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Video: Solving Story Problems Involving Linear Inequalities

Tim Burnham

We look at some story problems that require you to construct and solve linear inequalities. We show you how to define your variables and how number line diagrams can help you identify correct variable ranges for your solutions.


Video Transcript

In this video, we’re gonna write and solve some inequalities in order to answer questions raised in story problems or, as some people call them, real-life scenarios. In all of these, you have to carefully define your variables and think about how the information you’ve been given translates into equations or inequalities. Okay, let’s take a look at our first problem.

Barry has already saved ninety dollars. He can now save twenty-three dollars per week. Barry wants to buy a bike for four hundred and fifty dollars. Write and solve an inequality to determine how many weeks Barry needs to save for to be able to afford the bike. So first of all, we need to have a variable, let’s call it 𝑥, which represents the number of weeks that Barry needs to save for.

And although number of weeks is a time variable and time’s a continuous thing, in this particular case we’re only gonna deal with whole numbers of weeks because we can save a set amount per week. Presumably Barry’s being paid per week, so we can’t do one and a half weeks, one and three-quarter weeks. He’ll be paid in twenty-three dollars in that week So we’re only gonna allow integer values for 𝑥.

And now we’re gonna come up with a formula that tells us how much money Barry has got saved after 𝑥 weeks, and we’re gonna use the units of dollars for that because the bike is four hundred and fifty dollars. Well he starts off with ninety dollars so that’s ninety to start off with. And then every week, he gets an additional twenty-three dollars, so that’s twenty-three times the number of weeks.

So the formula for the amount that Barry has saved is ninety plus twenty-three times 𝑥, the number of weeks. Now if Barry is gonna be able to afford the bike, then he basically- the amount that he saved needs to be at least four hundred and fifty dollars, so greater than or equal to four hundred and fifty dollars.

If the amount saved is greater than or equal to four hundred and fifty, that means ninety plus twenty-three 𝑥 is greater than or equal to four hundred and fifty.

Now we can solve that inequality. So first of all, let’s subtract ninety from both sides. And that means twenty-three 𝑥 is greater than or equal to three hundred and sixty. Now if we divide both sides by twenty-three we’ll just get one 𝑥 on the left-hand side. So that means 𝑥 is greater than or equal to three hundred and sixty over twenty-three, which is fifteen and fifteen twenty-thirds.

So let’s just represent that on a number line. 𝑥 is greater than or equal to fifteen and fifteen twenty-thirds, so fifteen and fifteen twenty-thirds is about here. And it’s included in the range so the valid values for 𝑥 are fifteen and fifteen twenty-thirds onwards.

But earlier in the question, we said that 𝑥 is an integer; we’re only accepting whole numbers of weeks, so 𝑥 could be fourteen or fifteen or sixteen. So we’re looking for the smallest number of weeks which is in the valid region, so the valid region for values of 𝑥 in order for Barry to be able to afford the bike is to the right of fifteen and fifteen twenty-thirds.

And the first integer that crops up in that region is sixteen. So Barry needs to save for at least sixteen weeks in order to be able to afford the bike.

Now just as a little kinda validation of that, after sixteen weeks we’ll have- he’ll have ninety plus twenty-three times sixteen; that’s four hundred and fifty-eight dollars, so he’ll have eight dollars to spare. After fifteen weeks, he would only have had four hundred and thirty-five dollars, so he would’ve been fifteen dollars short of his target for the bike. So that’s why the answer is sixteen weeks.

So number two. You have a problem with your heating system. That’s a bit sad. You need to choose between two local plumbers to get it fixed. Arnie charges a fifty-dollar call-out fee and twenty-seven dollars per hour. And Bob charges no call-out fee, but he charges thirty-six dollars per hour for his work. You’ve gotta write and solve an inequality to find out how many hours the job would have to take for Arnie to be cheaper than Bob.

So if the work only took one hour, then Arnie would charge his fifty-dollar call-out fee plus his twenty-seven dollars for the first hour’s work, but Bob would only charge thirty-six dollars for his one hour’s work. So if it’s just one hour, then clearly Bob’s cheaper. But what we’ve gotta find out is how many hours would have to go by before it would actually be cheaper to use Arnie.

So we’re gonna set up a variable called 𝑥 and that represents the number of hours of work that need to be done. And in most cases, people will charge you for a whole number of hours. So we’re gonna make 𝑥 an integer so people will really charge for one hour or two hour three hour. We’re not gonna do the continuous time thing and get fractions of hours.

So let’s come up with an expression for the cost of Arnie’s work. Well it’s the initial fifty dollars plus twenty-seven dollars for every hour’s work. So twenty-seven times 𝑥. And that’s in dollars. And Bob’s cost is just thirty-six times- so thirty-six dollars for every hour, so thirty-six times 𝑥 dollars

And so if Arnie is to be cheaper than Bob, then this has got to be less than this. So fifty plus twenty-seven 𝑥 has gotta be less than thirty-six 𝑥.

So let’s rearrange and solve this. Well first of all, we’ve got twenty-seven 𝑥 on the left-hand side and thirty-six 𝑥 on the right-hand side. So if I subtract twenty-seven 𝑥 from both sides, that means I’m just left with fifty on the left-hand side and nine 𝑥 on the right-hand side. So if I now divide both sides by nine, it means that fifty over nine is less than 𝑥.

Well fifty over nine is the same as five and five-ninths so five and five-ninths is less than 𝑥 or 𝑥 is greater than five and five-ninths. That’s probably the easier way to put it. So again let’s just represent that on our number line just to visualise what’s going on here. 𝑥 has to be greater than but not equal to five and five-ninths, so five and five-ninths is in here somewhere.

and 𝑥 has to be bigger than that value, so anything to the right. If the number of hours is more than five and five-ninths, then Arnie will be cheaper than Bob. So the first whole number that we come across after five and five-ninths is six. So for five hours’ work, Bob will still be a bit cheaper. But by the time we get to six hours’ work, Arnie will be slightly cheaper.

So there’s our answer: Arnie’s cheaper for six or more hours. Let’s just do a quick check before we stop. For six hours using those numbers, Arnie would cost two hundred and twelve dollars and Bob would cost two hundred and sixteen dollars. And for five hours, Arnie would cost a hundred and eighty-five dollars but Bob would cost a hundred and eighty dollars, so Bob would be cheaper. So yep, that looks like we’re right.

So let’s summarise the basic process. The first important thing was to define our variables very carefully thinking about, “Is it gonna be an integer or is it gonna be a real number, continuous real number?” And we also have to pick out the relevant information from the question and create ourselves som- maybe some equations or maybe an inequality straight away. In this case, we were able to create two expressions for the cost of Arnie’s work and for the cost of Bob’s work. And then our inequality was in comparing one to the other. The next thing is to go ahead and solve that, so rearrange and solve that inequality.

And finally, presenting your answers on a number line on some sort of diagram really helps you to understand what the correct answer is. And finally finally finally, make sure you put a box around your answer and make it clear what your answer is.