Writing inequalities to represent real-world problems | Grade 8 (TX) | Khan Academy

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.

Right, and 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.

All right, 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 to the pool. I guess n is the number of days that she visits. So in the special deal, she's going to spend $20 up front whether or not she visits anymore, plus—oh, not eight—$4 per day times n. So plus 4N because N is a number of days.

Now, in the standard scenario, she doesn't pay any money up front. With the special—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 + 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 + 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, 0 days. Well, 20 is not less than zero, so zero days does not tell you how—if you only visit zero days the special deal is not going to be a better scenario.

So we're going to 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.