Is |a - b| < |a| + |b|? (1) a/b <0 (2) a^2b < 0

Dear DH99,

I'm happy to respond.

This is a very clever question. Let's think about the prompt.

If a & b have the same sign, then subtracting produces a difference of a smaller absolute value.
\(5 - 3 = 2\\
(-5) - (-3) = -2\)
These are examples of choices that would produce "yes" answers for the prompt question.

If a & b have opposite signs, the subtracting produces a difference that has the same absolute value as the sum of the absolute values of a & b separately.
\(5 - (-3) = 8\) and \(|8| = 8 = |5| + |-3|\)
\((-5) - 3 = -8\) and \(|-8| = 8 = |-5| + |3|\)
These are examples of choices that would produce "no" answers for the prompt question.

So really, the prompt question is: do a and b have the same sign?

Statement #1: \(\frac{a}{b}\)<0
When is a fraction negative? It's negative when the numerator and denominator have opposite signs. Thus, a & b have opposite signs. We can give a definitive "no" to the prompt question. Because we can give a definitive answer, we know that Statement #1, alone and by itself, must be sufficient.

Statement #2: \(a^2b\)<0
I assume that this was entered correctly, and that DH99 meant \(a^2b\) and not \(a^{2b}\).

Taking this statement at face value, we know \(a^2\) is always positive, regardless of the sign of a, so we know that b just be negative. Thus, we know the sign of b, but the sign of a could be anything. We are not able to give a definitive answer to the prompt. Thus, we know that Statement #2, alone and by itself, must be insufficient.

OA = (A)

Does all this make sense?
Mike