We're on problem 141. On Jane's credit card account, the average daily balance for a 30 day billing cycle-- when I see these long paragraphs it starts to give you a headache-- the average daily balance for a 30 day billing cycle is the average of the daily balances at the end of each of the 30 days. OK, so they take the average balance of each of the days and then they average it. OK. At the beginning of a certain 30 day billing cycle, Jane's credit card account had a balance of $600. At the beginning. What problem is this? 141. So at the beginning, she had a balance of$600. Fair enough. Jane made a payment of $300 on the account during the billing cycle. So this is the beginning of the month, and then at some point-- this is 30 days later-- at some point she made a$300 payment. Fair enough. Continuing. If no other amounts were added or subtracted from the account during the billing cycle, what was the average daily balance on Jane's account for the billing cycle? OK, so I think I'm visualizing this right. If we say that this is days, where this is day 0 and this is day 30. So essentially, at the end of every day, so at the end of day 1-- so at the beginning of a certain billing cycle Jane's credit card account had a balance of $600. So we don't know. She might have paid it on day 1. So I'm not going to put any labels here. But she starts at 600. And that's her balance until on some day she pays off half of it, and her balance goes down like that. So the question is, what is the average? So it's going to be$600 times a certain number of days, plus $300 times a certain number of days, divided by 30. So let me write that down. The average is going to be equal to$600 times however many days she carried the $600 balance plus$300 times however many days she kept the $300 balance. And that's going to be 30 minus x. If she paid after 15 days, she would have had a$600 balance for 15 days and then she would have had the $300 for the remainder of the days. If she paid after one day, then you would have the$300 balance for 29 days. And all of that divided by 30. That's the average. And I try to do that from the get-go, because I just want to get my algebraic brain around the problem so it becomes less abstract. So statement number one. Let me keep that in the screen. Statement number one tells us Jane's payment cycle was credited on the 21st day of the billing cycle. So that means she had a $600-- that x in this example-- that she had a$600 balance for 20 days. And then on the 21st day her balance would have gone to $300. So the average is going to be equal to$600-- she had a $600 balance for the first 20 days. Times 20 plus 300 times the remainder days. So 300 for 10 days. And that's all divided by the number of days. So statement one, alone, is enough to figure out her average balance for the month. Statement two. The average daily balance through the 25th day of the billing cycle was$540. This is interesting. So $540 is the average through the 25th day. So if we average the first 25 days-- so her balance was$600 for x days, plus \$300 for the remainder. Only the first 25 days. So it's x minus 25. So you can actually take this equation and solve for x. It's actually a linear equation. And I'm sure this time. Multiply both sides by 25, you get a number. And then you can distribute this out. Add all the x terms. Solve for x. And then once you solve for x, you can just use this equation up here to figure out the average daily balance for 30 days. So actually, each of these statements independently are sufficient to figure out this problem. Actually, it was very critical that we kind of thought about in this term from the get-go. Otherwise this would have been a very hard problem to get your hand around. But it's interesting. I, strangely, really like that problem. Anyway, next problem. 142. If x is an integer, they're asking is 9 to the x plus 9 to the minus x equal to b? Who knows? Problem one says 3 to the x plus 3 to the minus x is equal to the square root of b plus 2. So right from the get-go I don't see-- I mean there's 3's there's 9's, there seems to be some relationship. Let's square both sides of this equation, see if it can reduce to something that's useful here. So the right hand side is easy. The left hand side, if you square it you get 3 to the 2x. And then you get plus 2 times these multiplied by each other. 3 to the x, 3 to the minus x plus 3 to the minus 2x is equal to b plus 2. I just squared both sides. 3 to the x times 3 to the minus x, that just equals 1, right? You could add the exponents, that's equal to 3 to the 0. That equals 1. Or you could view that as 3 to the x divided by 3 to the x. Either way, that equals 1. Then you're left with 2 on both sides of the equation. So you can get rid of that. So then we're left with 3 to the 2x plus 3 to the minus 2x is equal to b. But then we could rewrite this. Think about this. 3 to the 2x, that's the same thing as 3 squared to the x. And this is the same thing as 3 to the minus 2 to the x is equal to b. And I think now bells are ringing in your head. 3 squared, that's the same thing as 9 to the x plus-- I should do 3 squared to the minus x. That's easier. Plus 9 to the minus x is equal to b. So statement number one actually reduced to what we were trying to prove. So statement number one, at least alone, is sufficient. Let's see what statement number two gets us. Statement number two says x is greater than 0 is equal to b. Let's see. The original was 9 to the x plus 9 to the minus x is equal to b. So they're now saying that b is equal to 0. So how does that help us? If b is equal to 0-- let's subtract 9 to the minus x from both sides. You get 9 to the x is equal to minus 9 to the minus x. Now let me multiply both sides by negative 9 to the x. I just want to make this-- minus 9 to the x times minus 9 to the x. I'm just trying to simplify it. So what do you have here? The minus sign, minus 9 to the 2x, right? Just add the exponents. Is equal to-- a minus times a minus is a positive-- and then a minus x and a positive x. Add them together, you get 0. So 9 and 0 is 1. That simplifies to minus 9 to the 2x is equal to 1. So can this ever be true? Let me think about that. The only way something that's non 1 or non negative 1, the only way that when you raise it to a power you can get a 1 is if you raise it to a 0th power. So this is only true if x is equal to 0. But they tell us that x is greater than 0. So, actually, this is an interesting case. Using the information they gave us in problem number two-- I just want to make sure I'm not missing something. Statement one was sufficient. Can we prove-- well, we don't have enough information to prove the statement. We have enough information to disprove the statement but not to prove it. So I'm going to stick with a. I was going to say that we can say whether the statement is true or false, but we definitely don't have enough information with just statement b to prove-- so they want to know, right. You have to be able to prove the statement is true. Statement two actually proves that it is false. It doesn't prove that it's true. It answers the question but says no. So statement one, alone, is sufficient to say that that statement is true. And I'll continue in the next video.