Hello!

I know how to solve this task but I have certain things that I do not understand.

Solution.

if

Because |x-3| occurs on the right side of this "if statement", it will be helpful to factor the left side to introduce a factor of |x-3|.

if

Using the triangle inequality:

By multiplying the statement above with |x-3| (which is positive) we got:

Now clearly

and

By multiplying both inequalities (I am not sure if this step is valid, please confirm me) I got:

We can conclude that:

if

Now isn't

??

Why in my book they choose

?

And why they choose

and with that restriction

and

.

Finally I don't understand why

.

Could somebody possibly explain?

Thanks a lot.