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.
PLEASE POST YOUR SOURCE FOR THIS QUESTION, Mo2men. It is a copyright violation to print a question without crediting the writer! If you wrote the question yourself, please mention that.
Is 1/(a - b) < b + a ?
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Statement 1:Mo2men wrote:Is 1/(a - b) < b + a ?
(1) (a + b)(a - b) < 1
(2) ab > a - b
a² - b² < 1
a² < b² + 1.
Case 1: b=2, a=1
Plugging these values into 1/(a-b) < b+a, we get:
1/(1-2) < 2+1
-1 < 3.
YES.
Case 2: b=-2, a=-1
Plugging these values into 1/(a-b) < b+a, we get:
1/[-1-(-2)] < -2 + (-1)
1 < -3.
NO.
Since the answer is YES in Case 1 but NO in Case 2, INSUFFICIENT.
Cases 1 and 2 satisfy both statements.
Since the answer is YES in Case 1 but NO in Case 2, the two statements combined are INSUFFICIENT.
The correct answer is E.
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Followed here and elsewhere by over 1900 test-takers.
I have worked with students based in the US, Australia, Taiwan, China, Tajikistan, Kuwait, Saudi Arabia -- a long list of countries.
My students have been admitted to HBS, CBS, Tuck, Yale, Stern, Fuqua -- a long list of top programs.
As a tutor, I don't simply teach you how I would approach problems.
I unlock the best way for YOU to solve problems.
For more information, please email me (Mitch Hunt) at [email protected].
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