DH99 wrote:

Is |a| - |b| < |a-b| ?

Statement 1: \(b^a\)<0

Statement 2: |b|>|a|

Dear

DH99,

I'm happy to respond.

This is an intriguing question. I don't agree with the answer: I wonder if you posted the correct OA.

Statement #1 is particularly interesting.

\(b^a\)<0

This means a few things. The number b must be negative and it doesn't have to be an integer. The number a cannot be zero, but it could be any positive or negative odd integer.

Example #1: \(a = 3\) and \(b = -2\)

This yields \(b^a\) = \((-2)^3\) = \(- 8\) < \(0\), so these choices are consistent with Statement #1

Then \(|a| - |b| = |3| - |-2| = 3 - 2 = 1\)

and \(|a - b| = |3 - (-2)| = |5| = 5\)

This choice gives a "yes" answer to the prompt question.

Example #2: \(a = -3\) and \(b = -2\)

This yields \(b^a\) = \((-2)^{-3}\) = \(- \tfrac{1}{8}\) < \(0\), so these choices are consistent with Statement #1

Then \(|a| - |b| = |-3| - |-2| = 3 - 2 = 1\)

and \(|a - b| = |-3 - (-2)| = |-1| = 1\)

This choice gives a "no" answer to the prompt question.

Two different choices consistent with the statement yield two different answer to the prompt question, so statement #1, alone and by itself, is

insufficient.

We can do algebraic work with Statement #1

\(|b|>|a|\)

Subtract |b| from both sides

\(0 > |a| - |b|\)

So this difference now is always negative. Of course, a single absolute value is always positive (or zero of a = b, but that's excluded by this statement). Thus,

\(|a| - |b| < 0 < |a - b|\)

We absolutely have a "yes" answer to this statement.

Because we have arrived at a definitive answer, we know that statement #2, alone and by itself, is

sufficient.

I find the answer

(B).

Again, I am not sure whether something was copied incorrect, but my answer doesn't match the OA posted.

Does all this make sense?

Mike

_________________

Mike McGarry

Magoosh Test Prep

Education is not the filling of a pail, but the lighting of a fire. — William Butler Yeats (1865 – 1939)