# Video: Solving Linear Inequalities in a Given Context

The cost of producing “doomflots” is \$325 for equipment plus another \$32 for materials and labor for each one produced. How many “doomflots” need to be produced such that the average cost per “doomflot” is less than \$50.

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### Video Transcript

The cost of producing “doomflots” is 325 dollars for equipment plus another 32 dollars for materials and labor for each one produced. How many “doomflots” need to be produced such that the average cost per “doomflot” is less than 50 dollars?

So in order to actually solve this problem, what we want to do is actually take our key information and use it to form an inequality. And in order for us to actually create the inequality, what we need is a variable, so in this case, I’m gonna call it 𝑥. And 𝑥 is gonna be our number of “doomflots.”

So first of all, we’re gonna have 325 because that’s 325 dollars because that’s the cost of equipment and then we’re gonna have add 32𝑥 and that’s because 32 dollars is the cost of materials and labor for each one produced. And 𝑥 is the number of “doomflots” produced.

And then, if we look at the question, it says that we want to work out how many “doomflots” need to be produced such that the average cost per “doomflot” is less than 50 dollars. So therefore, we’re gonna put less than sign.

And just to remind ourselves how the more than and less than signs work, if we got 𝑎 is less than 𝑏, it means that the 𝑎 is where the small end of our inequality sign is. If 𝑎 is greater than 𝑏, it’s where the open end is.

And then, we have 50𝑥 on the right-hand side of our inequality. That’s because we’re looking for it to be less than 50 dollars per “doomflot” that’s produced and 𝑥 is our number of “doomflots.” So now, in order to solve our inequality, we solve it in the same way we’d solve an equation. So we substract 32𝑥 first of all from each side.

So when we do that, we get 325 is less than 18𝑥 and that’s because 50 minus 32 is 18. So 50𝑥 minus 32𝑥 is gonna give us 18𝑥. So next, what we’re gonna do is actually divide each side by 18 because we wanna find out one 𝑥. So in that case, what we do is we do 325 divided by 18 is less than 𝑥. So therefore, if we actually solve the inequality, we get 𝑥 is greater than 18.05. I’ve just swapped it round the other side so that we have the 𝑥 on the left-hand side.

So let’s think about this. We’ve got 𝑥 is greater than 18.05 and 𝑥 is the number of “doomflots” produced. And therefore, we can say that if the cost of producing “doomflots” is 325 dollars for equipment plus another 32 dollars for materials and labor for each one produced, then the number of “doomflots” that need to be produced such that the average cost for “doomflot” is less than 50 dollars must be 19 because 19 is the nearest number of “doomflots” greater than 18.05.

So now, even that we’ve got our answer, what I want to do is actually check to see if it is the right answer. And to do that, what I’m gonna do is actually substitute in a value of 18 for our number of “doomflots” and a value of 19 for our number of “doomflots” to see what answers we get.

So first of all, if we’ve got 18 “doomflots,” we’re gonna have 32 multiplied by 18 because that’s the cost of materials and labor multiplied by the number of “doomflots” plus our 325 dollars because that’s our amount for the equipment and then divide it by the total number of “doomflots” which is 18 because we want to find out an average cost per “doomflot.” And this gives us an average cost per “doomflot” of 50 dollars and six cents, which shows that 18 isn’t enough “doomflots” because 50 dollars and six cents or 50.06 is greater than 50.

So let’s try 19. Well, when we try 19, we have 32 multiplied by 19 plus 325 all divided by 19, which is gonna give us 49 dollars and 11 cents. Well, 49 dollars and 11 cents is actually less than 50 dollars. So therefore, yes, this is confirmed that 19 is the number of “doomflots” that must be produced.