Video: Writing and Solving Linear Inequalities with Unknowns on Both Sides in Word Problems

A moving company charges a flat rate of $150 and an additional $5 for each box. If a taxi service would charge $20 for each box, how many boxes would you move for it to be cheaper than the moving company.

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Video Transcript

A moving company charges a flat rate of 150 dollars and an additional five dollars for each box. If a taxi service would charge 20 dollars for each box, how many boxes would you move for it to be cheaper than the moving company?

So the first thing we need to do to be able to solve this problem is to actually form an expression for the cost of the moving company and also an expression for the cost of the taxi service. So we’re gonna do that straight right now. Well, for the cost of the moving company, we know we’re first gonna have 150, because that’s our flat rate, plus five 𝑥. And that’s five cause it’s five dollars for each box, where 𝑥 is the number of boxes.

And now we can have a look at the taxi service to get an expression for the cost of that. And the expression for the taxi service’s cost is going to be 20𝑥 because it’s 20 dollars for each box. And again, 𝑥 is the number of boxes, so therefore 20𝑥.

Now we can get on with solving the problem. And the keyword here is cheaper, because what we want to know, we want to know when the taxi service is actually cheaper than the moving company. And to do that, we can set up an inequality. So I’ve just added an extra little line in here just to think about it in words, what does it mean?

So the inequality would be that we want the taxi service to be less than the moving company. And we use this sign for less than, not the one with the line underneath for less than or equal to, because again going back to that keyword, it’s we want to know when it is cheaper for the taxi service than the moving company.

So then now our inequality is going to be 20𝑥 is less than 150 plus five 𝑥, cause 20𝑥 is our taxi service and 150 plus five 𝑥 is that cost from moving company. We solve this as we would solve an equation.

So first of all, we’re going to subtract five 𝑥 from each side, which gives us 15𝑥 is less than 150. And then the final step would be to divide both sides by 15, which gives us an answer of 𝑥 is less than 10. Therefore, an answer to our question, we could say that you would have to move less than 10 boxes for the taxi service to be cheaper.

And we can actually check that by actually looking at the cost of both the moving company and the taxi service when you have nine boxes, so 𝑥 is equal to nine, or when you have 10 boxes, when 𝑥 is equal to 10.

Well, with nine boxes, the moving company would be equal to 150 plus five multiplied by nine, which will give you a total cost of 195 dollars. And the taxi service would be equal to 20 multiplied by nine, which will be equal to 180 dollars. Okay, so yes, we can see here that, with nine boxes, the taxi service would be cheaper.

Now if we look across to 10 boxes, the moving company is going to be 150 plus five multiplied by 10, which gives us a total cost of 200 dollars. And the taxi service would be 20 multiplied by 10, which again would give us 200 dollars. So therefore, would not be cheaper than the moving company. So then we can say yes, with confidence, we are correct for saying that you would have to move less than 10 boxes for the taxi service to be cheaper.

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