# Video: Writing and Solving Linear Inequalities in a Real-World Context

Anthony and nine of his friends want to buy a birthday cake. They plan on splitting the cost, and they want to spend at most \$8.50 per person. Write an inequality to find the cost of the cake they should buy, and then solve it.

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### Video Transcript

Anthony and nine of his friends want to buy a birthday cake. They plan on splitting the cost and they want to spend at most eight dollars 50 per person. Write an inequality to find the cost of the cake they should buy and then solve it.

Okay, so for the first part, what we need to look at is the fact that we need to write an inequality to find the cost of the cake they should buy. So what’s the key information?

So we know that Anthony and nine of his friends want to buy a birthday cake, and we also know that they want to spend at most eight dollars 50 per person. So the first thing to note is actually we’re gonna make an assumption with this question, because we’re gonna assume there is an even split.

So actually the money, so the cost of the cake, is divided evenly. Also, it’s worth noting that the first bit is where there could be a mistake. So be careful. It says Anthony and nine of his friends, so a common mistake here would be, alright, nine is the number that we’re looking at. But it’s not, because it’s Anthony and nine of his friends, so actually there are 10 people who’re gonna share the cost.

Well, to actually help us set up our inequality and then to solve it, what we’re gonna do is call 𝑥 the cost of the cake. So we’re saying that 𝑥 is equal to the cost of our cake. So therefore, if we have the cost of the cake and we divide it by 10 — so we say 𝑥 over 10 — then this is gonna be less than or equal to 8.5.

And why is that? Well, we know it’s less than or equal to because in the question it says they want to spend at most eight dollars 50 per person. So that means that actually also includes eight dollars 50.

So just to quickly recap the inequality notation, we know that 𝑥 is greater than when it’s near to the open side of our inequality sign, but it’s less than when it’s next to the point or so the closed side of our inequality sign. And also if it’s got a line underneath our inequality sign, this means “or equal to,” which means it also includes that value.

So now we’ve actually completed the first part of the question, because 𝑥 over 10 is less than or equal to 8.5 is the inequality written. Now what we need to do is actually find the cost of the cake they should buy by actually solving our inequality.

Now to solve this inequality, we’re gonna deal with it in the same way that we’d actually deal with any equation. So the first thing we’re gonna do here is actually multiply each side of our inquality by 10. And the reason we do that is because actually what we want is we want whole 𝑥, but at the moment we’ve got 𝑥 over 10. So to get 𝑥, we multiply 𝑥 over 10 by 10. So then when we actually multiply each side of our inequality by 10, we get 𝑥, as we already explained, is less than or equal to 85. And that’s because 8.5 multiplied by 10 gives us 85. So we can say that we’ve actually answered the second part of the question as well.

So therefore, we can say that if Anthony and nine of his friends want to buy a birthday cake and they plan on splitting the cost, and we’re saying that it’s an even split and they want to spend at most eight dollars 50 per person, then the inequality to actually find the cost of the cake is 𝑥 over 10 is less than or equal to 8.5. And then the cost of the cake they should buy is 𝑥 is less than or equal to 85 dollars.