How to solve absolute inequality problems with complete understanding

[Steven G]

|x-a| means the distance between x and a.
so |x-2|>5 means that the distance between x and 2 is more than 5. So go to 2 on a number and go more than 5 to the right (so x>7) and go more than 5 to the left of 2 (so x<-3) and we are done.

If we have |x-3|<2 then this means that the difference between x and 3 is less than 2. If you go to 3 and move around on the number line but do not go more than 2 away then you are between 1 and 5, so 1<x<5

We did not use any boring memorized steps, but rather we took the easy way out and understood what the problems means.

 

[pka]

|x-a| means the difference between x and a.

Actually \(\displaystyle |x-a|\) means the distance between \(\displaystyle x~\&~a\).
Please do definitions correctly if we are going to do them at all.

 

[stapel]

|x-a| means the difference between x and a.

No; "x - a" means "the difference of x and a". Perhaps you mean "distance"...? (Although, technically, the absolute value is defined in terms of square roots of squared values.)

We did not use any boring memorized steps, but rather we took the easy way out and understood what the problems means.

I'm sorry, but I don't understand what point you're trying to make...? In my experience, most instructors first explain "absolute value" in the intuitive sense of "distance apart", so what you've posted is nothing new. But the "too hard" (for you?) method you discount is needed later, when "distance" concepts won't work so well. Responsible instructors generally (eventually) cover both methods.

 

[Steven G]

Actually \(\displaystyle |x-a|\) means the distance between \(\displaystyle x~\&~a\).
Please do definitions correctly if we are going to do them at all.

You are correct and I knew that but at 1 am in the morning things sometimes do not come out correctly. Thanks for pointing this out.

 

[Steven G]

Responsible instructors generally (eventually) cover both methods.

I agree with this statement 100%.
What I have seen is that after talking about distance and absolute values many instructors then give rules for solving them to be memorized. This is what I do not like.

 

[lookagain]

There are things any of us do not like.
I think I hate this one the most:
3x - 6 = x + 2
3x - x - 6 = x - x + 2
2x - 6 = 2
2x - 6 - 2 = 2 - 2
2x - 8 = 0
2x - 8 + 8 = 0 + 8
2x = 8
2x / 2 = 8 / 2
x = 4

May be ok for a teacher to do this once or twice to "illustrate", but why force
students to keep showing all those steps...YES, some teachers do!

*None* of the elementary algebra teachers should be showing that way above
ever in class, anyway, because it's inefficient. Once the option is taken to anchor
down the variable term on a side, there should never be shown needless steps of bringing
the constants together on the same side as the variable term in this linear equation.

Now it has to be undone to get the constant on the other side.


This is all that should be shown if the steps are to be detailed:

3x - 6 = x + 2
3x - x - 6 = x - x + 2
2x - 6 = 2
2x - 6 + 6 = 2 + 6
2x = 8
2x/2 = 8/2
x = 4