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A carpenter is using a lathe to shape the final leg of a

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hand-crafted table.

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A lathe is this carpentry tool that spins things around, and

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so it can be used to make things that are, I guess you

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could say, almost cylindrical in shape, like a leg for a

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table or something like that.

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In order for the leg to fit, it needs to be 150 millimeters

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wide, allowing for a margin of error of 2.5 millimeters.

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So in an ideal world, it'd be exactly 150 millimeters wide,

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but when you manufacture something, you're not going to

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get that exact number, so this is saying that we can be 2 and

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1/2 millimeters above or below that 150 millimeters.

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Now, they want us to write an absolute value inequality that

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models this relationship, and then find the range of widths

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that the table leg can be.

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So the way to think about this, let's let w be the width

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of the table leg.

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So if we were to take the difference between w and 150,

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what is this?

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This is essentially how much of an error

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did we make, right?

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If w is going to be larger than 150, let's say it's 151,

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then this difference is going to be 1 millimeter, we were

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over by 1 millimeter.

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If w is less than 150, it's going to be a negative number.

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If, say, w was 149, 149 minus 150 is going to be negative 1.

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But we just care about the absolute margin.

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We don't care if we're above or below, the margin of error

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says we can be 2 and 1/2 above or below.

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So we just really care about the absolute value of the

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difference between w and 150.

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This tells us, how much of an error did we make?

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And all we care is that error, that absolute error, has to be

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a less than 2.5 millimeters.

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And I'm assuming less than-- they're saying a margin of

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error of 2.5 millimeters-- I guess it could be less

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than or equal to.

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We could be exactly 2 and 1/2 millimeters off.

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So this is the first part.

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We have written an absolute value inequality that models

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this relationship.

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And I really want you to understand this.

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All we're saying is look, this right here is the difference

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between the actual width of our leg and 150.

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Now we don't care if it's above or below, we just care

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about the absolute distance from 150, or the absolute

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value of that difference, so we took the absolute value.

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And that thing, the difference between w a 150, that absolute

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distance, has to be less than 2 and 1/2.

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Now, we've seen examples of solving this before.

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This means that this thing has to be either, or it has to be

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both, less than 2 and 1/2 and greater than

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negative 2 and 1/2.

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So let me write this down.

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So this means that w minus 150 has to be less than 2.5 and w

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minus 150 has to be greater than or equal to negative 2.5.

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If the absolute value of something is less than 2 and

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1/2, that means its distance from 0 is less than 2 and 1/2.

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For something's distance from 0 to be less than 2 and 1/2,

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in the positive direction it has to be less than 2 and 1/2.

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But it also cannot be any more negative than negative 2 and

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1/2, and we saw that in the last few videos.

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So let's solve each of these.

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If we add 150 to both sides of these equations, if you add

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150-- and we can actually do both of them simultaneously--

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let's add 150 on this side, too, what do we get?

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What do we get?

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The left-hand side of this equation just becomes a w--

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these cancel out-- is less than or equal to 150 plus 2.5

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is 152.5, and then we still have our and.

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And on this side of the equation-- this cancels out--

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we just have a w is greater than or equal to negative 2.5

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plus 150, that is 147.5.

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So the width of our leg has to be greater than 147.5

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millimeters and less than 152.5 millimeters.

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We can write it like this.

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The width has to be less than or equal to 152.5 millimeters.

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Or it has to be greater than or equal to, or we could say

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147.5 millimeters is less than the width.

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And that's the range.

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And this makes complete sense because we can only be 2 and

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1/2 away from 150.

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This is saying that the distance between w and 150 can

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only at most be 2 and 1/2.

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And you see, this is 2 and 1/2 less than 150, and this is 2

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and 1/2 more than 150.

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