Final answer:
The sum of the roots that satisfy the equation || x-2 | -6 | = 8 is 4, which comes from case 1 as case 2 does not provide any valid roots.
Step-by-step explanation:
To solve for the sum of the roots that satisfy the modular equation || x-2 | -6 | = 8, we'll break it down into two cases based on the definition of the absolute value. For an absolute value equation |y| = a, y can be equal to a or -a.
Case 1: If we assume that | x-2 | - 6 = 8, then | x-2 | = 14. This implies that x - 2 = 14 or x - 2 = -14, giving us x = 16 or x = -12.
Case 2: If we assume that | x-2 | - 6 = -8, this case is not possible because the absolute value is always non-negative, and thus | x-2 | - 6 cannot be a negative number.
So we only have two solutions from case 1, which are x = 16 and x = -12. The sum of the roots is 16 + (-12) = 4.