With INEQUALITIES, we have to remember: if we multiply or divide both sides by a negative, the inequality sign flips. In this problem, you might be tempted to simplify the inequality in the question stem by multiplying both sides by (a - b), but since we don't know the sign of either variable, we won't know whether to flip the inequality sign.

If we want to simplify, we will have to give ourselves 2 cases:

Case 1: (a - b) is positive
Simplify:
1/(a - b) < b + a
1 < (b + a)(a - b)
1 < a^2 - b^2

Case 2: (a - b) is negative
Simplify:
1/(a - b) > b + a
1 > (b + a)(a - b)
1 > a^2 - b^2

Rephrased question: is 1 < a^2 - b^2 if (a - b) is positive, or is 1 > a^2 - b^2 if (a - b) is negative?

(1) (a + b)(a - b) < 1
If (a - b) is positive, this will give us a "yes" answer to the question, but if (a - b) is negative, it will give us a "no" answer. Since we don't know, this is INSUFFICIENT.

(2) ab > a - b
This can't be simplified or rearranged to match our target question, so we'll have to test numbers that fit:

test 1: a = 1, b = 2
(1)(2) > 1 - 2 --> this fits the statement
question: Is 1/(a - b) < b + a ?
1/(1 - 2) < 2 + 1
1/(-1) < 3
answer: yes

test 1: a = -1, b = -2
(-1)(-2) > -1 - (-2)
2 > 1 --> this fits the statement
question: Is 1/(a - b) < b + a ?
1/(-1 - (-2)) < -2 + -1
1/(1) < -3
answer: no

INSUFFICIENT

(1) & (2) Together:
Again, we'll have to test values here. Both sets of values that we tested for statement (2) still fit here, though. If we simplify (a + b)(a - b) < 1 to a^2 - b^2 < 1, we can see that since both sets of values that we tested for (2) have the same absolute value, they'll both fit.

And since the sets that we tested for (2) alone gave us a "yes" and a "no" value respectively, we know that combining the two statements will still be INSUFFICIENT.

The answer is E.

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