Problem. 95. Um, we're filing, ah, tax return. And the charge for filing that tax return is C equals 50 plus 40 t where C is the charge in dollars and t is the time and ours all right. And we have the greatest integer function in T. Okay, So if that's ah desk, more fraction, it gets rounded down to the nearest integer. And they want to know what values of T is the charge over to 35. Okay, so we want see greater than 2. 35. This is our function right here. Dysfunction right here. See? Okay. So I am gonna substitute that function in place of C. All right. So in place of, say, 50 plus 40 t is greater than 2. 35 minus 50 on both sides. Fifties cancel 40 t is greater than 1 85 and dividing both sides by 40. We have tea is greater than 4.6 to 5. Now, remember, for tea, we have the greatest integer function. Okay, so this decimal is always going to get cut off because it rounds it down to the nearest integer. All right. So because the decimal is always rounded down. We need to round tee up because he has to be greater than this decimal. But if it even if see this is greater than 4.625 let's say it's 4.9. It's still going to get rounded down to four. So that's not gonna work. So we need to round it up. So we get rid of that decimal, and we say T is greater than five. Okay, and that makes perfect sense. If you plug in five for tea, you'll get 40 times five, which is 200 plus 50 is 2 50. Okay, but if T is less than five, it will get rounded down to four. You know that four times 40 is 1 60 plus 50. That's to 10. That's no good. That's not greater than 2 35. So it has to be five or more so, really, I should have said greater than or equal to five. Okay? And they also wanted us to sketch the graph. So I just counted by ones for the X axis that's ours, and I counted by forties for the Y axis. That's cost. It's because it goes up $40 every hour. Okay, so up to one hour T is gonna be zero, and it's just $50 upto one hour. So we start. We start right here upto one hour. Okay, That's gonna be $50 all right? And it goes up by 40 each time. And you know what? I sink. I think it will be better if I rewrite this y axis and start with a 50. So let's do that. I'm gonna start with 50 on the white axis. That will make things easier. All right. So this will be 50 then we'll count by 40 each time. 90 1 30 1 70 to 10 2. 50 and so on. And we'll pretend Will say it started at 10. All right. The X axis started at zero. Okay, let's try this again. And I think I'll make it in red so you could see it better. Okay, so for the first hour, it's $50 for the first hour. All right? And if x 50 equals one all the way up to 1.999 it's gonna be $90. Okay, So I'm gonna make a open, open circle here, and it jumps up to $90 at one all right, and then make an open circle at two, and it jumps upto 1 $30. I make an open circle at three, and it jumps upto 1 $70. Open circle at four. It jumps up 2 to 10 and so on. It keeps going. Keeps going up and up like steps. All right, so that's this. That's the sketch of the graph. Each time it goes up to the next hole, our it jumps up by $40. Okay, but any fraction of an hour gets rounded down because this is the greatest integer function, which means it's rounded down.