Absolute value inequalities word problem

a carpenter is using a lathe to shape the final leg of a handcrafted table a lathe is this carpentry tool that spins things around and so it can be used to make things that are I guess you could say almost cylindrical in shape like a leg for a table or something like that in order for the leg to fit it needs to be 150 millimeters wide allowing for a margin of error of 2.5 millimeters so in an ideal world you're you would be exactly 150 millimeters wide but when you manufacture something you're not going to get that exact number so this is saying that we can be two and a half millimeters above or below that 150 millimeters now they want us to write an absolute value inequality that models this relationship and then find the range of widths that the table leg can be so way to think about this let's let W W be the width the width of the table leg so if we were to take the difference between W and 150 what is this this is essentially how much of an error did we make right if W is going to be larger than 150 let's say it's 151 then this difference is going to be one millimeter we were over by one millimeter if W is less than 150 it's going to be a negative number if say W is 149 149 minus 150 it's going to be negative one but we just care about the absolute margin we don't care if we're above or below the margin of error says we can be two and a half above or below so we just really care about the absolute value of the distance of the difference between W and 150 this tells us how much of an error did we make and we all we care is that error that absolute error has to be less than has to be less than 2.5 millimeters and I'm assuming less than they're saying a margin of error of 2.5 mm I guess it could be less than or equal to we could be exactly two and a half millimeters off so this is the first part we have written we have written an absolute value inequality that models this relationship but I really want you to understand this all we're saying is look this right here is the difference between the actual width of our leg and 150 now we don't care if it above or below we just care about the absolute distance from 150 or the absolute value of that difference so we took the absolute value and that thing the difference between W 150 that absolute distance has to be less than two and a half now we've seen examples of solving this before this means that this thing has to be either or X has to be both less than two and a half and greater than negative two and a half so let me write this down so this means that W minus 150 has to be less than 2.5 and W minus 150 has to be greater than or equal to negative two point five if the absolute value of something is less than two-and-a-half that means its distance from zero is less than two-and-a-half for some things distance from zero to be less than two and a half and the positive direction has to be less than two and a half but it also cannot be any more negative than negative two and a half and we saw that in the last few videos so let's solve each of these if we add 150 to both sides of these equations if you add 150 and we can actually do it both of them simultaneously let's add 150 on this side too what do we get what do we get the left-hand side of this equation just becomes a W these cancel out is less than or equal to 150 plus 2.5 is 150 2.5 and then we still have our and and on this side of the equation this cancels out we just have a w is greater than or equal to negative two point five plus 150 that is one hundred forty seven point five so the width of our leg has to be greater than 140 seven point five millimeters and less than 150 two point five millimeters we can write it like this the width has to be less than or equal to 150 two point five millimeters or it has to be greater than or equal to or we could say one hundred and forty seven point five millimeters is less than the width and that's the range and this makes complete sense because it can only be two and a half away from 150 this is saying that the distance between W and 150 can only at most be two and a half and you see this is two and a half less than 150 and this is two and a half more than 150