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GMAT: Data sufficiency 41

154-155, pg. 290. Created by Sal Khan.

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Video transcript

We're on problem 154. They ask us, is x negative? So the question is x less than 0. Statement number 1 tells us-- let's see, they say x to the third times 1 minus x squared is less than 0. Now let's simplify it. Let's see, x to the third minus x to the fifth is less than 0. We could add x to the fifth to both sides. x to the third is less than x to the fifth. Let's divide both sides by x squared. That won't change the inequality since x squared is definitely going to be positive. Any integer squared, even if it's negative, is a positive. So you divide both sides by x squared, you get x is less than x to the third. Now let's see, does this answer that question? Let's see what x's would satisfy this condition. So we can already say that x cannot be equal to 0, right? Anything where this would have been an inequality doesn't work. So x can't be equal 0, it can't be able to 1, because 1 is equal 1 to the third, and it can't be equal to negative 1. So we know that immediately. We know that for regular positive numbers, for numbers greater than 1, taking it to the third power is always going to be greater than the number itself. If you took 2 to the third power, that's greater than 2. 3 to the third power is greater than 3. 1.5 to the third power is greater than 1.5. So we know that this statement right here implies that x is greater than 1. Now let's see, is there a situation where x can be negative? Well, let's see. First off, x is between 0 and 1. 1/2 is not less than 1/8. So it doesn't work between 0 and 1. But what if x were between negative 1 and 0? Let's try an example. Negative 1/2 is less than negative 1/8. It's more negative. Negative 1/2 is less than negative 1/8, right? You could try that with any fraction. Because when you take it to the third power, it becomes a smaller fraction, but it becomes a smaller negative fraction. So x will be less than it, because it's more negative. So this is true. If this is true, then x is either greater than 1, or it's between negative 1 and 0. Let's see, does it work if x is less than negative 1. Negative 2, is negative 2 less than negative 8? Nope. Right. So these are the only ranges where it works. This statement implies this. But this still isn't enough information to tell us whether x is less than 0. x could be greater than 1, which is definitely greater than 0, or x could be less than 0. So we don't know yet just from statement 1. Statement 2 tells us x squared minus 1 is less than 0. So add 1 to both sides, that tells us that x squared is less than 1. So when could something squared be less than 1? Well, it's going to have to be between negative 1 and 1, right? It's essentially going to have be a positive or a negative fraction less than 1. That's the only way that when you square it, you get a number less than 1. If it was greater than 1, you're definitely going to get a number greater than 1 and it's going to be positive either way. If it's 1, it's going to be 1, and then 0 is in our range. So statement 2 tells us this, but it includes both positive and negative numbers. So it doesn't answer our question whether x is less than 0. But if we use both statements combined, what is the intersection? What is the overlap of this and this? Well, if x has to be between negative 1 and 1 and it has to be one of these, the only one that it overlaps with is this one. You cannot have x being greater than 1 and x being in this range. You can have x being in this range and x being in this range, since this is a subset of this. So this is actually the more restrictive. And so since we go to that, that actually tells us that x has to be negative. So both statements combined are sufficient to answer this question, but individually they're not. The last question, 155. Marsha's bucket can hold a maximum of how many liters of water? So we want to know the capacity of her bucket. Statement number 1 says, the bucket currently contains 9 liters of water. That's useless. We don't know how-- that could be the bucket, 9 liters could be that. Or maybe 9 liters is the full capacity. That tells us nothing about what the capacity of the bucket is. If I told you I had a bite of the sandwich, that tells you nothing as far as how large of a sandwich I can eat. So statement 1 doesn't seem that useful. Statement 2, if 3 liters of water are added to the bucket when it is 1/2 full of water. So 1/2 times the capacity, right? When it's 1/2 full of water, you add 3 liters. So if 3 liters of water are added to the bucket when it is 1/2 full of water, the amount of water in the bucket will increase by 1/3. To increase something by 1/3, that is like multiplying it by 1 1/3 That's 1 1/3 times the capacity. That's what they're saying. The amount of water in the bucket will increase by 1. Sorry. You're going to have 1 1/3 times what you started off with, which was half your capacity. That make sense? We say we're starting with 1/2 of our capacity and we're adding 3 liters. They're saying that that will increase the amount we have by 1 1/3. Let me write that. Let's say we start with x, and we're adding 3 liters. They're saying that will equal 1 1/3 or 4/3 times x, right? This shows that x is increasing by 1/3. You could view this as 1 plus 1/3 times x. This is an increase of 1/3. They tell us that our starting point is 1/2 of capacity. So that's where I get the 1/2 c-- that's our starting point-- plus 3 is equal to 4/3 times the starting point, or where we're adding water to, 1/2 times c. We could easily solve for c here. So you get a 2, this cancels out. Then you get 1/2 c plus 3 is equal to 2/3 c. Let's see, we could subtract 1/2 c from both sides, and you get 3 is equal to 2/3 minus 1/2 c. Then you can just do that fraction, and you get c is equal to 3 over-- well, what is this? 1/2-- so if you do it over a 6, 4 minus 3, so this is 1/6. So this turns into 1/6. So c is equal to 18. Wait a minute. You didn't have to do all that. You should have just realized that as soon as you can write this algebraic equation down, it's a linear equation with one unknown, and that one unknown is what we're trying to solve for, you have enough information to solve the problem. So statement 2, alone, is sufficient to solve this problem. We're all done with data sufficiency. See you all in the next section.