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What is a variable Introduction to algebra Algebra I Khan Academy.mp3 | From scenario to scenario, it can change. So for example, in one scenario, maybe it's lunchtime, I'm getting really big tips. So tips is equal to, let's say it's equal to $30. And so the total amount I might make in that hour is going, we can go back to this expression right over here, it's going to be 10 plus, instead... |
What is a variable Introduction to algebra Algebra I Khan Academy.mp3 | And so the total amount I might make in that hour is going, we can go back to this expression right over here, it's going to be 10 plus, instead of writing tips here, I'll write 30, because that's what my tips are in that hour. And so that is going to be equal to, it's going to be equal to 40. Let me do that, let me do... |
What is a variable Introduction to algebra Algebra I Khan Academy.mp3 | It's going to be equal to $40. But let's say right after that, the restaurant slows down, we're out of the lunch hour for whatever reason, maybe the restaurant next door has a big sale or something. And so the next hour, my tips go down dramatically. My tips go down to $5 for that hour. Now I go back to this expression... |
What is a variable Introduction to algebra Algebra I Khan Academy.mp3 | My tips go down to $5 for that hour. Now I go back to this expression. The total I make is my hourly wage plus the $5 in tips, plus the $5 in tips, which is equal to $15. As you see, this entire expression, the 10 plus tips, it changed depending on what the value of the variable tips is. Now, you won't see whole words ... |
What is a variable Introduction to algebra Algebra I Khan Academy.mp3 | As you see, this entire expression, the 10 plus tips, it changed depending on what the value of the variable tips is. Now, you won't see whole words typically used in algebra as variables. We get lazy, and so instead, we tend to use just easier to write symbols. And so in this context, instead of writing tips, maybe we... |
What is a variable Introduction to algebra Algebra I Khan Academy.mp3 | And so in this context, instead of writing tips, maybe we could have just written 10 plus t, where t represents the tips that we get in an hour. And so then we would say, okay, what happens when t is equal to 30? Well, if t is equal to 30, then we'd have, let me write, so what happens when t is equal to 30? Well, then ... |
What is a variable Introduction to algebra Algebra I Khan Academy.mp3 | Well, then we have a situation, t is equal to 30. This evaluates to 10 plus 30, which would be 40. What would happen if t is equal to five? Well, then this would evaluate to 10 plus five, which is equal to 15. Now I wanna be clear, we didn't even have to use t. We didn't even really have to use a letter, although in tr... |
What is a variable Introduction to algebra Algebra I Khan Academy.mp3 | Well, then this would evaluate to 10 plus five, which is equal to 15. Now I wanna be clear, we didn't even have to use t. We didn't even really have to use a letter, although in traditional algebra, you almost do use a letter. We could have written it as 10 plus x, where x is your tips per hour. X might not be as natur... |
How to evaluate an expression with variables Introduction to algebra Algebra I Khan Academy.mp3 | A local hospital is holding a raffle as a fundraiser. The individual cost of participating in the raffle is given by the following expression. 5t plus 3, or 5 times t plus 3, where t represents the number of tickets someone purchases. Evaluate the expression when t is equal to 1, t is equal to 8, and t is equal to 10. ... |
How to evaluate an expression with variables Introduction to algebra Algebra I Khan Academy.mp3 | Evaluate the expression when t is equal to 1, t is equal to 8, and t is equal to 10. So let's first take the situation where t is equal to 1. Then this expression right over here becomes, and I'll use that same color, becomes 5 times 1 plus 3. And we know from order of operations, you do the multiplication before you d... |
How to evaluate an expression with variables Introduction to algebra Algebra I Khan Academy.mp3 | And we know from order of operations, you do the multiplication before you do the addition. So this will be 5 times 1 is 5 plus 3. And then this is clearly equal to 8. Now let's do it when t is equal to 8. So when t is equal to 8, this expression becomes, and I'll do the same colors again, 5 times 8 plus 3. And once ag... |
How to evaluate an expression with variables Introduction to algebra Algebra I Khan Academy.mp3 | Now let's do it when t is equal to 8. So when t is equal to 8, this expression becomes, and I'll do the same colors again, 5 times 8 plus 3. And once again, 5 times 8 is 40. And then we have the plus 3 there. So this is equal to 43. And so we have the last situation. With t is equal to 10, I'll do that in blue. |
How to evaluate an expression with variables Introduction to algebra Algebra I Khan Academy.mp3 | And then we have the plus 3 there. So this is equal to 43. And so we have the last situation. With t is equal to 10, I'll do that in blue. So we have 5 times 10. So 5t is 5 times 10. Instead of a t, we put a 10 there. |
How to evaluate an expression with variables Introduction to algebra Algebra I Khan Academy.mp3 | With t is equal to 10, I'll do that in blue. So we have 5 times 10. So 5t is 5 times 10. Instead of a t, we put a 10 there. 5 times 10 plus 3. That's a slightly differentiated green, but I think you get the idea. 5 times 10 is 50. |
How to evaluate an expression with variables Introduction to algebra Algebra I Khan Academy.mp3 | Instead of a t, we put a 10 there. 5 times 10 plus 3. That's a slightly differentiated green, but I think you get the idea. 5 times 10 is 50. And then we're going to have to add 3 to that. And that is equal to 53. And we're done. |
Why a negative times a negative is a positive Pre-Algebra Khan Academy.mp3 | And you know how to add and subtract negative numbers. But now you are faced with a conundrum. What happens when you multiply negative numbers? Either when you multiply a positive number times a negative number, or when you multiply two negative numbers. So for example, you aren't quite sure what should happen if you w... |
Why a negative times a negative is a positive Pre-Algebra Khan Academy.mp3 | Either when you multiply a positive number times a negative number, or when you multiply two negative numbers. So for example, you aren't quite sure what should happen if you were to multiply, and I'm just picking two numbers where one is positive and one is negative. What would happen if you were to multiply five time... |
Why a negative times a negative is a positive Pre-Algebra Khan Academy.mp3 | You're not quite sure about this just yet. You're also not quite sure about what would happen if you multiply two negative numbers. So let's say negative two times negative six. This is also unclear to you. What you do know, because you are a mathematician, is however you define this, or whatever this should be, it sho... |
Why a negative times a negative is a positive Pre-Algebra Khan Academy.mp3 | This is also unclear to you. What you do know, because you are a mathematician, is however you define this, or whatever this should be, it should hopefully be consistent with all of the other properties of mathematics that you already know, and preferably all of the other properties of multiplication. That would make y... |
Why a negative times a negative is a positive Pre-Algebra Khan Academy.mp3 | And later we can think about other ways to get the intuition for what these might be and why it actually makes sense. But to make this consistent with the rest of the mathematics that you know, you go into a little bit of a thought experiment. You say, well what should five times three plus negative three be? Well you ... |
Why a negative times a negative is a positive Pre-Algebra Khan Academy.mp3 | Well you already have a philosophy of adding negative numbers or adding positive to negative numbers. You know that negative three is the opposite of three. If you add three to negative three, you're going to get zero. So this is going to be equal to five times zero, based on how you already thought about adding a nega... |
Why a negative times a negative is a positive Pre-Algebra Khan Academy.mp3 | So this is going to be equal to five times zero, based on how you already thought about adding a negative to a positive. And anything times zero is going to be zero. So this expression right over here should be zero. But on the other hand you say, well I want multiplying positive and negative numbers to be consistent w... |
Why a negative times a negative is a positive Pre-Algebra Khan Academy.mp3 | But on the other hand you say, well I want multiplying positive and negative numbers to be consistent with the distributive property. So I should be able to distribute this five. I should be able to distribute this five, and for math to be consistent, and math should be consistent, I should get the exact same answer. S... |
Why a negative times a negative is a positive Pre-Algebra Khan Academy.mp3 | So let's distribute this five. So if we get five times three, so five times three is, let me write it out. This is going to be five times three, and let me write it with a multiplication, not the dot. I'll write the x sign for multiplying. Five times three, so I distribute it there, plus five times negative three. And ... |
Why a negative times a negative is a positive Pre-Algebra Khan Academy.mp3 | I'll write the x sign for multiplying. Five times three, so I distribute it there, plus five times negative three. And I'll do that in yellow. Five times negative three. Five times negative three. And this whole thing we just said should be equal to zero. This should be equal to zero. |
Why a negative times a negative is a positive Pre-Algebra Khan Academy.mp3 | Five times negative three. Five times negative three. And this whole thing we just said should be equal to zero. This should be equal to zero. Well five times three, those are two positive numbers, we know what that should be. That is going to be 15. So now we get this thing. |
Why a negative times a negative is a positive Pre-Algebra Khan Academy.mp3 | This should be equal to zero. Well five times three, those are two positive numbers, we know what that should be. That is going to be 15. So now we get this thing. 15 plus whatever five times negative three is, plus whatever five times negative three is, needs to be equal to zero in order to be consistent with all of t... |
Why a negative times a negative is a positive Pre-Algebra Khan Academy.mp3 | So now we get this thing. 15 plus whatever five times negative three is, plus whatever five times negative three is, needs to be equal to zero in order to be consistent with all of the other mathematics that we know. Well what plus 15 is going to be equal to zero? Well the opposite of 15. In order for this to be true, ... |
Why a negative times a negative is a positive Pre-Algebra Khan Academy.mp3 | Well the opposite of 15. In order for this to be true, in order for this to be consistent with all of the other mathematics we know, this right over here needs to be equal to negative 15. And so you say five times negative three, in order to be consistent with all the other mathematics we know, needs to be equal to neg... |
Why a negative times a negative is a positive Pre-Algebra Khan Academy.mp3 | And that's also consistent with the intuition of adding negative three repeatedly five times. Now a slightly harder to conceive idea is multiplying two negatives. But we can do the exact same thought experiment. We want whatever this answer to be to be consistent with the rest of mathematics that we know. So we can say... |
Why a negative times a negative is a positive Pre-Algebra Khan Academy.mp3 | We want whatever this answer to be to be consistent with the rest of mathematics that we know. So we can say, so we can do the same thought experiment. What would negative two times six plus negative six be equal to? Well six plus negative six is going to be zero. Negative two times zero, anything times zero needs to b... |
Why a negative times a negative is a positive Pre-Algebra Khan Academy.mp3 | Well six plus negative six is going to be zero. Negative two times zero, anything times zero needs to be equal to zero. But then once again we can distribute. We can distribute negative two times six. So we get negative two, negative two times six plus negative two times negative six. Plus negative two times negative s... |
Why a negative times a negative is a positive Pre-Algebra Khan Academy.mp3 | We can distribute negative two times six. So we get negative two, negative two times six plus negative two times negative six. Plus negative two times negative six. And once again all of that's going to need to be equal to zero. Now based on the thought experiment we just did, we said well this needs to be equal to neg... |
Why a negative times a negative is a positive Pre-Algebra Khan Academy.mp3 | And once again all of that's going to need to be equal to zero. Now based on the thought experiment we just did, we said well this needs to be equal to negative 12. Or we could view this as going to the six twice in the left direction on the number line which would get us to negative 12. Or you could say repeatedly add... |
Why a negative times a negative is a positive Pre-Algebra Khan Academy.mp3 | Or you could say repeatedly adding negative two six times. That would also get you to negative 12. And now we also saw it over here that we multiplied a positive times a negative. We got the negative. So this could be, or we know that this is going to be equal to negative 12. And so we had negative 12 plus whatever thi... |
Why a negative times a negative is a positive Pre-Algebra Khan Academy.mp3 | We got the negative. So this could be, or we know that this is going to be equal to negative 12. And so we had negative 12 plus whatever this business is, whatever this business is, is going to have to be equal to zero, is going to have to be equal to zero in order to be consistent with all of the other mathematics tha... |
Example 3 Factoring quadratics as a perfect square of a sum (a+b)^2 Algebra I Khan Academy.mp3 | And a good place to start is to say, well, are there any common factors for all of these terms? And when you look at them, well, these first two are divisible by 4, these last two are divisible by 3, but not all of them are divisible by any one number. Well, but you could factor out a negative 1. But even if you factor... |
Example 3 Factoring quadratics as a perfect square of a sum (a+b)^2 Algebra I Khan Academy.mp3 | But even if you factor out a negative 1, so you say this is the same thing as negative 1, times positive 4t squared plus 12t plus 9, you still end up with a non-1 coefficient out here and on the second degree term, on the t squared term. So you might want to immediately start grouping this. And if you did factor it by ... |
Example 3 Factoring quadratics as a perfect square of a sum (a+b)^2 Algebra I Khan Academy.mp3 | You would get the right answer. But there is something you might be able to see, or there is something about this equation that might pop out at you that might make it a little bit simpler to solve. And to understand that, let's take a little bit of a break here on the right-hand side and just think about what happens ... |
Example 3 Factoring quadratics as a perfect square of a sum (a+b)^2 Algebra I Khan Academy.mp3 | Well, you have a times a, which is a squared. Then you have a times that b, which is plus ab. Then you have b times a, which is the same thing as ab. And then you have b times b, or you have b squared. And so if you add these middle two terms right here, you're left with a squared plus 2ab plus b squared. This is the s... |
Example 3 Factoring quadratics as a perfect square of a sum (a+b)^2 Algebra I Khan Academy.mp3 | And then you have b times b, or you have b squared. And so if you add these middle two terms right here, you're left with a squared plus 2ab plus b squared. This is the square of a binomial. Now, does this right here, does 4t squared plus 12t plus 9 fit this pattern? Well, if 4t squared is a squared, so if this right h... |
Example 3 Factoring quadratics as a perfect square of a sum (a+b)^2 Algebra I Khan Academy.mp3 | Now, does this right here, does 4t squared plus 12t plus 9 fit this pattern? Well, if 4t squared is a squared, so if this right here is a squared, if that is a squared right there, then what does a have to be? If this is a squared, then a would be equal to the square root of this. It would be 2t. And if this is b squar... |
Example 3 Factoring quadratics as a perfect square of a sum (a+b)^2 Algebra I Khan Academy.mp3 | It would be 2t. And if this is b squared, let me do that in a different color. If this right here is b squared, if the 9 is b squared right there, then that means that b is equal to 3, equal to the positive square root of the 9. Now, this number right here, and actually it doesn't have to just be equal to 3. It might h... |
Example 3 Factoring quadratics as a perfect square of a sum (a+b)^2 Algebra I Khan Academy.mp3 | Now, this number right here, and actually it doesn't have to just be equal to 3. It might have been negative 3 as well. It could be plus or minus 3. But this number here, is it 2 times ab? That's the middle term that we care about. Is it 2 times ab? Well, if we multiply 2t times 3, we get 6t. |
Example 3 Factoring quadratics as a perfect square of a sum (a+b)^2 Algebra I Khan Academy.mp3 | But this number here, is it 2 times ab? That's the middle term that we care about. Is it 2 times ab? Well, if we multiply 2t times 3, we get 6t. And then we multiply that times 2, you get 12t. This right here, 12t, is equal to 2 times 2t times 3. It is 2 times ab. |
Example 3 Factoring quadratics as a perfect square of a sum (a+b)^2 Algebra I Khan Academy.mp3 | Well, if we multiply 2t times 3, we get 6t. And then we multiply that times 2, you get 12t. This right here, 12t, is equal to 2 times 2t times 3. It is 2 times ab. And if this was a negative 3, we would look to see if this was a negative 12, but this does work for a positive 3. So this does fit the pattern of a perfect... |
Example 3 Factoring quadratics as a perfect square of a sum (a+b)^2 Algebra I Khan Academy.mp3 | It is 2 times ab. And if this was a negative 3, we would look to see if this was a negative 12, but this does work for a positive 3. So this does fit the pattern of a perfect square. This is a special type of, or you could view this as a square of a binomial. So if you wanted to factor this, the stuff on the inside, yo... |
Example 3 Factoring quadratics as a perfect square of a sum (a+b)^2 Algebra I Khan Academy.mp3 | This is a special type of, or you could view this as a square of a binomial. So if you wanted to factor this, the stuff on the inside, you still have that negative 1 out there, the 4t squared plus 12t plus 9, you could immediately say, well that's going to be a plus b times a plus b. Or 2t plus 3 times 2t plus 3. Or yo... |
Example 3 Factoring quadratics as a perfect square of a sum (a+b)^2 Algebra I Khan Academy.mp3 | Or you could just say it's 2t plus 3 squared. It fits this pattern. And of course, you can't forget about this negative 1 out here. You could have also solved it by grouping, but this might be a quicker thing to recognize. This is a number squared. That's another number squared. If you take each of those numbers that y... |
Talking bird solves systems with substitution Algebra II Khan Academy.mp3 | And this makes you a little bit self-conscious, a little bit insecure, so you tell the king, what is the bird talking about? And the king says, well, the bird says that he thinks that there's another way to do the problem. And you're not used to taking advice from birds, and so being a little bit defensive, you say, we... |
Talking bird solves systems with substitution Algebra II Khan Academy.mp3 | And so the bird whispers a little bit more in the king's ear and says, okay, well, I'll have to do the writing because the bird does not have any hands, or at least can't manipulate chalk. And so the king, the bird continues to whisper in the king's ear, and the king translates and says, well, the bird says, let's use ... |
Talking bird solves systems with substitution Algebra II Khan Academy.mp3 | And that's essentially going to be a constraint of one variable in terms of another. So let's see if we can do that. So here, if we want to solve for m, we could subtract 400w from both sides, and we would have 100m is, if we subtract 400w from the left, this 400w goes away. If we subtract 400w from the right, we have ... |
Talking bird solves systems with substitution Algebra II Khan Academy.mp3 | If we subtract 400w from the right, we have is equal to negative 400w plus 1,100. So what got us from here to here is just subtracting 400w from both sides. And if we want to solve for m, we just divide both sides by 100. So we just divide all of the terms by 100, and then we get m is equal to, m is equal to negative 4... |
Talking bird solves systems with substitution Algebra II Khan Academy.mp3 | So we just divide all of the terms by 100, and then we get m is equal to, m is equal to negative 400 divided by 100 is negative 4w. 1,100 divided by 100 is 11, plus 11. So now we've constrained m in terms of w, and this is what the bird is saying, using the king as his translator. Why don't we take this constraint and ... |
Talking bird solves systems with substitution Algebra II Khan Academy.mp3 | Why don't we take this constraint and substitute it back for m in the first equation, and then we will have one equation with one unknown. And so the king starts to write at the bird's direction, 200, 200, so he's looking at that first equation now. He says 200, 200, and instead of putting an m there, the bird says, we... |
Talking bird solves systems with substitution Algebra II Khan Academy.mp3 | So instead of writing an m, we substitute the value for m, we substitute for m the expression negative 4w plus 11. So negative 4w plus 11, and then we have the rest of it, plus 300w, plus 300w is equal to 1,200. So just to be clear, everywhere we saw an m, we replaced it with this right over here in that first equation... |
Talking bird solves systems with substitution Algebra II Khan Academy.mp3 | So the first thing, you start to scratch your head, and say, is this a legitimate thing to do? Will I get the same answer as I got when I solved the same problem with elimination? And I want you to sit and think about that for a second. But then the bird starts whispering in the king's ear, and the king just progresses... |
Talking bird solves systems with substitution Algebra II Khan Academy.mp3 | But then the bird starts whispering in the king's ear, and the king just progresses to just work through the algebra. This is one equation with one unknown now. So the first step would be to distribute the 200. So 200 times negative 4w is negative 800w. 200 times 11 is 2,200, plus 2,200. And then we have the plus 300w,... |
Talking bird solves systems with substitution Algebra II Khan Academy.mp3 | So 200 times negative 4w is negative 800w. 200 times 11 is 2,200, plus 2,200. And then we have the plus 300w, plus 300w is equal to positive 1,200. Now we just need to solve for w. We first might wanna group this negative 800w with this 300w. Negative 800 of something plus 300 of something is going to be negative 500w.... |
Talking bird solves systems with substitution Algebra II Khan Academy.mp3 | Now we just need to solve for w. We first might wanna group this negative 800w with this 300w. Negative 800 of something plus 300 of something is going to be negative 500w. And then we still have this plus 2,200. Plus 2,200 is equal to 1,200. Now to solve for w, we'd wanna subtract 2,200 from both sides so subtract 2,2... |
Talking bird solves systems with substitution Algebra II Khan Academy.mp3 | Plus 2,200 is equal to 1,200. Now to solve for w, we'd wanna subtract 2,200 from both sides so subtract 2,200, subtract 2,200. On the left-hand side, you're left just with the negative 500w, negative 500w. And on the right-hand side, you are left with negative 1,000. And this is starting to look interesting because if ... |
Talking bird solves systems with substitution Algebra II Khan Academy.mp3 | And on the right-hand side, you are left with negative 1,000. And this is starting to look interesting because if we divide both sides by negative 500, we get w is equal to 2, which is the exact same answer that we got when we tried to solve, when we tried to figure out how many bags of chips each woman would eat on av... |
Solving a two-step inequality Linear inequalities Algebra I Khan Academy.mp3 | They're easy to think about, oh it's a little bit more than 8. Let's convert this to a improper fraction. So 8 and 1 third is equal to, the denominator is going to be 3. 3 times 8 is 24, plus 1 is 25. So this thing over here is the same thing as 25 over 3. Let me just rewrite the whole thing. So it's 2 thirds is greate... |
Solving a two-step inequality Linear inequalities Algebra I Khan Academy.mp3 | 3 times 8 is 24, plus 1 is 25. So this thing over here is the same thing as 25 over 3. Let me just rewrite the whole thing. So it's 2 thirds is greater than negative 4y minus 25 over 3. Now the next thing I want to do, just because dealing with fractions are a bit of a pain, is multiply both sides of this inequality by... |
Solving a two-step inequality Linear inequalities Algebra I Khan Academy.mp3 | So it's 2 thirds is greater than negative 4y minus 25 over 3. Now the next thing I want to do, just because dealing with fractions are a bit of a pain, is multiply both sides of this inequality by some quantity that will eliminate the fractions. And the easiest one I can think of is multiply both sides by 3. That will ... |
Solving a two-step inequality Linear inequalities Algebra I Khan Academy.mp3 | That will get rid of the 3's in the denominator. So let's multiply both sides of this equation by 3. That's the left hand side. And then I'm going to multiply the right hand side. 3, I'll put in parentheses like that. And, well one point that I want to point out is that I did not have to swap the inequality sign becaus... |
Solving a two-step inequality Linear inequalities Algebra I Khan Academy.mp3 | And then I'm going to multiply the right hand side. 3, I'll put in parentheses like that. And, well one point that I want to point out is that I did not have to swap the inequality sign because I multiplied both sides by a positive number. If the 3 was a negative number, if I multiplied both sides by negative 3 or nega... |
Solving a two-step inequality Linear inequalities Algebra I Khan Academy.mp3 | If the 3 was a negative number, if I multiplied both sides by negative 3 or negative 1 or negative whatever, I would have had to swap the inequality sign. Anyway, let's simplify this. So the left hand side, we have 3 times 2 thirds, which is just 2. 2 is greater than, and then we can distribute this 3. 3 times negative... |
Solving a two-step inequality Linear inequalities Algebra I Khan Academy.mp3 | 2 is greater than, and then we can distribute this 3. 3 times negative 4y is negative 12y. And then 3 times negative 25 over 3 is just negative 25. Now, we want to get all of our constant terms on one side of the inequality and all of our variable terms. The only variable here is y on the other side. The y is already s... |
Solving a two-step inequality Linear inequalities Algebra I Khan Academy.mp3 | Now, we want to get all of our constant terms on one side of the inequality and all of our variable terms. The only variable here is y on the other side. The y is already sitting here, so let's guess it's 25 on the other side of the inequality. And we can do that by adding 25 to both sides of this equation. So let's ad... |
Solving a two-step inequality Linear inequalities Algebra I Khan Academy.mp3 | And we can do that by adding 25 to both sides of this equation. So let's add 25 to both sides of this equation. Adding 25. And the left hand side, 2 plus 25 is 27. And we're going to get 27 is greater than. The right hand side of the inequality is negative 12y. And the negative 25 plus 25, those cancel out. |
Solving a two-step inequality Linear inequalities Algebra I Khan Academy.mp3 | And the left hand side, 2 plus 25 is 27. And we're going to get 27 is greater than. The right hand side of the inequality is negative 12y. And the negative 25 plus 25, those cancel out. That was the whole point. So we're left with 27 is greater than negative 12y. Now, to isolate the y, we can multiply, or you can eithe... |
Solving a two-step inequality Linear inequalities Algebra I Khan Academy.mp3 | And the negative 25 plus 25, those cancel out. That was the whole point. So we're left with 27 is greater than negative 12y. Now, to isolate the y, we can multiply, or you can either multiply both sides by negative 1 twelfth, or you could say let's just divide both sides by negative 12. Now, because I'm multiplying or ... |
Solving a two-step inequality Linear inequalities Algebra I Khan Academy.mp3 | Now, to isolate the y, we can multiply, or you can either multiply both sides by negative 1 twelfth, or you could say let's just divide both sides by negative 12. Now, because I'm multiplying or dividing by a negative number here, I'm going to need to swap the inequality. So let me write this. If I divide both sides of... |
Solving a two-step inequality Linear inequalities Algebra I Khan Academy.mp3 | If I divide both sides of this equation by negative 12, then it becomes 27 over negative 12 is less than. I'm swapping the inequality. Let me do this in a different color. Is less than negative 12y over negative 12. Notice, when I divide both sides of the inequality by a negative number, I swap the inequality. I swap t... |
Solving a two-step inequality Linear inequalities Algebra I Khan Academy.mp3 | Is less than negative 12y over negative 12. Notice, when I divide both sides of the inequality by a negative number, I swap the inequality. I swap the greater than becomes a less than. When it was positive, I didn't have to swap it. So 27 divided by negative 12, well, they're both divisible by 3. So we're going to get,... |
Solving a two-step inequality Linear inequalities Algebra I Khan Academy.mp3 | When it was positive, I didn't have to swap it. So 27 divided by negative 12, well, they're both divisible by 3. So we're going to get, if we do the numerator and the denominator by 3, we get negative 9 over 4 is less than. These cancel out. y. So y is greater than negative 9 fourths, or negative 9 fourths is less than... |
Solving a two-step inequality Linear inequalities Algebra I Khan Academy.mp3 | These cancel out. y. So y is greater than negative 9 fourths, or negative 9 fourths is less than y. And if you wanted to write that, let me write this. So our answer is y is greater than negative 9 fourths. I just swapped the order. You can say negative 9 fourths is less than y. |
Solving a two-step inequality Linear inequalities Algebra I Khan Academy.mp3 | And if you wanted to write that, let me write this. So our answer is y is greater than negative 9 fourths. I just swapped the order. You can say negative 9 fourths is less than y. Or if you want to visualize that a little bit better, 9 fourths is 2 and 1 fourth. So we could also say y is greater than negative 2 and 1 f... |
Solving a two-step inequality Linear inequalities Algebra I Khan Academy.mp3 | You can say negative 9 fourths is less than y. Or if you want to visualize that a little bit better, 9 fourths is 2 and 1 fourth. So we could also say y is greater than negative 2 and 1 fourth if we want to put it as a mixed number. And if we wanted to graph it on the number line, let me draw a number line right here, ... |
Solving a two-step inequality Linear inequalities Algebra I Khan Academy.mp3 | And if we wanted to graph it on the number line, let me draw a number line right here, a real simple one. Maybe this is 0. Negative 2 is right over, let's say negative 1. Negative 2. Then say negative 3 is right there. Negative 2 and 1 fourth is going to be right here. And it's greater than. |
Solving a two-step inequality Linear inequalities Algebra I Khan Academy.mp3 | Negative 2. Then say negative 3 is right there. Negative 2 and 1 fourth is going to be right here. And it's greater than. So we're not going to include that in the solution set. So we're going to make an open circle right there. And everything larger than that is a valid y, is a y that will satisfy the inequality. |
Why a negative times a negative makes intuitive sense Pre-Algebra Khan Academy.mp3 | You want to have a slightly deeper intuition than just having to accept it so that it's consistent with the distributive property and whatever else. And so you try another thought experiment. You say, well, what is just basic multiplication doing? So if I say two times, if I say two times three, one way to conceptualiz... |
Why a negative times a negative makes intuitive sense Pre-Algebra Khan Academy.mp3 | So if I say two times, if I say two times three, one way to conceptualize this basic multiplication here is really repeated addition. So you could view this as two threes, so two threes would literally be three plus three, and notice there are two of them, there are two of these, or you could view this as three twos. A... |
Why a negative times a negative makes intuitive sense Pre-Algebra Khan Academy.mp3 | And either way you conceptualize them, you get the same exact answer. This is going to be equal to six. Fair enough, you knew that before you even tried to tackle negative numbers. Now let's try to make one of these negatives and see what, one of these negative and see what happens. Let's do two, two times, two times n... |
Why a negative times a negative makes intuitive sense Pre-Algebra Khan Academy.mp3 | Now let's try to make one of these negatives and see what, one of these negative and see what happens. Let's do two, two times, two times negative three. And I'm going to make the negative in a different color. Two times negative, two times negative three. Well, one way you could view this is just the same analogy here... |
Why a negative times a negative makes intuitive sense Pre-Algebra Khan Academy.mp3 | Two times negative, two times negative three. Well, one way you could view this is just the same analogy here. It's negative three twice. So it would be, it would be negative three, and I'll try to color code it, negative three, and then another negative, another negative three, or you could say negative three minus th... |
Why a negative times a negative makes intuitive sense Pre-Algebra Khan Academy.mp3 | So it would be, it would be negative three, and I'll try to color code it, negative three, and then another negative, another negative three, or you could say negative three minus three. Or, and now this is the interesting thing, instead of over here, since you had two times positive three, you added two three times, b... |
Why a negative times a negative makes intuitive sense Pre-Algebra Khan Academy.mp3 | So this would be subtracting two, subtracting two, mind another, subtract another two right over here, subtract another two, and then you subtract another two, another two right over there. Let me color code it, make sure I don't get the colors messed up. And you have another two right over there. And notice, you did i... |
Why a negative times a negative makes intuitive sense Pre-Algebra Khan Academy.mp3 | And notice, you did it, once again, you did it three times. So this was a negative three, essentially you are subtracting two three times. And either way you conceptualize it right over here, you are going to get negative six. You are going to get negative six as the answer. Now, so you're already starting to feel good... |
Why a negative times a negative makes intuitive sense Pre-Algebra Khan Academy.mp3 | You are going to get negative six as the answer. Now, so you're already starting to feel good about this part right over here, negative times a positive, negative times a positive, or a positive times a negative, give you a negative. Now let's take to the really unintuitive one. A negative times a negative, all of a su... |
Why a negative times a negative makes intuitive sense Pre-Algebra Khan Academy.mp3 | A negative times a negative, all of a sudden the negatives kind of cancel out and give you a positive. Why is that the case? Well, we can just build from this example right over here. Let's say that we had negative two, let's say that we had negative two, let me do it in a different color. Let's say that we had negativ... |
Why a negative times a negative makes intuitive sense Pre-Algebra Khan Academy.mp3 | Let's say that we had negative two, let's say that we had negative two, let me do it in a different color. Let's say that we had negative two, I already used that color, negative two times negative three, times negative three. So now we can do it a couple of ways. Actually, I'll do this one first. We're still multiplyi... |
Why a negative times a negative makes intuitive sense Pre-Algebra Khan Academy.mp3 | Actually, I'll do this one first. We're still multiplying something by negative three, so we're going to repeatedly subtract that thing three times, whatever that thing is. But now that thing isn't a positive two, that thing that we're going to subtract three times is a negative two. So let me make it clear. This says ... |
Why a negative times a negative makes intuitive sense Pre-Algebra Khan Academy.mp3 | So let me make it clear. This says we're going to subtract something three times. We're going to subtract something three times. So subtracting something, subtracting something, subtracting something three times. That's what this part right over here tells us. And we're going to do this, we're going to do it exactly th... |
Why a negative times a negative makes intuitive sense Pre-Algebra Khan Academy.mp3 | So subtracting something, subtracting something, subtracting something three times. That's what this part right over here tells us. And we're going to do this, we're going to do it exactly three times. Over here was a positive two that we subtracted three times. Now we're going to do a negative two. Now we're going to ... |
Why a negative times a negative makes intuitive sense Pre-Algebra Khan Academy.mp3 | Over here was a positive two that we subtracted three times. Now we're going to do a negative two. Now we're going to do a negative two. And we know from subtracting negative numbers, we've already built this intuition that subtracting a negative is the same thing, it's like taking away someone's debt. It's the same th... |
Why a negative times a negative makes intuitive sense Pre-Algebra Khan Academy.mp3 | And we know from subtracting negative numbers, we've already built this intuition that subtracting a negative is the same thing, it's like taking away someone's debt. It's the same thing as adding a positive. And so this is going to be the same thing as two plus two plus two, which will once again give you positive six... |
Why a negative times a negative makes intuitive sense Pre-Algebra Khan Academy.mp3 | You can use the same logic over here. Now instead of adding negative three twice, and really I could have written this as negative three in this example, negative three, let me write it this way, negative three, negative three, and we added it, we added it, and I'll write a plus out here to make it clear. Over here we ... |
Why a negative times a negative makes intuitive sense Pre-Algebra Khan Academy.mp3 | We added negative three two times. Over here, since we now have a negative two, we're going to subtract negative three twice. So we're going to subtract something, and we're going to subtract something again, and that something is going to be our negative three. It's going to be our negative three. So negative, negativ... |
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