Related range of values of f(x) necessary/sufficient/ with absolute values

Hi Forum.

I've asked a lot of times about absolute values.

But here I go again:

Consider two conditions

and

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?

is

** if ** and

** if**

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

**Sufficient**

If you mow the lawn, you receive 10 dollars.

**____________________Necessary**

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--->B

A implies B

A sufficient, B necessary

I appreciate any help!

Thanks!(Bow)