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# Absolute value inequalities word problem

Video transcript

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.