Its amazing how students come to me only knowing the absolute value is always going to be positive but they don't always know why. Due to this lack of understanding, they sometimes have difficulty solving |x-7| = 5. Most of my students know x = 12 in |12 - 7| = 5 but they have difficulty with the other possibility.
Unfortunately, this carries over to solving absolute value inequalities such as |3x-3|> 6. The students who do not have the basic understanding down, will have difficulty solving this type of problem because they do not realize they have to solve two separate equations.
They have to look at this as 3x-3< -6 and 3x-3> 6. My students are able to solve for the positive value but not the second value. They have almost the same problem if the problem is |3x-3| < 6. I think its because they do not grasp the concept fully. I believe my students arrive in high school without a solid foundation in absolute values and inequalities.
So far, I've not had any students ask me when absolute value is used in real life. The most obvious one is distance. The distance is the same whether you go to the next town over or come back from the next town over. I tell the students, the positive or negative represents the direction.
Another situation which uses absolute value has to do with the speed limit. In most cases you are allowed to go x miles per hour over the speed limit or below the speed limit. Remember the signs on the freeway which state you may not go below.
Another absolute value application is the commission charged when converting from one country's money to another or back. The commission is the same regardless of which way the value is converted.
These three are the examples which pop up again and again but examples of absolute inequalities are much harder to find. Or at least real examples rather than seeming contrived. The best examples I found had to do with eggs or fruit or something similar where their weight varies a bit and the total can be no more than a certain amount.
The other reasonable example is situations when things are plus or minus a certain range such as plus or minus 1.5%. Statistical deviations fall into this category but those are really the only ones I found that seemed believable. I think the final example dealt with sales commissions and similar situations where the amount you receive is based on selling less than or equal to a certain amount.
Let me know what you think. I'd love to hear from people with real examples for absolute inequalities because I want to be ready for the day I'm asked the usual question. I also want to make sure my examples for inequalities are correct. Please feel free to comment.