Writing inequalities to represent real-world problems

- [Instructor] We're told at the beginning of summer the city pool advertises a special offer. Swimmers can pay an initial fee of $20, and then the daily admission will be $4 per-day. Without the special offer, the standard price is $8 per-day. Irene wants to know after how many days of visiting the pool will the special offer be a better deal. She defines N as the number of visits to the pool. Write an inequality to represent the situation. So, like always, pause this video and see if you can do this on your own before we do this together. Alright, now let's tackle this together. Let's think about how much Irene is going to spend in the special deal case. Special deal. And then let's also think about how much she's going to spend in the standard case if she doesn't do the special deal. So in the special deal case, let's read the details again, it's an initial fee of $20, and then the daily admission will be $4 per-day. And then N is the number of visits per the pool. I guess N is the number of days that she visits. So in the special deal she's going to spend $20 upfront, whether or not she visits anymore, plus eight, no, not eight, $4 per-day times N, so plus 4N, 'cause N is the number of days. Now in the standard scenario, she doesn't pay any money upfront. Without the special offer, the standard price is $8 per-day. So that's just going to be eight times the number of days. And what we want is an inequality to represent after how many days of visiting the pool will the special offer be a better deal. So a better deal means that the special offer needs to cost less. So one way to think about it is 20 plus 4N, where N is the number of days, that needs to cost less in order for it to be a better deal than the standard situation. So 20 plus 4N needs to be less than 8N. And we're done. We could try to simplify this and even solve this inequality, or try to simplify it, but this is all we wanted. We just wanted an inequality to represent this situation. And you could see here if Irene visited, say, zero days, well, 20 is not less than zero. So zero days does not tell you, if you only visit zero days, the special deal is not going to be a better scenario. So we're gonna have to figure out after how many days does it start to become a better scenario. And if she visits enough, it will be. And you could figure that out by simplifying this.