Mathmatics Question?? Absolute Value and Inequalities

Ok, I received some help earlier.

I'm hoping for some kind souls to help again.

Now I'm completely lost on a problem.

Solve and Express in interval notation.

1<=|x|<=4

I came up with this

-4<=x<=4 and x<=-1 and 1<=x

the book says it is

[-4,-1]U[1,4]

I know what interval notation is, but I'm lost on the how they grouped the different x values.

What step am I missing to get the books answer?

[edit1]

Do we default grouping the negative values together and the positive values together?

I'm not sure, but let me try to explain why I see it grouping in this way.

-4 is less then or equal to x which is less then or equal to 4

this covers all values from -4 to 4

x is also less then or equal to -1

x is greater than or equal to 1

-4 to 4 is my outside boundries

so

-4 to -1 is one of the boundries

1 to 4 is the upper side boundry.

Is this the way they are looking at it or is there a fail safe rule that I can use?

[/edit1]