two, this is zero, yep. Negative four, if you increase by two, negative four, negative two, zero, two, four, this would be a six. This would be a negative six. They intentionally left those numbers off, so we had to figure out that hey, look, between negative four and two, to go from negative four to two, you have to increase by six and we only have one, two, three hashmarks. So each of those hashmarks must be increasing by two. Well anyway, now we know, now we know what all the points in the number line are. Evaluate E minus F. Well we know, we know that E is equal to negative two. And we know, we know that F is equal to four. So this is going to be the same thing, E is equal to negative two minus, minus positive four. Minus positive four. Let me do it in that same blue color. Minus four. Well negative two minus four, we have the number line in front of us, this is just going to be negative six. If you start at negative two, you subtract, if you subtract two you get to negative four, you subtract another two, you get to negative six. So we are done. Let's do one more of these. Let's evaluate T plus negative U. Where once again T and U are on this number line and it looks like each of these hashmarks we're incrementing by three as we go up. Zero, three, six. And if we go down, zero, negative three, negative six. So it's clear that U is equal to three. U is equal to three and it is also clear that T is equal to negative six. T is equal to negative six. So T plus negative U is going to be, so it's going to be negative six plus, negative six plus negative U, U is three. U is three. You want to be very careful, you might say oh well U is positive so I'm just gonna put a positive number here, but remember, it's negative U. Wherever you see the U, replace it with the three. So it's negative three. So this is gonna be six plus negative three, or sorry, negative six plus negative three, which we can rewrite as negative six minus three. All of this business can be rewritten as negative six minus three. Negative six minus three. If you're at negative six and you go three to the left, you go three more negative, you're going to end up at negative nine. And we're all done.