Question on Inequalities

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Spartacus2000: Junior | Next Rank: 30 Posts; Posts: 26; Joined: Sun May 01, 2011 2:31 pm

by Spartacus2000 » Tue Aug 09, 2011 6:49 pm

Is 1/a - b < b - a?

(1) a < b
(2) 1 < |a - b|

Thanks

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Source: Beat The GMAT — Data Sufficiency | Tue Aug 09, 2011 6:49 pm

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by [email protected] » Tue Aug 09, 2011 7:55 pm

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gmatboost: Master | Next Rank: 500 Posts; Posts: 312; Joined: Tue Aug 02, 2011 3:16 pmFollowed by: 33 members; Location: New York City; Thanked: 130 times; GMAT Score:780

by gmatboost » Tue Aug 09, 2011 10:33 pm

Best to translate prompt to
a + 1/a < 2b?
in my opinion.

Plugging in numbers works well here. Whenever there are no limitations on negative numbers I like to start there.

Try these and see what happens:

If a = -5
b = -3, or b = -2

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Spartacus2000: Junior | Next Rank: 30 Posts; Posts: 26; Joined: Sun May 01, 2011 2:31 pm

by Spartacus2000 » Wed Aug 10, 2011 6:49 am

This might be a stupid question but can you illustrate how you re-translated the question stem?

Thanks,

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gmatboost: Master | Next Rank: 500 Posts; Posts: 312; Joined: Tue Aug 02, 2011 3:16 pmFollowed by: 33 members; Location: New York City; Thanked: 130 times; GMAT Score:780

by gmatboost » Wed Aug 10, 2011 7:48 am

No, it's a good question.

Is 1/a - b < b - a?

When I see this, my immediate thought it to try to combine the a's on one side and the b's on the other side. It's not this is always going to be the best possible thing to do, but I find that in general it helps me see what's going on more clearly.

So, start by adding b to both sides, in order to cancel out the (-b) on the left:
1/a - b + b < b - a + b

Simplify:
1/a < 2b - a

Now, do the same with the (-a) on the right:
1/a + a < 2b - a + a
1/a + a < 2b

Definitely ask if you have further questions.

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Spartacus2000: Junior | Next Rank: 30 Posts; Posts: 26; Joined: Sun May 01, 2011 2:31 pm

by Spartacus2000 » Wed Aug 10, 2011 6:27 pm

Ok, this is my fault but I should added the () - that's why I could not get the translation

The question is:

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

Sorry about that.

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gmatboost: Master | Next Rank: 500 Posts; Posts: 312; Joined: Tue Aug 02, 2011 3:16 pmFollowed by: 33 members; Location: New York City; Thanked: 130 times; GMAT Score:780

by gmatboost » Thu Aug 11, 2011 7:21 am

Okay, no worries. For this question, it is critical to recognize that (a-b) and (b-a) are opposites of one another.

In other words, (a-b) = -(b-a)
Or, (b-a) = -(a-b)

From this, we can conclude that one of them is always positive, and one of them is always negative.
(a-b) cannot be zero because we can't have 1/0.

Therefore, in order for
1/(a-b) < b-a

We need (a-b) to be the negative one, and (b-a) to be the positive one.
That way, the inequality will be negative < positive
In other words, we need a < b.

Statement 1:
Tells us exactly what we need. Sufficient.

Statement 2:
1 < |a - b|
Here, either one could be the bigger one, which means that (a-b) could be positive, or could be negative. Insufficient.

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Spartacus2000: Junior | Next Rank: 30 Posts; Posts: 26; Joined: Sun May 01, 2011 2:31 pm

by Spartacus2000 » Thu Aug 11, 2011 3:32 pm

Thanks.

That was really helpful.

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