How many solutions does the equation |x−A|=B have?

|x−A|=B

(1) A = 8
Many solutions are possible.
Insufficient

(2) B = -8
modulus can never be negative
—> 0 solutions
Sufficient

IMO B

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Forget conventional ways of solving math questions. For DS problems, the VA (Variable Approach) method is the quickest and easiest way to find the answer without actually solving the problem. Remember that equal numbers of variables and independent equations ensure a solution.
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The first step of the VA (Variable Approach) method is to modify the original condition and the question. We then recheck the question. We should simplify conditions if necessary.

Since we have \(|x-A| ≥ 0\), \(|x-A| = -8\) does not have a solution.
Then we have an answer 'no', which is a unique answer.
Since 'no' is also a unique answer by CMT (Common Mistake Type) 1, Condition 1) is sufficient.

Condition 1)
If we have \(A = 8\), \(B = 8\), then we have \(x = 0\) or \(x = 16\) and we have two solutions.
If we have \(A = 8\), \(B = -8\), then we don't have a solution.
Since condition 1) does not yield a unique answer, it is not sufficient.

Therefore, B is the answer.

But isn't the question asking for the number of solutions rather than for an unique solution?

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It's from Beat the gmat course.

Whether "No solution" is a "Definite Solution"? Nope.