Related range of values of f(x) necessary/sufficient/ with absolute values
I've asked a lot of times about absolute values.
But here I go again:
Consider two conditions
Where x is a real number, and a is a positive real number.
The range of a, so that is a necessary condition for
The range of a, so that is a sufficient condition for
This problem is mainly about logic, right?
Here is something that I thought, after calculating the range of , that is
If we have
-2________________________________5 ___________________________a=0 ______________________________ a=3
|_________________________________________________ _| <-|x-2|<a
Edit: This didn't work so well but, the a=0 is right below -2 and the a=3 is right below 5
This is our sufficient condition: 0<a<3
I'm not sure what the absolute value does here.
What does the absolute value mean?
Absolute values are only the distance to x=0, right?
After calculating -correctly- you get a nicer solution that is:
But how do we get such solution?
This relates to two range of values so this got a little out of control.
Somehow confusing, I wish I knew somewhere to find exercises of this kind!
I'm confused about that necessary/sufficient part
If you mow the lawn, you receive 10 dollars.
The range is sufficient for f(x) to work, or something like that?
But where does necessity comes in this? I know that if something is necessary it is not sufficient.
A implies B
A sufficient, B necessary
I appreciate any help!