, which means that  must be positive. 

If  is nonnegative, then . If  is negative, then it follows that . Either way, . Therefore,  cannot be the least. 

We now show that we cannot eliminate  or  as the least.

For example, if , then  is the least;  we test both statements:

, which is true.

, which is also true.

If , then  is the least; we test both statements:

, which is true.

, which is also true.

Therefore, the correct response is  or  only.

, so  must be positive. Therefore, since , equivalently, , so  must be positive, and

If  is negative or zero, it is the least of the three. If  is positive, then the statement becomes

,

and  is still the least of the three. Therefore,  must be the greatest of the three.

If , then either  or . Solve separately:

or 

The solution set, in interval notation, is .

Since , evaluate

, setting  :

Since , then select the pattern

Since , evaluate

, setting :

, so the correct choice is that .

, which means that  must be positive. 

If  is nonnegative, then . If  is negative, then it follows that . Either way, . Therefore,  cannot be the least. 

Now examine the statemtn . If , then  - but we are given that  and  are distinct. Therefore,  is nonzero, , and 

and

.

 cannot be the least either.

, so  must be positive. Therefore, since , it follows that , so  must be positive, and

If  is negative or zero, it is the least of the three. If  is positive, then the statement becomes

,

and  is still the least of the three. Therefore,  must be the least of the three, and the correct choice is "None of the other responses is correct."

When dealing with absolute value bars, it is important to understand that whatever is inside of the absolute value bars can be negative or positive. This means that an inequality can be made.

In this particular case if , then, equivalently, 

From here, isolate the variable by adding seven to each side.

In interval notation, this is .

First you plug in  for  and squre it.  

This gives the expression  which is equal to .  

Since the equation is within the absolute value lines, you must make it the absolute value which is the amount of places the number is from zero.  

This makes your answer .

The absolute value of a number is its distance from zero.  Absolute value is represented with  or absolute value bars .  To solve this absolute value equation remove the absolute value bars and set the equation equal to positive and negative eleven.

 and  are the two values of  which make this statement true.

and

Remember, because Absolute Value measures the distance from zero, the absolute value of a number whether negative or positive is always                    non-negative.

Absolute value measures distance from that number to the point of origin or zero.

However, there is a negative sign outside the Absolute value bars, which indicates the multiplication by 

 becomes 

Therefore  is the correct solution.