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- A carpenter is using a lathe to shape the final leg of a
- hand-crafted 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, it'd 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 2 and
- 1/2 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 the way to think about this, let's let w be 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 1 millimeter, we were
- over by 1 millimeter.
- If w is less than 150, it's going to be a negative number.
- If, say, w was 149, 149 minus 150 is going to be negative 1.
- 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 2 and 1/2 above or below.
- So we just really care about the absolute value of the
- difference between w and 150.
- This tells us, how much of an error did we make?
- And all we care is that error, that absolute error, has to be
- a less than 2.5 millimeters.
- And I'm assuming less than-- they're saying a margin of
- error of 2.5 millimeters-- I guess it could be less
- than or equal to.
- We could be exactly 2 and 1/2 millimeters off.
- So this is the first part.
- We have written an absolute value inequality that models
- this relationship.
- And 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's 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 a 150, that absolute
- distance, has to be less than 2 and 1/2.
- Now, we've seen examples of solving this before.
- This means that this thing has to be either, or it has to be
- both, less than 2 and 1/2 and greater than
- negative 2 and 1/2.
- 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 2.5.
- If the absolute value of something is less than 2 and
- 1/2, that means its distance from 0 is less than 2 and 1/2.
- For something's distance from 0 to be less than 2 and 1/2,
- in the positive direction it has to be less than 2 and 1/2.
- But it also cannot be any more negative than negative 2 and
- 1/2, 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 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 152.5, 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 2.5
- plus 150, that is 147.5.
- So the width of our leg has to be greater than 147.5
- millimeters and less than 152.5 millimeters.
- We can write it like this.
- The width has to be less than or equal to 152.5 millimeters.
- Or it has to be greater than or equal to, or we could say
- 147.5 millimeters is less than the width.
- And that's the range.
- And this makes complete sense because we can only be 2 and
- 1/2 away from 150.
- This is saying that the distance between w and 150 can
- only at most be 2 and 1/2.
- And you see, this is 2 and 1/2 less than 150, and this is 2
- and 1/2 more than 150.