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## Algebra (all content)

### Course: Algebra (all content) > Unit 8

Lesson 3: Solving absolute value inequalities- Intro to absolute value inequalities
- Solving absolute value inequalities 1
- Solving absolute value inequalities 2
- Solving absolute value inequalities: fractions
- Solving absolute value inequalities: no solution
- Absolute value inequalities word problem

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# Absolute value inequalities word problem

Sal solves a word problem about a carpenter by writing an appropriate absolute value inequality and solving it. Created by Sal Khan and Monterey Institute for Technology and Education.

## Want to join the conversation?

- couldnt you just add 2.50 to 150 and subtract 2.50 from 150 and get the anwser?(13 votes)
- No, this would result in the right numbers, but the wrong inequality. You would have gotten w <= 152.5 and w <= 147.5. Which is equal to W being less than 147.5 to -infinity.(6 votes)

- How would you solve an equation like this: 8 | x - 3 / 2 | > 40 ? *This is only an example.(3 votes)
- Chang,

I would solve it like this:

First, divide both sides of the equation by the 8, leaving:

|x-3/2|>5

Now, you have to deal with the absolute value symbol. The easiest way to do this is to turn the one equation into two equations. Absolute values have the effect of changing what is inside the symbols into a positive number if they weren't already, so we have to consider the case where they were already positive, and the case where what's inside the symbol was negative to begin with. This will give you two answers.

The first case: Everything was already positive:

|x-3/2|> 5 becomes x-3/2>5. So you add 3/2 to both sides and one of the two answers is x>6 1/2.

Second case: (x-3/2) was a negative number, whose absolute value was greater than 5. That means that (x-3/2) was a number that was less than negative 5:

|x-3/2|>5 becomes (x-3/2) < -5 (NOTICE THE DIRECTION OF THE INEQUALITY SWITCHED).

Now add 3/2 to both sides and x < -6 1/2.

So the two answers are x>6 1/2 and x<-6 1/2.(10 votes)

- From the initial question we are told that there is a margin of error of 2.5mm, so my question is why was the answer w<=152.5 and w>=147.5 which would give an overall difference of 5mm not 2.5 right?(5 votes)
- The answer is 147.5<=w<=152.5 because having a margin of error of 2.5 mm means being 2.5 mm off from 150 going both ways on a number line.(6 votes)

- How do you write inequalities?

For example: "Passengers have to be at least 12 years of age."(5 votes)- An inequality is written on a keyboard as:

Less than:`<`

Greater than:`>`

Less than or equal to:`<=`

Greater than or equal to:`>=`

When converting sentences to inequalities, it's helpful to plug in a couple numbers to see if they fit. If they need to be**at least**12, then that means they can**be**12. That's a flag telling you that the equal sign will be included in the answer. An 11 year old would not be able to ride the bus, but a thirteen year old would. Essentially, when a person's age is**greater**than 12, the equation also works.

Therefore:`age >= 12`

(6 votes)

- At1:44, how do we know that it is a less than or equal to sign? How can you tell which sign it is going to be?(5 votes)
- In this case, Sal was assuming that it is a <= sign because the question seems to indicate that the error margin can be less than 2.5 or equal to 2.5, since the question does not specify.(5 votes)

- I can't seem to find practice problems for Absolute value inequalities? Only lessons?(4 votes)
- I don't know if there are exercises currently on the site for this skill.(2 votes)

- At1:39, the term of "absolute value" is supplemented by the term "absolute error". Neither of these terms is explained in detail. I can assume I understand what "absolute value" means but I would appreciate a little elaboration, as well as the use of the two lines on either side of the figure. I may have missed the lesson on this but it is the first time I have ever seen these symbols. Please explain their usage with a few examples outside of the present one, for comparison.(1 vote)
- Is there a video for proving inequlities using substitution(2 votes)
- I think so, because if there isn't, then proving inequalities using substitution is a easy skill, but it isn't, so it must.(2 votes)

- Can it also be W + 150 >= 2.5?(1 vote)
- No. The problem states the width needs to be 150 mm. This means we want the width to = 150 mm. It also says we are allowed some margin of error in our measurements by 2.5mm. This means the actual measurement can range from 150-2.5 up to 150+2.5. Thus, the inequality could be written as: 150-2.5 <= w <= 150+2.5.

Your version wouldn't create this type of relationship. Your version would create the inequality: -150-2.5 <= w <= -150+2.5

It's impossible for a table leg to have a negative number as its width.(3 votes)

- |w-150| means it doesn't matter if will be up to 2.5 milimeters missing or there will be up to 2.5 milimeters left or leaving(I don't know, i'm foreign 7;^(] ) and the second part just says that it must be someway about 2.5 milimeters.(2 votes)
- That's correct. By the way, I would say "up to 2.5 millimeters LEFT OVER". Congratulations on writing such good English, most English-speakers don't know how to write so well. Where are you from?(1 vote)

## 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.