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Is 1/(a+b) < 1/a + 1/b ?

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Is 1/(a+b) < 1/a + 1/b ?

by gmattesttaker2 » Sun Feb 02, 2014 11:16 pm
Hello,

Can you please tell me if my approach is OK here:

If ab ≠ 0 and a + b ≠ 0, is 1/(a+b) < 1/a + 1/b?

(1) |a| + |b| = a + b

(2) a > b

OA: A


To find if:

1/(a+b) < 1/a + 1/b ?

=> 1/(a + b) < (a + b)/(ab) ?


1) |a| + |b| = a + b

=> Is: 1/(|a| + |b|) < (|a| + |b|)/(ab) ?

Since |a| + |b| is always positive, when we cross-multiply by |a| + |b|, the sign won't change:

=> Is: ab < ( |a| + |b| )^2 ?

This is always true. Hence, sufficient


2) a > b


To find if:

1/(a+b) < 1/a + 1/b ?

=> 1/(a + b) < (a + b)/(ab) ?


Here we don't know whether (a + b) is positive or (a + b) is negative.

If ( a + b ) > 0
=> Is: ab < (a+b)^2 ?

If ( a + b ) < 0
=> Is: ab > (a+b)^2 ?

At this point I conclude that Statement 2 is in-sufficient. I was wondering if this is OK or if I need to solve this further. Thanks for your help.

Best Regards,
Sri
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by ceilidh.erickson » Mon Feb 03, 2014 8:16 am
Your approach is very good here. There are only a few things I would add:

You rephrased the question perfectly: 1/(a + b) < (a + b)/(ab) ?

What we need to know to answer that question is primarily information about positive v. negative.

(1) |a| + |b| = a + b

Not only does this tell us that |a| + |b| is always positive, thus allowing us to cross-multiply, it also tells us that a and b are each positive. That's the only way the the sum of the absolute values will equal the sum of the terms. Since they are both positive, then the product ab will also be positive. (I think you intuited this, but it's worth stating).

When we have the question: is ab < (a + b)^2 ?, we can rephrase:
ab < a^2 + 2ab + b^2 ?
0 < a^2 + ab + b^2 ?

Since these terms are positive, then that sum must be greater than 0. Sufficient.

(2) a > b

This tells us about a relative to b, but it tells us nothing about whether the terms are positive or negative. It's not just that be don't know if (a + b) is pos/neg, but we also don't know about product ab. We don't even need to do the algebra to quickly see that sometimes we'll get a "yes" answer and sometimes a "no" answer to the question.
Ceilidh Erickson
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by Matt@VeritasPrep » Mon Feb 03, 2014 4:25 pm
Yup, you're on the right track here!

I'd do it much the same way. Start by rephrasing the question as

"Is (a+b)/ab > 1/(a+b)?"

S1 tells us that (a+b) is positive, so we can rephrase the question as

"Is (a+b)²/ab > 1?"

If ab is positive, then the question is really

"Is (a+b)² > ab?"
or
"Is a² + 2ab + b² > ab?"
or
"Is a² + ab + b² > 0?"
or
"Is a² + b² > -ab?"

Since ab is positive, -ab is negative. a² + b² is never negative, so in this case the answer is YES, (a+b)² > ab, implying that 1/a + 1/b > 1/(a+b).

The second case is if ab is negative. For this to happen, exactly one of the two variables must be negative. But then |a| + |b| could not equal a + b, which we know is true, so this scenario is impossible.

S2 just says that a > b, but it doesn't tell us anything about the positivity or negativity of each number, so we can't do much with it: as you say, it leads to two different questions but provides no information with which to answer those questions.

If you ever get stuck on one of these conceptually on the test, I'd just pick some numbers to illustrate likely scenarios (both positive, both negative, one pos one neg, etc.) and see if they all give the same result. If they do, the statement is likely sufficient; if they don't, the statement is definitely insufficient. You won't be certain of your answer, but you'll improve your odds of being right and buy time for other hard questions to come.
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