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Course: Algebra (all content)>Unit 8

Lesson 3: Solving absolute value inequalities

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.

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• couldnt you just add 2.50 to 150 and subtract 2.50 from 150 and get the anwser?
• 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.
• How would you solve an equation like this: 8 | x - 3 / 2 | > 40 ? *This is only an example.
• 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.
• 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?
• 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.
• How do you write inequalities?

For example: "Passengers have to be at least 12 years of age."
• 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`
• At , 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?
• 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.
• I can't seem to find practice problems for Absolute value inequalities? Only lessons?
• I don't know if there are exercises currently on the site for this skill.
• At , 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.
• Is there a video for proving inequlities using substitution
• I think so, because if there isn't, then proving inequalities using substitution is a easy skill, but it isn't, so it must.