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Example of adding fractions with unlike denominators word problem Pre-Algebra Khan Academy.mp3 | So let's think about that. We're going to add the 2 5ths of a gallon of red paint, and we're going to add that to half a gallon of yellow paint. And we want to see if this gets to being one whole gallon. So whenever we add fractions, right over here, we're not adding the same thing. Here we're adding 2 5ths. Here we're adding 1 halves. So in order to be able to add these two things, we need to get to a common denominator. |
Example of adding fractions with unlike denominators word problem Pre-Algebra Khan Academy.mp3 | So whenever we add fractions, right over here, we're not adding the same thing. Here we're adding 2 5ths. Here we're adding 1 halves. So in order to be able to add these two things, we need to get to a common denominator. And the common denominator, or the best common denominator to use, is the number that is the smallest multiple of both 5 and 2. And since 5 and 2 are both prime numbers, the smallest number is just going to be their product. 10 is the smallest number that we can think of that is divisible by both 5 and 2. |
Example of adding fractions with unlike denominators word problem Pre-Algebra Khan Academy.mp3 | So in order to be able to add these two things, we need to get to a common denominator. And the common denominator, or the best common denominator to use, is the number that is the smallest multiple of both 5 and 2. And since 5 and 2 are both prime numbers, the smallest number is just going to be their product. 10 is the smallest number that we can think of that is divisible by both 5 and 2. So let's rewrite each of these fractions with 10 as a denominator. So 2 5ths is going to be something over 10, and 1 half is going to be something over 10. And to help us visualize this, let me draw a grid with tenths in it. |
Example of adding fractions with unlike denominators word problem Pre-Algebra Khan Academy.mp3 | 10 is the smallest number that we can think of that is divisible by both 5 and 2. So let's rewrite each of these fractions with 10 as a denominator. So 2 5ths is going to be something over 10, and 1 half is going to be something over 10. And to help us visualize this, let me draw a grid with tenths in it. So that's that, and that's that right over here. So each of these are in tenths. These are 10 equal segments this bar is divided into. |
Example of adding fractions with unlike denominators word problem Pre-Algebra Khan Academy.mp3 | And to help us visualize this, let me draw a grid with tenths in it. So that's that, and that's that right over here. So each of these are in tenths. These are 10 equal segments this bar is divided into. So let's try to visualize what 2 5ths looks like on this bar. Well, right now it's divided into tenths. If we were to divide this bar into fifths, then we're going to have 1. |
Example of adding fractions with unlike denominators word problem Pre-Algebra Khan Academy.mp3 | These are 10 equal segments this bar is divided into. So let's try to visualize what 2 5ths looks like on this bar. Well, right now it's divided into tenths. If we were to divide this bar into fifths, then we're going to have 1. Actually, let me do it in that same color. So it's going to be 1 division, 2, 3, 4. So notice, if you go between the red marks, these are each a fifth of the bar. |
Example of adding fractions with unlike denominators word problem Pre-Algebra Khan Academy.mp3 | If we were to divide this bar into fifths, then we're going to have 1. Actually, let me do it in that same color. So it's going to be 1 division, 2, 3, 4. So notice, if you go between the red marks, these are each a fifth of the bar. If you go between the red marks, and we have two of them. So we're going to go 1 and 2. This right over here, this part of the bar represents 2 5ths of it. |
Example of adding fractions with unlike denominators word problem Pre-Algebra Khan Academy.mp3 | So notice, if you go between the red marks, these are each a fifth of the bar. If you go between the red marks, and we have two of them. So we're going to go 1 and 2. This right over here, this part of the bar represents 2 5ths of it. Now let's do the same thing for 1 half. So let's divide this bar exactly in half. So let me do that. |
Example of adding fractions with unlike denominators word problem Pre-Algebra Khan Academy.mp3 | This right over here, this part of the bar represents 2 5ths of it. Now let's do the same thing for 1 half. So let's divide this bar exactly in half. So let me do that. So I'm going to divide it exactly in half. And 1 half literally represents one of the two equal sections. So this is 1 1 half. |
Example of adding fractions with unlike denominators word problem Pre-Algebra Khan Academy.mp3 | So let me do that. So I'm going to divide it exactly in half. And 1 half literally represents one of the two equal sections. So this is 1 1 half. Now, to go from fifths to tenths, you're essentially taking each of the equal sections and you are multiplying by 2. So to go from fifths to tenths, you're multiplying by 2. You have 5 equal sections. |
Example of adding fractions with unlike denominators word problem Pre-Algebra Khan Academy.mp3 | So this is 1 1 half. Now, to go from fifths to tenths, you're essentially taking each of the equal sections and you are multiplying by 2. So to go from fifths to tenths, you're multiplying by 2. You have 5 equal sections. You split each of those into 2, so you have twice as many. You now have 10 equal sections. So those two sections that were shaded in, well, you were going to multiply by 2 the same way. |
Example of adding fractions with unlike denominators word problem Pre-Algebra Khan Academy.mp3 | You have 5 equal sections. You split each of those into 2, so you have twice as many. You now have 10 equal sections. So those two sections that were shaded in, well, you were going to multiply by 2 the same way. Those two are going to turn into 4 tenths. And you see it right over here when we shaded it initially. If you look at the tenths, you have 1 tenth, 2 tenths, 3 tenths, and 4 tenths. |
Example of adding fractions with unlike denominators word problem Pre-Algebra Khan Academy.mp3 | So those two sections that were shaded in, well, you were going to multiply by 2 the same way. Those two are going to turn into 4 tenths. And you see it right over here when we shaded it initially. If you look at the tenths, you have 1 tenth, 2 tenths, 3 tenths, and 4 tenths. Let's do the same logic over here. If you have 2 halves and you want to make them into 10 tenths, you have to take each of the halves and split them into 5 sections. You're going to have 5 times as many sections. |
Example of adding fractions with unlike denominators word problem Pre-Algebra Khan Academy.mp3 | If you look at the tenths, you have 1 tenth, 2 tenths, 3 tenths, and 4 tenths. Let's do the same logic over here. If you have 2 halves and you want to make them into 10 tenths, you have to take each of the halves and split them into 5 sections. You're going to have 5 times as many sections. So to go from 2 to 10, we multiply by 5. So similarly, that one shaded in section, that in yellow, that's going to turn into 5. That 1 half is going to turn into 5 tenths. |
Example of adding fractions with unlike denominators word problem Pre-Algebra Khan Academy.mp3 | You're going to have 5 times as many sections. So to go from 2 to 10, we multiply by 5. So similarly, that one shaded in section, that in yellow, that's going to turn into 5. That 1 half is going to turn into 5 tenths. So we're going to multiply by 5. Another way to think about it, whatever we did to the numerator or whatever we did to the denominator, we have to do to the numerator. Whatever we did to the denominator, we have to do to the numerator. |
Example of adding fractions with unlike denominators word problem Pre-Algebra Khan Academy.mp3 | That 1 half is going to turn into 5 tenths. So we're going to multiply by 5. Another way to think about it, whatever we did to the numerator or whatever we did to the denominator, we have to do to the numerator. Whatever we did to the denominator, we have to do to the numerator. Otherwise, we're changing the value of the fraction. So 1 times 5 is going to get you to 5. And you see that over here when we shaded it in. |
Example of adding fractions with unlike denominators word problem Pre-Algebra Khan Academy.mp3 | Whatever we did to the denominator, we have to do to the numerator. Otherwise, we're changing the value of the fraction. So 1 times 5 is going to get you to 5. And you see that over here when we shaded it in. That 1 half, if you look at the tenths, is equal to 1, 2, 3, 4, 5 tenths. And now we are ready to add these two things. 4 tenths plus 5 tenths, well, this is going to be equal to a certain number of tenths. |
Example of adding fractions with unlike denominators word problem Pre-Algebra Khan Academy.mp3 | And you see that over here when we shaded it in. That 1 half, if you look at the tenths, is equal to 1, 2, 3, 4, 5 tenths. And now we are ready to add these two things. 4 tenths plus 5 tenths, well, this is going to be equal to a certain number of tenths. It's going to be equal to 4 plus 5 tenths. And we can once again visualize that. Let me draw our grid again. |
Example of adding fractions with unlike denominators word problem Pre-Algebra Khan Academy.mp3 | 4 tenths plus 5 tenths, well, this is going to be equal to a certain number of tenths. It's going to be equal to 4 plus 5 tenths. And we can once again visualize that. Let me draw our grid again. So 4 plus 5 tenths. I'll do it actually on top of the paint can right over here. So let me color in 4 tenths. |
Example of adding fractions with unlike denominators word problem Pre-Algebra Khan Academy.mp3 | Let me draw our grid again. So 4 plus 5 tenths. I'll do it actually on top of the paint can right over here. So let me color in 4 tenths. So 1, 2, 3, 4. And then let me color in the 5 tenths. And notice that was exactly the 4 tenths here, which is exactly the 2 fifths. |
Example of adding fractions with unlike denominators word problem Pre-Algebra Khan Academy.mp3 | So let me color in 4 tenths. So 1, 2, 3, 4. And then let me color in the 5 tenths. And notice that was exactly the 4 tenths here, which is exactly the 2 fifths. Let me color in the 5 tenths. 1, 2, 3, 4, and 5. And so how many total tenths do we have? |
Example of adding fractions with unlike denominators word problem Pre-Algebra Khan Academy.mp3 | And notice that was exactly the 4 tenths here, which is exactly the 2 fifths. Let me color in the 5 tenths. 1, 2, 3, 4, and 5. And so how many total tenths do we have? We have a total of 1, 2, 3, 4, 5, 6, 7, 8, 9. 9 of the tenths are now shaded in. We had 9 tenths of a gallon of paint. |
Example of adding fractions with unlike denominators word problem Pre-Algebra Khan Academy.mp3 | And so how many total tenths do we have? We have a total of 1, 2, 3, 4, 5, 6, 7, 8, 9. 9 of the tenths are now shaded in. We had 9 tenths of a gallon of paint. So now to answer their question, will they have the gallon they need? No, they have less than a whole. A gallon would be 10 tenths. |
Example of adding fractions with unlike denominators word problem Pre-Algebra Khan Academy.mp3 | We had 9 tenths of a gallon of paint. So now to answer their question, will they have the gallon they need? No, they have less than a whole. A gallon would be 10 tenths. They only have 9 tenths. So no, they do not have enough of a gallon. Now another way you could have thought about this. |
Example of adding fractions with unlike denominators word problem Pre-Algebra Khan Academy.mp3 | A gallon would be 10 tenths. They only have 9 tenths. So no, they do not have enough of a gallon. Now another way you could have thought about this. You could have said, hey, look, 2 fifths is less than 1 half. And you could even visualize that right over here. So I have something less than 1 half plus 1 half. |
Example of adding fractions with unlike denominators word problem Pre-Algebra Khan Academy.mp3 | Now another way you could have thought about this. You could have said, hey, look, 2 fifths is less than 1 half. And you could even visualize that right over here. So I have something less than 1 half plus 1 half. I'm not going to get a whole. So either way, you could think about it. But this way, at least we can think it through with actually adding the fractions. |
Percentage of a whole number Decimals Pre-Algebra Khan Academy.mp3 | So you could view this as 30 over 100 times 6 is the same thing as 30% of 6. Or you could view this as 30 hundredths times 6, so 0.30 times 6. And we could solve both of these, and you'll see that we'll get the same answer. If you do this multiplication right over here, 30 over 100, and you could view this times 6 over 1, this is equal to 180 over 100. And let's see, we can simplify. We can divide the numerator and the denominator by 10. And then we can divide the numerator and the denominator by 2. |
Percentage of a whole number Decimals Pre-Algebra Khan Academy.mp3 | If you do this multiplication right over here, 30 over 100, and you could view this times 6 over 1, this is equal to 180 over 100. And let's see, we can simplify. We can divide the numerator and the denominator by 10. And then we can divide the numerator and the denominator by 2. And we will get 9 fifths, which is the same thing as 1 and 4 fifths. And then if we wanted to write this as a decimal, 4 fifths is 0.8. And if you want to verify that, you could verify that 5 goes into 4. |
Percentage of a whole number Decimals Pre-Algebra Khan Academy.mp3 | And then we can divide the numerator and the denominator by 2. And we will get 9 fifths, which is the same thing as 1 and 4 fifths. And then if we wanted to write this as a decimal, 4 fifths is 0.8. And if you want to verify that, you could verify that 5 goes into 4. And there's going to be a decimal, so let's throw some decimals in there. It goes into 4 zero times. We don't have to worry about that. |
Percentage of a whole number Decimals Pre-Algebra Khan Academy.mp3 | And if you want to verify that, you could verify that 5 goes into 4. And there's going to be a decimal, so let's throw some decimals in there. It goes into 4 zero times. We don't have to worry about that. It goes into 48 times. 8 times 5 is 40. Subtract, you have no remainder, and you just have zeros left here. |
Percentage of a whole number Decimals Pre-Algebra Khan Academy.mp3 | We don't have to worry about that. It goes into 48 times. 8 times 5 is 40. Subtract, you have no remainder, and you just have zeros left here. So 4 fifths is 0.8. You got the 1 there. This is the same thing as 1.8, which you would have gotten if you divided 5 into 9. |
Percentage of a whole number Decimals Pre-Algebra Khan Academy.mp3 | Subtract, you have no remainder, and you just have zeros left here. So 4 fifths is 0.8. You got the 1 there. This is the same thing as 1.8, which you would have gotten if you divided 5 into 9. You would have gotten 1.8. So 30% of 6 is equal to 1.8. And we can verify it doing this way as well. |
Percentage of a whole number Decimals Pre-Algebra Khan Academy.mp3 | This is the same thing as 1.8, which you would have gotten if you divided 5 into 9. You would have gotten 1.8. So 30% of 6 is equal to 1.8. And we can verify it doing this way as well. So if we were to multiply 0.30 times 6, let's do that. And I could just write that literally as 0.3 times 6. Well, 3 times 6 is 18. |
Percentage of a whole number Decimals Pre-Algebra Khan Academy.mp3 | And we can verify it doing this way as well. So if we were to multiply 0.30 times 6, let's do that. And I could just write that literally as 0.3 times 6. Well, 3 times 6 is 18. I have only one digit behind the decimal amongst both of these numbers that I'm multiplying. I only have the 3 to the right of the decimal, so I'm only going to have one number to the right of the decimal here. So I'll just count one number. |
Multi-step inequalities Linear inequalities Algebra I Khan Academy.mp3 | Let's do a few more problems that bring together the concepts that we learned in the last two videos. So let's say we have the inequality 4x plus 3 is less than negative 1. So let's find all of the x's that satisfy this. So the first thing I'd like to do is get rid of this 3. So let's subtract 3 from both sides of this equation. So the left-hand side is just going to end up being 4x. These 3's cancel out. |
Multi-step inequalities Linear inequalities Algebra I Khan Academy.mp3 | So the first thing I'd like to do is get rid of this 3. So let's subtract 3 from both sides of this equation. So the left-hand side is just going to end up being 4x. These 3's cancel out. That just ends up with a 0. No reason to change the inequality just yet. We're just adding and subtracting from both sides. |
Multi-step inequalities Linear inequalities Algebra I Khan Academy.mp3 | These 3's cancel out. That just ends up with a 0. No reason to change the inequality just yet. We're just adding and subtracting from both sides. In this case, subtracting. That doesn't change the inequality as long as we're subtracting the same value. We have negative 1 minus 3. |
Multi-step inequalities Linear inequalities Algebra I Khan Academy.mp3 | We're just adding and subtracting from both sides. In this case, subtracting. That doesn't change the inequality as long as we're subtracting the same value. We have negative 1 minus 3. That is negative 4. Negative 1 minus 3 is negative 4. And then we'll want to divide both sides of this equation by 4. |
Multi-step inequalities Linear inequalities Algebra I Khan Academy.mp3 | We have negative 1 minus 3. That is negative 4. Negative 1 minus 3 is negative 4. And then we'll want to divide both sides of this equation by 4. Let's divide both sides of this equation by 4. Once again, when you multiply or divide both sides of an inequality by a positive number, it doesn't change the inequality. So the left-hand side is just x. x is less than negative 4 divided by 4 is negative 1. x is less than negative 1. |
Multi-step inequalities Linear inequalities Algebra I Khan Academy.mp3 | And then we'll want to divide both sides of this equation by 4. Let's divide both sides of this equation by 4. Once again, when you multiply or divide both sides of an inequality by a positive number, it doesn't change the inequality. So the left-hand side is just x. x is less than negative 4 divided by 4 is negative 1. x is less than negative 1. Or we could write this in interval notation. All of the x's from negative infinity to negative 1, but not including negative 1. So we put a parenthesis right there. |
Multi-step inequalities Linear inequalities Algebra I Khan Academy.mp3 | So the left-hand side is just x. x is less than negative 4 divided by 4 is negative 1. x is less than negative 1. Or we could write this in interval notation. All of the x's from negative infinity to negative 1, but not including negative 1. So we put a parenthesis right there. Let's do a slightly harder one. Let's say we have 5x is greater than 8x plus 27. Let's get all our x's on the left-hand side. |
Multi-step inequalities Linear inequalities Algebra I Khan Academy.mp3 | So we put a parenthesis right there. Let's do a slightly harder one. Let's say we have 5x is greater than 8x plus 27. Let's get all our x's on the left-hand side. The best way to do that is subtract 8x from both sides. You subtract 8x from both sides. The left-hand side becomes 5x minus 8x. |
Multi-step inequalities Linear inequalities Algebra I Khan Academy.mp3 | Let's get all our x's on the left-hand side. The best way to do that is subtract 8x from both sides. You subtract 8x from both sides. The left-hand side becomes 5x minus 8x. That's negative 3x. We still have a greater than sign. We're just adding or subtracting the same quantity to both sides. |
Multi-step inequalities Linear inequalities Algebra I Khan Academy.mp3 | The left-hand side becomes 5x minus 8x. That's negative 3x. We still have a greater than sign. We're just adding or subtracting the same quantity to both sides. These 8x's cancel out and you're just left with a 27. So you have negative 3x is greater than 27. Now, to just turn this into an x, we want to divide both sides by negative 3. |
Multi-step inequalities Linear inequalities Algebra I Khan Academy.mp3 | We're just adding or subtracting the same quantity to both sides. These 8x's cancel out and you're just left with a 27. So you have negative 3x is greater than 27. Now, to just turn this into an x, we want to divide both sides by negative 3. But remember, when you multiply or divide both sides of an inequality by a negative number, you swap the inequality. So if we divide both sides of this by negative 3, we have to swap this inequality. It will go from being a greater than sign to a less than sign. |
Multi-step inequalities Linear inequalities Algebra I Khan Academy.mp3 | Now, to just turn this into an x, we want to divide both sides by negative 3. But remember, when you multiply or divide both sides of an inequality by a negative number, you swap the inequality. So if we divide both sides of this by negative 3, we have to swap this inequality. It will go from being a greater than sign to a less than sign. And this is a bit of a way that I remember greater than. The left-hand side just looks bigger. This is greater than. |
Multi-step inequalities Linear inequalities Algebra I Khan Academy.mp3 | It will go from being a greater than sign to a less than sign. And this is a bit of a way that I remember greater than. The left-hand side just looks bigger. This is greater than. If you just imagine this height, that height is greater than that height right there, which is just a point. I don't know if that confuses you or not. This is less than. |
Multi-step inequalities Linear inequalities Algebra I Khan Academy.mp3 | This is greater than. If you just imagine this height, that height is greater than that height right there, which is just a point. I don't know if that confuses you or not. This is less than. This little point is less than the distance of that big opening. That's how I remember it. But anyway, 3x over negative 3. |
Multi-step inequalities Linear inequalities Algebra I Khan Academy.mp3 | This is less than. This little point is less than the distance of that big opening. That's how I remember it. But anyway, 3x over negative 3. So anyway, now that we divided both sides by a negative number, by negative 3, we swapped the inequality from greater than to less than. In the left-hand side, the negative 3's cancel out. You get x is less than 27 over negative 3, which is negative 9. |
Multi-step inequalities Linear inequalities Algebra I Khan Academy.mp3 | But anyway, 3x over negative 3. So anyway, now that we divided both sides by a negative number, by negative 3, we swapped the inequality from greater than to less than. In the left-hand side, the negative 3's cancel out. You get x is less than 27 over negative 3, which is negative 9. Or in interval notation, it would be everything from negative infinity to negative 9, not including negative 9. If you wanted to do it as a number line, it would look like this. This would be negative 9. |
Multi-step inequalities Linear inequalities Algebra I Khan Academy.mp3 | You get x is less than 27 over negative 3, which is negative 9. Or in interval notation, it would be everything from negative infinity to negative 9, not including negative 9. If you wanted to do it as a number line, it would look like this. This would be negative 9. Maybe this would be negative 8. Maybe this would be negative 10. You would start at negative 9, not include it because we don't have an equal sign here. |
Multi-step inequalities Linear inequalities Algebra I Khan Academy.mp3 | This would be negative 9. Maybe this would be negative 8. Maybe this would be negative 10. You would start at negative 9, not include it because we don't have an equal sign here. You go all the way, everything less than that, all the way down as we see to negative infinity. Let's do a nice, hairy problem. Let's say we have 8x minus 5 times 4x plus 1 is greater than or equal to negative 1 plus 2 times 4x minus 3. |
Multi-step inequalities Linear inequalities Algebra I Khan Academy.mp3 | You would start at negative 9, not include it because we don't have an equal sign here. You go all the way, everything less than that, all the way down as we see to negative infinity. Let's do a nice, hairy problem. Let's say we have 8x minus 5 times 4x plus 1 is greater than or equal to negative 1 plus 2 times 4x minus 3. This might seem very daunting, but if we just simplify it step by step, you'll see it's no harder than any of the other problems we've tackled. Let's just simplify this. You get 8x minus, let's distribute this negative 5. |
Multi-step inequalities Linear inequalities Algebra I Khan Academy.mp3 | Let's say we have 8x minus 5 times 4x plus 1 is greater than or equal to negative 1 plus 2 times 4x minus 3. This might seem very daunting, but if we just simplify it step by step, you'll see it's no harder than any of the other problems we've tackled. Let's just simplify this. You get 8x minus, let's distribute this negative 5. Let me say 8x and then distribute the negative 5. Negative 5 times 4x is negative 20x. Negative 5, when I say negative 5, I'm talking about this whole thing. |
Multi-step inequalities Linear inequalities Algebra I Khan Academy.mp3 | You get 8x minus, let's distribute this negative 5. Let me say 8x and then distribute the negative 5. Negative 5 times 4x is negative 20x. Negative 5, when I say negative 5, I'm talking about this whole thing. Negative 5 times 1 is negative 5. Then that's going to be greater than or equal to negative 1 plus 2 times 4x is 8x. 2 times negative 3 is negative 6. |
Multi-step inequalities Linear inequalities Algebra I Khan Academy.mp3 | Negative 5, when I say negative 5, I'm talking about this whole thing. Negative 5 times 1 is negative 5. Then that's going to be greater than or equal to negative 1 plus 2 times 4x is 8x. 2 times negative 3 is negative 6. Now we can merge these two terms. 8x minus 20x is negative 12x minus 5 is greater than or equal to, we can merge these constant terms, negative 1 minus 6, that's negative 7. Then we have this plus 8x left over. |
Multi-step inequalities Linear inequalities Algebra I Khan Academy.mp3 | 2 times negative 3 is negative 6. Now we can merge these two terms. 8x minus 20x is negative 12x minus 5 is greater than or equal to, we can merge these constant terms, negative 1 minus 6, that's negative 7. Then we have this plus 8x left over. I like to get all my x terms on the left-hand side, so let's subtract 8x from both sides of this equation. Let's subtract 8x from both sides of this equation. From both sides, I'm subtracting 8x. |
Multi-step inequalities Linear inequalities Algebra I Khan Academy.mp3 | Then we have this plus 8x left over. I like to get all my x terms on the left-hand side, so let's subtract 8x from both sides of this equation. Let's subtract 8x from both sides of this equation. From both sides, I'm subtracting 8x. This left-hand side, negative 12 minus 8, that's negative 20. Negative 20x minus 5. Once again, no reason to change the inequality just yet. |
Multi-step inequalities Linear inequalities Algebra I Khan Academy.mp3 | From both sides, I'm subtracting 8x. This left-hand side, negative 12 minus 8, that's negative 20. Negative 20x minus 5. Once again, no reason to change the inequality just yet. All we're doing is simplifying the sides or adding and subtracting from them. The right-hand side becomes, this thing cancels out, 8x minus 8x, that's 0. You're just left with a negative 7. |
Multi-step inequalities Linear inequalities Algebra I Khan Academy.mp3 | Once again, no reason to change the inequality just yet. All we're doing is simplifying the sides or adding and subtracting from them. The right-hand side becomes, this thing cancels out, 8x minus 8x, that's 0. You're just left with a negative 7. Now I want to get rid of this negative 5, so let's add 5 to both sides of this equation. The left-hand side, you're just left with a negative 20x. The 5 and these 5s cancel out. |
Multi-step inequalities Linear inequalities Algebra I Khan Academy.mp3 | You're just left with a negative 7. Now I want to get rid of this negative 5, so let's add 5 to both sides of this equation. The left-hand side, you're just left with a negative 20x. The 5 and these 5s cancel out. No reason to change the inequality just yet. Negative 7 plus 5, that's negative 2. Now we're at an interesting point. |
Multi-step inequalities Linear inequalities Algebra I Khan Academy.mp3 | The 5 and these 5s cancel out. No reason to change the inequality just yet. Negative 7 plus 5, that's negative 2. Now we're at an interesting point. We have negative 20x is greater than or equal to negative 2. If this was an equation or really any type of even inequality, we want to divide both sides by negative 20, but we have to remember, when you multiply or divide both sides of an inequality by a negative number, you have to swap the inequality, so let's remember that. If we divide this side by negative 20 and we divide this side by negative 20, all I did is took both of these sides, divided by negative 20. |
Multi-step inequalities Linear inequalities Algebra I Khan Academy.mp3 | Now we're at an interesting point. We have negative 20x is greater than or equal to negative 2. If this was an equation or really any type of even inequality, we want to divide both sides by negative 20, but we have to remember, when you multiply or divide both sides of an inequality by a negative number, you have to swap the inequality, so let's remember that. If we divide this side by negative 20 and we divide this side by negative 20, all I did is took both of these sides, divided by negative 20. We have to swap the inequality. The greater than or equal to has to become a less than or equal sign. Of course, these cancel out. |
Multi-step inequalities Linear inequalities Algebra I Khan Academy.mp3 | If we divide this side by negative 20 and we divide this side by negative 20, all I did is took both of these sides, divided by negative 20. We have to swap the inequality. The greater than or equal to has to become a less than or equal sign. Of course, these cancel out. You get x is less than or equal to, the negatives cancel out. 2 over 20 is 1 over 10. If we were writing it in interval notation, the upper bound would be 1 over 10. |
Multi-step inequalities Linear inequalities Algebra I Khan Academy.mp3 | Of course, these cancel out. You get x is less than or equal to, the negatives cancel out. 2 over 20 is 1 over 10. If we were writing it in interval notation, the upper bound would be 1 over 10. Notice we're including it because we have an equal sign, less than or equal, so we're including 1 over 10, and we're going to go all the way down to negative infinity, everything less than or equal to 1 over 10. This is just another way of writing that. Just for fun, let's draw the number line right here. |
Multi-step inequalities Linear inequalities Algebra I Khan Academy.mp3 | If we were writing it in interval notation, the upper bound would be 1 over 10. Notice we're including it because we have an equal sign, less than or equal, so we're including 1 over 10, and we're going to go all the way down to negative infinity, everything less than or equal to 1 over 10. This is just another way of writing that. Just for fun, let's draw the number line right here. This is maybe 0. That is 1. 1 over 10 might be over here. |
Multi-step inequalities Linear inequalities Algebra I Khan Academy.mp3 | Just for fun, let's draw the number line right here. This is maybe 0. That is 1. 1 over 10 might be over here. Everything less than or equal to 1 over 10. We're going to include the 1 tenth and everything less than that is included in the solutions. You could try out any value less than 1 over 10 and verify that it will satisfy this inequality. |
Negative exponents Exponents, radicals, and scientific notation Pre-Algebra Khan Academy.mp3 | So that will give us 16. Now I will ask you a more interesting question. What do you think two to the negative, negative four powers, and I encourage you to pause the video and think about that. Well you might be tempted to say, oh, maybe it's negative 16 or something like that, but remember what the exponent operation is trying to do. This is, one way of viewing it is, this is telling us how many times are we going to multiply two times negative one. But here we're gonna multiply negative four times. Well what does negative traditionally mean? |
Negative exponents Exponents, radicals, and scientific notation Pre-Algebra Khan Academy.mp3 | Well you might be tempted to say, oh, maybe it's negative 16 or something like that, but remember what the exponent operation is trying to do. This is, one way of viewing it is, this is telling us how many times are we going to multiply two times negative one. But here we're gonna multiply negative four times. Well what does negative traditionally mean? Negative traditionally means the opposite. So here this is how many times you're going to multiply. Maybe when we make it negative, this says, how many times are we gonna, starting with a one, how many times are we going to divide by two? |
Negative exponents Exponents, radicals, and scientific notation Pre-Algebra Khan Academy.mp3 | Well what does negative traditionally mean? Negative traditionally means the opposite. So here this is how many times you're going to multiply. Maybe when we make it negative, this says, how many times are we gonna, starting with a one, how many times are we going to divide by two? So let's think about that a little bit. So this could be viewed as one times, and we're gonna divide by two four times. Well dividing by two is the same thing as multiplying by 1 1⁄2. |
Negative exponents Exponents, radicals, and scientific notation Pre-Algebra Khan Academy.mp3 | Maybe when we make it negative, this says, how many times are we gonna, starting with a one, how many times are we going to divide by two? So let's think about that a little bit. So this could be viewed as one times, and we're gonna divide by two four times. Well dividing by two is the same thing as multiplying by 1 1⁄2. So we could say that this is one times 1 1⁄2, times 1 1⁄2, times, so let me just do it in one color, it's gonna take, so one times 1 1⁄2, times 1 1⁄2, times 1 1⁄2, times 1 1⁄2. Notice, multiplying by 1 1⁄2 four times is the exact same thing as dividing by two four times. And in this situation, this would get you, well 1 1⁄2, well one times 1 1⁄2 is just 1 1⁄2, times 1 1⁄2 is 1 1⁄4, times 1 1⁄2 is 1 1⁄8, times 1 1⁄2 is one over 16. |
Negative exponents Exponents, radicals, and scientific notation Pre-Algebra Khan Academy.mp3 | Well dividing by two is the same thing as multiplying by 1 1⁄2. So we could say that this is one times 1 1⁄2, times 1 1⁄2, times, so let me just do it in one color, it's gonna take, so one times 1 1⁄2, times 1 1⁄2, times 1 1⁄2, times 1 1⁄2. Notice, multiplying by 1 1⁄2 four times is the exact same thing as dividing by two four times. And in this situation, this would get you, well 1 1⁄2, well one times 1 1⁄2 is just 1 1⁄2, times 1 1⁄2 is 1 1⁄4, times 1 1⁄2 is 1 1⁄8, times 1 1⁄2 is one over 16. And so you probably see the relationship here. If you're, this is essentially, you're starting with the one and you're dividing by two four times, you could also say, you could also say that two, two, I'm gonna do the same colors, two to the negative four, two to the negative four, is the same thing as one over two to the fourth power. One over two to the fourth power. |
Negative exponents Exponents, radicals, and scientific notation Pre-Algebra Khan Academy.mp3 | And in this situation, this would get you, well 1 1⁄2, well one times 1 1⁄2 is just 1 1⁄2, times 1 1⁄2 is 1 1⁄4, times 1 1⁄2 is 1 1⁄8, times 1 1⁄2 is one over 16. And so you probably see the relationship here. If you're, this is essentially, you're starting with the one and you're dividing by two four times, you could also say, you could also say that two, two, I'm gonna do the same colors, two to the negative four, two to the negative four, is the same thing as one over two to the fourth power. One over two to the fourth power. Let me color code it nicely so you realize what the negative is doing. So this negative right over here, let me do that in a better color, I'll do it in magenta, something that jumps out. So this negative right over here, this is what's causing us to go one over. |
Negative exponents Exponents, radicals, and scientific notation Pre-Algebra Khan Academy.mp3 | One over two to the fourth power. Let me color code it nicely so you realize what the negative is doing. So this negative right over here, let me do that in a better color, I'll do it in magenta, something that jumps out. So this negative right over here, this is what's causing us to go one over. So two to the negative four is the same thing based on the way we've defined it just up right here, as one over, or the reciprocal of two to the fourth, or one over two to the fourth. And so you could view this as being one over two times, so two times two times two times two, if you just view two to the fourth as taking four twos and multiplying them, or if you use this idea right over here, you could view it as starting with a one and multiplying it by two four times, either way, you are going to get one over, one over 16, one over 16. So let's do a few more examples of this just so that we make sure things are clear to us. |
Negative exponents Exponents, radicals, and scientific notation Pre-Algebra Khan Academy.mp3 | So this negative right over here, this is what's causing us to go one over. So two to the negative four is the same thing based on the way we've defined it just up right here, as one over, or the reciprocal of two to the fourth, or one over two to the fourth. And so you could view this as being one over two times, so two times two times two times two, if you just view two to the fourth as taking four twos and multiplying them, or if you use this idea right over here, you could view it as starting with a one and multiplying it by two four times, either way, you are going to get one over, one over 16, one over 16. So let's do a few more examples of this just so that we make sure things are clear to us. So let's try three to the negative third power. So remember, whenever you see that negative, what my brain always does is say, I need to take the reciprocal here. So this is going to be equal to, and I'm gonna highlight the negative again, this is going to be one over three to the third power, one over three to the third power, which would be equal to, well, one over three times three, or you could say one over three times three times three, or one times three times three times three, is going to be 27. |
Negative exponents Exponents, radicals, and scientific notation Pre-Algebra Khan Academy.mp3 | So let's do a few more examples of this just so that we make sure things are clear to us. So let's try three to the negative third power. So remember, whenever you see that negative, what my brain always does is say, I need to take the reciprocal here. So this is going to be equal to, and I'm gonna highlight the negative again, this is going to be one over three to the third power, one over three to the third power, which would be equal to, well, one over three times three, or you could say one over three times three times three, or one times three times three times three, is going to be 27. So this is going to be 1 27th. Let's try another example. I'll do two or three more. |
Negative exponents Exponents, radicals, and scientific notation Pre-Algebra Khan Academy.mp3 | So this is going to be equal to, and I'm gonna highlight the negative again, this is going to be one over three to the third power, one over three to the third power, which would be equal to, well, one over three times three, or you could say one over three times three times three, or one times three times three times three, is going to be 27. So this is going to be 1 27th. Let's try another example. I'll do two or three more. So let's take a negative number to a negative exponent just to see if we can confuse ourselves. So let's take the number negative four, negative four, and let's take it, I don't want my numbers to get too big too fast, so let's take, let's just take negative two, let's take negative two, and let's take it to the negative three power. Negative, I wanna make my negatives in magenta. |
Negative exponents Exponents, radicals, and scientific notation Pre-Algebra Khan Academy.mp3 | I'll do two or three more. So let's take a negative number to a negative exponent just to see if we can confuse ourselves. So let's take the number negative four, negative four, and let's take it, I don't want my numbers to get too big too fast, so let's take, let's just take negative two, let's take negative two, and let's take it to the negative three power. Negative, I wanna make my negatives in magenta. Negative three power. Negative, negative three power. So at first this might be daunting, do the negatives cancel, and that'll just be the remnants in your brain that you're trying to think of multiplying negatives, do not apply that here. |
Negative exponents Exponents, radicals, and scientific notation Pre-Algebra Khan Academy.mp3 | Negative, I wanna make my negatives in magenta. Negative three power. Negative, negative three power. So at first this might be daunting, do the negatives cancel, and that'll just be the remnants in your brain that you're trying to think of multiplying negatives, do not apply that here. Remember, you see a negative exponent, that just means the reciprocal of the positive exponent. So, one over negative two, negative two to the third power, to the positive third power, and this is equal to, this is equal to one over negative two, negative two times negative two times negative two. Times negative two, or you could view it as one times negative two times negative two times negative two. |
Negative exponents Exponents, radicals, and scientific notation Pre-Algebra Khan Academy.mp3 | So at first this might be daunting, do the negatives cancel, and that'll just be the remnants in your brain that you're trying to think of multiplying negatives, do not apply that here. Remember, you see a negative exponent, that just means the reciprocal of the positive exponent. So, one over negative two, negative two to the third power, to the positive third power, and this is equal to, this is equal to one over negative two, negative two times negative two times negative two. Times negative two, or you could view it as one times negative two times negative two times negative two. which is going to give you 1 over negative 8, or negative 1 eighth. Let me scroll over a little bit. I don't want to have to start squinching things. |
Negative exponents Exponents, radicals, and scientific notation Pre-Algebra Khan Academy.mp3 | Times negative two, or you could view it as one times negative two times negative two times negative two. which is going to give you 1 over negative 8, or negative 1 eighth. Let me scroll over a little bit. I don't want to have to start squinching things. So this is equal to negative 1 eighth. Let's do one more example, just in an attempt to confuse ourselves. Let's take 5 eighths and raise this to the negative 2 power. |
Negative exponents Exponents, radicals, and scientific notation Pre-Algebra Khan Academy.mp3 | I don't want to have to start squinching things. So this is equal to negative 1 eighth. Let's do one more example, just in an attempt to confuse ourselves. Let's take 5 eighths and raise this to the negative 2 power. So once again, this negative, oh, I got a fraction. This is a negative here. Remember, this just means 1 over 5 eighths to the second power. |
Negative exponents Exponents, radicals, and scientific notation Pre-Algebra Khan Academy.mp3 | Let's take 5 eighths and raise this to the negative 2 power. So once again, this negative, oh, I got a fraction. This is a negative here. Remember, this just means 1 over 5 eighths to the second power. So this is just going to be the same thing as 1 over 5 eighths squared, which is going to be the same thing. So this is going to be equal to, I'm trying to color code it, 1 over 5 eighths times 5 eighths, which is 25 over 64. 1 over 25 over 64 is just going to be 64 over 25. |
Negative exponents Exponents, radicals, and scientific notation Pre-Algebra Khan Academy.mp3 | Remember, this just means 1 over 5 eighths to the second power. So this is just going to be the same thing as 1 over 5 eighths squared, which is going to be the same thing. So this is going to be equal to, I'm trying to color code it, 1 over 5 eighths times 5 eighths, which is 25 over 64. 1 over 25 over 64 is just going to be 64 over 25. So another way to think about it is you're going to take the reciprocal of this and raise it to the positive exponent. So another way you could have thought about this is 5 eighths to the negative 2 power. Let me just take the reciprocal of this, 8 fifths, and raise it to the positive 2 power. |
Negative exponents Exponents, radicals, and scientific notation Pre-Algebra Khan Academy.mp3 | 1 over 25 over 64 is just going to be 64 over 25. So another way to think about it is you're going to take the reciprocal of this and raise it to the positive exponent. So another way you could have thought about this is 5 eighths to the negative 2 power. Let me just take the reciprocal of this, 8 fifths, and raise it to the positive 2 power. So all of these statements are equivalent. And that would have applied even when you are dealing with non-fractions as your base right over here. So 2, you could say, well, this is going to be the same thing. |
Solving systems of equations graphically Algebra II Khan Academy.mp3 | Just in case we encounter any more trolls who want us to figure out what types of money they have in their pockets, we have devised an exercise for you to practice with. And this is to solve systems of equations visually. So they say right over here, graph this system of equations and solve. And they give us two equations, this first one in blue. Y is equal to 7 5ths x minus five, and then this one in green. Y is equal to 3 5ths x minus one. So let's graph each of these, and we'll do it in the corresponding color. |
Solving systems of equations graphically Algebra II Khan Academy.mp3 | And they give us two equations, this first one in blue. Y is equal to 7 5ths x minus five, and then this one in green. Y is equal to 3 5ths x minus one. So let's graph each of these, and we'll do it in the corresponding color. So first let's graph this first equation. So the first thing I see is that it's y-intercept is negative five. Or another way to think about it, when x is equal to zero, y is going to be negative five. |
Solving systems of equations graphically Algebra II Khan Academy.mp3 | So let's graph each of these, and we'll do it in the corresponding color. So first let's graph this first equation. So the first thing I see is that it's y-intercept is negative five. Or another way to think about it, when x is equal to zero, y is going to be negative five. So let's try this out. So when x is equal to zero, y is going to be equal to negative five. So that makes sense. |
Solving systems of equations graphically Algebra II Khan Academy.mp3 | Or another way to think about it, when x is equal to zero, y is going to be negative five. So let's try this out. So when x is equal to zero, y is going to be equal to negative five. So that makes sense. And then we see its slope is 7 5ths. This was conveniently placed in slope intercept form for us. So it's rise over run. |
Solving systems of equations graphically Algebra II Khan Academy.mp3 | So that makes sense. And then we see its slope is 7 5ths. This was conveniently placed in slope intercept form for us. So it's rise over run. So for every time it moves five to the right, it's going to move seven up. So if it moves one, two, three, four, five to the right, it's going to move seven up. One, two, three, four, five, six, seven. |
Solving systems of equations graphically Algebra II Khan Academy.mp3 | So it's rise over run. So for every time it moves five to the right, it's going to move seven up. So if it moves one, two, three, four, five to the right, it's going to move seven up. One, two, three, four, five, six, seven. So it'll get right over there. Another way you could have done it is you could have just tested out some values. You could have said, oh, when x is equal to zero, y is equal to negative five. |
Solving systems of equations graphically Algebra II Khan Academy.mp3 | One, two, three, four, five, six, seven. So it'll get right over there. Another way you could have done it is you could have just tested out some values. You could have said, oh, when x is equal to zero, y is equal to negative five. When x is equal to five, 7 5ths times five is seven, minus five is two. So I think we've properly graphed this top one. Let's try this bottom one right over here. |
Solving systems of equations graphically Algebra II Khan Academy.mp3 | You could have said, oh, when x is equal to zero, y is equal to negative five. When x is equal to five, 7 5ths times five is seven, minus five is two. So I think we've properly graphed this top one. Let's try this bottom one right over here. So we have negative, when x is equal to zero, y is equal to negative one. So when x is equal to zero, y is equal to negative one. And the slope is 3 5ths. |
Solving systems of equations graphically Algebra II Khan Academy.mp3 | Let's try this bottom one right over here. So we have negative, when x is equal to zero, y is equal to negative one. So when x is equal to zero, y is equal to negative one. And the slope is 3 5ths. So if we move over five to the right, if we move over five to the right, we will move up three. So we will go right over there. And it looks like they intersect right at that point, right at the point x is equal to five, y is equal to two. |
Solving systems of equations graphically Algebra II Khan Academy.mp3 | And the slope is 3 5ths. So if we move over five to the right, if we move over five to the right, we will move up three. So we will go right over there. And it looks like they intersect right at that point, right at the point x is equal to five, y is equal to two. So we'll type in x is equal to five, y is equal to two. And you could even verify by substituting those values into both equations to show that it does satisfy both constraints. So let's check our answer. |
Finding intercepts from an equation Algebra I Khan Academy.mp3 | And we're told to find the intercepts of this equation. So we have to find the intercepts. And then use the intercepts to graph this line on the coordinate plane. So then graph the line. So whenever someone talks about intercepts, they're talking about where you're intersecting the x and the y axes. So let me label my axes here. So this is the x-axis, and that is the y-axis there. |
Finding intercepts from an equation Algebra I Khan Academy.mp3 | So then graph the line. So whenever someone talks about intercepts, they're talking about where you're intersecting the x and the y axes. So let me label my axes here. So this is the x-axis, and that is the y-axis there. And when I intersect the x-axis, what's going on? What is my y value when I'm at the x-axis? Well, my y value is 0. |
Finding intercepts from an equation Algebra I Khan Academy.mp3 | So this is the x-axis, and that is the y-axis there. And when I intersect the x-axis, what's going on? What is my y value when I'm at the x-axis? Well, my y value is 0. I'm not above or below the x-axis. So the x-intercept is when y is equal to 0. And then by that same argument, what's the y-intercept? |
Finding intercepts from an equation Algebra I Khan Academy.mp3 | Well, my y value is 0. I'm not above or below the x-axis. So the x-intercept is when y is equal to 0. And then by that same argument, what's the y-intercept? Well, if I'm somewhere along the y-axis, what's my x value? Well, I'm not to the right or the left, so my x value has to be 0. So the y-intercept occurs when x is equal to 0. |
Finding intercepts from an equation Algebra I Khan Academy.mp3 | And then by that same argument, what's the y-intercept? Well, if I'm somewhere along the y-axis, what's my x value? Well, I'm not to the right or the left, so my x value has to be 0. So the y-intercept occurs when x is equal to 0. So to figure out the intercepts, let's set y equal to 0 in this equation and solve for x. And then let's set x is equal to 0 and then solve for y. So when y is equal to 0, what does this equation become? |
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