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.

To solve an equation with an absolute value, recognize that there are two things that would make an absolute value equal 5: the inside of the absolute value could equal 5, or it could equal -5. So your job is to solve for both possibilities:

Possibility 1:

You can then subtract 3 from both sides to get:

And divide both sides by 2 to finish:

Possibility 2: 

Subtract 3 from both sides to get:

Divide both sides by 2:

Your two solutions are 1 and -4. Since the question wants the sum of the two answers, you can add them together to get your answer, -3.

Inequalities can be solved just like equations, with one important caveat: if you multiply or divide by a negative number, you have to flip the inequality sign. Here, as you will see, there is no need to do that, so you can solve this just like you would an equation. First, multiply both sides by 2 to eliminate the denominator:

Next, add 2 to both sides to isolate the  term:

Then divide both sides by 3 to get  alone: