An absolute value of a number must always assume a nonnegative value, so

, and

Therefore, 

and the range of  is the set .

Before we apply the absolute value to the two terms in the equation, we simplify what's inside of them first:

Now we can apply the absolute value to each term. Remember that taking the absolute value of a quantity results in solely its value, regardless of what its sign was before the absolute value was taken. This means that that absolute value of a number is always positive:

The key to answering this question is to note that this equation can be rewritten in piecewise fashion. 

If , since both  and  are nonnegative, we can rewrite  as

, or

.

On  , this has as its graph a line with positive slope, so it is an increasing function. The range of this part of the function is , or, since

, 

.

If , since  is negative and  is positive, we can rewrite  as

, or

 is a constant function on this interval and its range is .

If , since both  and  are nonpositive, we can rewrite  as

, or

.

On  , this has as its graph a line with negative slope, so it is a decreasing function. The range of this part of the function is , or, since 

, .

The union of the ranges is the range of the function - .

The key to answering this question is to note that this equation can be rewritten in piecewise fashion. 

If , since both  and  are positive, we can rewrite  as

, or

,

a constant function with range .

If , since  is negative and  is positive, we can rewrite  as

, or

This is decreasing, as its graph is a line with negative slope. The range is ,

or, since

and

,

.

If , since both  and  are negative, we can rewrite  as

, or

,

a constant function with range .

The union of the ranges is the range of the function -  - which is not among the choices.

This question plays a few tricks dealing with absolute values. To begin, we can recognize that any negative sign within an absolute value can basically be rendered positive. So this:

becomes:

In this case, we still have a negative that was outside of the absolute value sign. This term will stay negative, so we get:

This makes our answer .

To solve this absolute value inequality, we must remember that the absolute value of a function that is less than a certain number must be greater than the negative of that number. Using this knowledge, we write the inequality as follows, and then perform some algebra to solve for :

To solve this absolute value inequality, we must remember that the absolute value of a function that is greater than a certain number is also less than the negative of that number. With this in mind, we rewrite the inequality as follows and then solve for the possible intervals of :

   or   

   or   

   or