Question Video: Solving Linear Inequalities in a Real-World Context

Elizabeth has saved $31 in her piggy bank and her sister Hannah has saved $36. If Hannah saves $6 a week and Elizabeth saves $9 a week, after how many weeks will Elizabeth have saved more than Hannah?

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Video Transcript

Elizabeth has saved 31 dollars in her piggy bank and her sister Hannah has saved 36 dollars. If Hannah saves six dollars a week and Elizabeth saves nine dollars a week, after how many weeks will Elizabeth have saved more than Hannah?

In order to solve this problem, what we need to do is actually take a look at the information we’ve been given and set ourselves up an inequality. So first of all, I’m gonna start with Elizabeth. And what I’m gonna say is that we’re gonna call the number of weeks 𝑤. You can use any letter; I’ve just chosen 𝑤 because it seems relevant as we’re talking about weeks.

If we want to talk about how much money Elizabeth has, we start off with 31 because we’ve got 31 dollars in her piggy bank. And then this is gonna be plus nine 𝑤. And it’s nine 𝑤 because she saves nine dollars per week and 𝑤 is the number of weeks. Now, here’s where we use the inequality notation.

So if we check out the question, the question says after how many weeks will Elizabeth have saved more than Hannah. And then we have the inequality sign with the open end where the 31 plus nine 𝑤 is. And that’s because if we remind ourselves of our notation for inequalities, if we have 𝑎 is greater than or more than 𝑏, then we have the 𝑎 by the open end of the inequality sign. If we have 𝑎 is less than 𝑏, then we have 𝑎 at the closed or pointy end of our inequality sign.

So now, what we want to have a look at is actually Hannah and how much money that Hannah will have. Well, on the right-hand side of our inequality, we’re gonna have 36 and that’s because Hannah has saved 36 dollars plus six 𝑤 and that’s because she saves six dollars per week. And obviously, as we’ve mentioned before, 𝑤 is the number of weeks.

So therefore, our full inequality reads 31 plus nine 𝑤 is greater than 36 plus six 𝑤. So if we take a look at the question, what we’re trying to do is find out after how many weeks will Elizabeth have saved more than Hannah. So what we need to do is actually solve our inequality. And we solve an inequality in much the same way we’d solve an equation.

So the first thing we’re gonna do is actually subtract 31 from each side. And we’re gonna do that because we want the most 𝑤s left on the same side and that’s on the left-hand side of this inequality. And when we do that, we get nine 𝑤 and that’s because 31 minus 31 is zero is greater than and then we’ve got five and that’s because 36 minus 31 is five plus six 𝑤.

So now, what we’re gonna do is actually subtract six 𝑤 from each side of the inequality. So when we do that, we’ve got three 𝑤 is greater than five. And then, as we want to find out 𝑤, what we’re gonna do is divide each side of the inequality by three. And when we do that, we get 𝑤 is greater than five over three or five-thirds. And if we turn it into a decimal, it’ll be 𝑤 is greater than 1.67 and I’ve rounded it here to two decimal places.

So if we want to actually answer the question after how many weeks will Elizabeth have saved more than Hannah, we’ve got 𝑤 here as greater than 1.67. So therefore, we know that the number of weeks is greater than 1.67. So therefore, we can say that after two weeks, Elizabeth will have saved more than Hannah. And that’s because if 𝑤 was greater than 1.67, then the whole number of weeks that’s bigger than 1.67 and closest to it is two weeks.

What we can actually do is we can check that. And to check it, what we’re gonna do is actually see how much is saved either side of our 1.67. So we’ll see how much is saved after one week and how much is saved after two weeks.

Well, after one week, Elizabeth was gonna have saved 31 because that’s the original amount in her piggy bank plus then we’ve got nine multiplied by one because one was the number of weeks. So it’ll give us 40 dollars. And Hannah will have saved 36 because that was it- what was in her piggy bank plus six multiplied by one because she saves six dollars a week and it’s been one week and that’s gonna be equal to 42 dollars.

So as you can see, at this point, Hannah has saved more than Elizabeth. So now, we can move on to two weeks because this was the answer we got. Well, Elizabeth will have saved 49 dollars and that’s because we had 31 dollars that she already had plus nine multiplied by two because it’s nine dollars per week multiplied by two weeks. So it’s 31 plus 18 is 49. We also could have just added on nine to our first week total because she’s getting nine dollars a week in her savings. So as we said she’s got 49 dollars.

And for Hannah, showing the method that I’ve just discussed, I’ve just added on six to her first week total because she saves six dollars per week. And you can see that her total now after two weeks is 48 dollars. And now, we can see that Elizabeth has saved more.

And our answer is correct that Elizabeth will have saved more after two weeks.

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