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   

Remember that the absolute value of any number is its positive value, regardless of whether or not the number is negative before the absolute value is taken. We start by simplifying any expressions inside the absolute value signs:

Now we apply the absolute values and solve the expression:

In order to solve the given absolute value equation, we need to solve for  for the two ways in which this absolute value can be solved:

1.) 

2.) 

Solving Equation 1:

Solving Equation 2:

Therefore, there are two solutions to the absolute value equation:  and 

In order to solve the given absolute value equation, we need to solve for  in the two ways in which this absolute value can be solved:

1.)

2.) 

Solving Equation 1:

Solving Equation 2:

Therefore, there are two correct values of :  and .