If you're looking for a number exactly  units to either side of , you need an absolute value equation that gives you answers of  and .

One way forward is to simply solve each equation for  using the two-case method, assuming first that the expression within the absolute value signs positive and then that it is negative and solving the resulting equation for each case.

If you do so, choice  becomes: , or  and , or , which is correct.

Choice  becomes: , or , which doesn't work. 

Choice  becomes: , or , which doesn't match. 

Choice  becomes: , or , which doesn't work.

You could also quickly solve this question if you remember the general way number line absolute value problems like this one work. Each one will be in the general form , where the "Distance" referenced is the distance from the center value to one of the two values for .

So in this case the "center value" would be  and the "distance" would be , yielding an equation of .

Any time you have an absolute value equation and a question that is asking for the smallest possible value for the expression, remember that absolute value expressions can be zero but can never be less than 0. There are two ways to solve a problem like the one given. The first is by finding the value of  such that the expression within the absolute value signs is zero and then finding the closest integer to that number. The second is by inspection (simply using brute force to find the value for xx that will minimize the value given). 

If you set , and that . Since , the closest integer value is , which will yield a value of .

The other way to approach this problem is by inspection, or through brute force. Try plugging in a few values for  to see what happens.

If  then the expression is 

If , then the expression is .

If x=2x=2 then the expression becomes .

Notice that the function decreases and then increases around , indicating that (for integer values, at least)  will yield the smallest result for .

This problem is a perfect candidate to test the answer choices, as the calculations required are not particularly difficult but the algebraic setup can be quite challenging to conceptualize. If you look for the two numbers that are 66 units away from −4−4, they are: 

Then plug those values in as  and see which equations work. For answer choice ,  does equal , and  also equals , so  satisfies the equation.

Even if you're unsure of where to start on this problem, you should have a head start. The problem is testing absolute values, and you should know that the result of any absolute value is always nonnegative, . So the answer choices that include an absolute value equalling a negative number must be incorrect: that just cannot be possible.

To test the remaining choices, consider that the numbers that are exactly two units away from -3 are -3+2 = -1, and -3-2 = -5.  When you plug these numbers in for  in the answer choices, only one is valid:

 gives you:

 --> 

 --> 

Therefore this absolute value satisfies the given situation, and is correct.

It is important to recognize that absolute values must be nonnegative, . That means that for this given expression, the  can only go as low as , and then the second part of the expression asks you to add . So this expression can never equal zero: it's an absolute value added to 1, so the lowest this expression can be is 1.

To solve an absolute value like this, recognize that there are two outcomes inside an absolute value that would have it equal 3.  If the inside of an absolute value expression is 3, then the result is 3. Or if the inside of an absolute value expression equals -3, the absolute value will equal 3.  So you can solve this as two equations:

 and .

Solving for the first one, you have:

And solving for the second one, you have:

Therefore, the correct answer is: 

When you're solving equations involving absolute values, it's important to recognize that there are generally two solutions. Here if the inside of the absolute value equals 1, you've solved for  -- or if the inside of the absolute value equals -1 you've also solved for . So you should solve this as two equations:

Possibility 1

Here you can add 7 to both sides to get: 

And then divide both sides by 2:

Possibility 2

Add 7 to both sides:

 And divide both sides by 2:

The question asks for the sum of all answers, so add  to get the right answer, 

An important thing to know about absolute values is that their minimum value is zero; absolute values must be nonnegative.  So here if you take the result of an absolute value and then add 1 to it, it simply cannot equal 0. To do so, the absolute value itself would have to equal -1, and that is just not possible.

One helpful shortcut on this problem is just understanding that the result of an absolute value can never be negative. So an answer choice like  simply cannot be correct: it's not a valid equation.

Of course, there are three remaining choices so your guessing probability isn't high enough to quit now. To solve this, first think about which values are 5 units away from -3. Since -3 + 5 = 2, and -3 - 5 = -8, you have two values that you know fit the definition: 2 and -8.

Now plug those numbers into the answer choices to see which fit with one of the absolute values. You'll see that  fits:

, so this satisfies 

, so this satisfies 

When an absolute value is said to equal 7, that has two meanings: the expression inside the absolute value could equal 7, or it could equal -7.  Whatever the value is within the absolute value sign always becomes nonnegative, so both  and  are equal to 7.

So here you need to solve this problem for both cases:

Case 1

Add 1 to both sides to get:

Case 2

Add 1 to both sides to get 

Therefore, the correct answers are -6 and 8.