Video: Solving Linear Inequalities in a Real-World Context

A cell phone company offers the following two plans. Plan A: $15 per month and $2 for every 300 texts. Plan B: $25 per month and $0.50 for every 100 texts. How many texts would you need to send per month for plan B to save you money?

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Video Transcript

A cell phone company offers the following two plans. Plan A, which is 15 dollars per month and two dollars for every 300 texts. Plan B, 25 dollars per month and 0.50 dollars for every 100 texts. How many texts would you need to send per month for plan B to save you money?

Well, the way that we can set up and solve this problem is to first think, how much does it cost for plan A and plan B to send 100 texts? Well, if we take a look at plan A, it costs two-thirds of a dollar per 100 texts. And that’s because of its two dollars for every 300 texts. Just divide that by three. Then if we look at plan B, we can see that it’s half a dollar or 50 cents for every 100 texts.

Well, what we can do is we start set up an inequality. And we could do that because what we’re gonna have for plan A is 15 plus two-thirds 𝑥, where 𝑥 is the number of hundreds of texts. Well, we want to see where this is greater than. And we want to see where it’s greater than because what we’re trying to do is find out how many texts you’d need to send per month for plan B to save you money. So we want plan B to be less than. And then we’ve got for plan B 25 plus a half 𝑥.

So now what we do is we’ve converted both of our fractions into sixths to make it have a common denominator. Now we’ve got 15 plus four-sixths 𝑥 is greater than 25 plus three-sixths 𝑥. So now what we’re gonna do is subtract 15 and subtract three-sixths 𝑥 from both sides of the inequality. And we’re gonna do this because we want the 𝑥 on one side. And then we want the numeric values on the other. And when we do that, we’re gonna get a sixth 𝑥 is greater than 10.

So if we’ve got a sixth 𝑥 is greater than 10, if we multiply both sides by six, we’re gonna get 𝑥 is greater than 60. Well, as we said at the beginning, 𝑥 represents the number of hundreds of texts. So therefore, the number of texts that would need to be sent for plan B to save you money would be greater than 6000 texts. And we get that because we multiplied 60 by 100, which gives us 6000.

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