Learn about Special Cases of Absolute Value Equations In this lesson, let’s learn about special cases when we’re dealing with absolute value equations. We’ll just do two special equations. First one is |x+6| + 7 = 7. So since we see that this is the absolute value |x+6| and I've added 7 to it. Let’s subtract 7 from both sides. What do I get? So I get |x+6| = 0. Now since the absolute value is 0, only one value can have an absolute value of 0 because 0 doesn’t have +0 and –0. So if |x+6|=0, x = -6. So the answer in this case is x=-6. There is only one value that’s -6 which is the answer. The reason we have only one value instead of two like we normally do is because the mile marker is 0 where the marker is 0 which means that the distance from the 0 on the number line is 0. There can only be one value that is 0 units away from 0. There is no going left or no going right. Let’s look at the second problem. It says 8 = |x+2| +11. So again I'm going to solve this, |x+2|. Since we added 11 on the right side let’s subtract 11 from both sides. What do I get, 8 – 11 = -3 = x+2. So |x+2| = -3. Since an absolute value can never be a negative, what is an absolute value? Absolute value is the distance from 0 on the number line. So how can distance be negative? So this means that this equation has no solution because the absolute value is negative. Distance can not be negative which means any absolute value can not be negative. That means this has no solution.