\[ |c-6|=|c-5| \] The solution set is

When you have an equation like \( |c-6| = |c-5| \), it means the distances from \( c \) to 6 and from \( c \) to 5 are equal. This occurs when \( c \) is exactly midway between 5 and 6, which is 5.5. However, there are also two scenarios to consider: \( c - 6 = c - 5 \) (which doesn't provide new solutions) and \( c - 6 = -(c - 5) \). Solving the second yields \( c = 5.5 \). Thus, the solution set is \( \{5.5\} \). Another way to think about this is considering the geometric interpretation of the absolute value. The points represent locations on a number line. The value 5.5 is the balance point between 5 and 6, making it a sweet spot where both distances to 5 and 6 are the same. If you were to fold that number line at 5.5, both sides would align perfectly!