Using inequalities to solve problems | Solving equations & inequalities | Algebra I | Khan Academy

We're told that Kayla wants to visit a friend who lives eight kilometers away. She'll ride the subway as far as she can before walking the rest of the way. First, she needs to buy an access pass that costs five dollars and fifty cents. There is also a fee of 1.25 per stop. This is an expensive subway. Kayla doesn't want to spend more than 15 on the trip, so she wants to know the largest number of stops she can afford.

Let s represent the number of stops that Kayla buys. So first, pause this video and see if you can write an inequality that describes how many stops or that describes the situation that describes that she wants to take as many stops as she can, but she doesn't want to spend more than 15 dollars.

All right, now let's do this together. So first, let's just think about an expression for how much she spends. No matter what, she's going to spend five dollars and fifty cents. So we could write it like this: So five dollars and fifty cents, that's what she's going to spend even if she doesn't take any stops, and it's 1.25 per stop.

And s is the number of stops, so the amount she's going to spend just from the stops is going to be 1.25 times s. So it's going to be plus 1.25 s. This is the upfront she has to spend, and this is how much she's going to spend on stops. And this is an s right over here, and I wrote a 5 right next to it. They look kind of similar.

And we know that she doesn't want to spend more than 15, so she's willing to spend up to 15. So this total amount that she spends has to be less than or equal to 15 dollars, or if we didn't write it with the dollar symbols, we would write 5.50 plus 1.25 s is going to be less than or equal to or needs to be less than or equal to 15.

Now that we've written this inequality, what is the number of stops that Kayla can afford? What's the largest number of stops that she can afford? Pause this video and try to figure that out.

Well, to figure that out, we just have to solve for s and then figure out what the largest s is that satisfies the inequality once we've solved for s. So the first thing I would do is subtract 550 from both sides. When we do that, we are left with 1.25 s is less than or equal to 9.50.

Then I would divide both sides by 1.25, and since I'm dividing both sides by a positive value, it doesn't change the direction of the inequality. 9.50 divided by 1.25 is equal to 7.6. So we get that s needs to be less than or equal to 7.6.

So we can't take a fractional number of stops. The largest number of stops that Kayla can take is going to be seven stops. She can't take eight, and she can't take seven and a half or seven point six, so the largest number she can take is seven stops. So she can take as many as seven stops, and we are done.