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Negative Values

Problem Solving — algebra and arithmetic (GMAT Focus Edition)
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by kamalakarthi » Mon Aug 07, 2017 10:51 am
Hi,

This is from MCAT test.

In this below question, I got to the right answer after testing for various numbers. But I took long time to solve this question.
How can I improve my choosing the right numbers.

I tried with a=5 and b=3 then I tried with A=1 and B=-1 then I tried with A=-2 and B=-3





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Source: — Quantitative Reasoning |

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by [email protected] » Mon Aug 07, 2017 3:34 pm
Hi kamalakarthi,

In certain circumstance - when TESTing VALUES - it's sometimes best to 'target' a particular answer and try to select values that will directly eliminate it (as opposed to randomly selecting values and seeing what happens).

With these answer choices, it also helps to break each answer into 2 parts: the first absolute value and whatever is subtracted from it.

For Answers A, B and C, you should look to make the first part equal 0 and the second part NOT 0....
Answer A: A = 1, B = -1 ... |1 - 1| - |1 - (-1)| = 0 - 2 = -2 Eliminate Answer A.

Answer B: A = 1, B = -1 .... |1 - 1| - |1| = 0 - 1 = -1 Eliminate Answer B.

Answer C: A = 1, B = -2.... |2 - 2| - |1 -2| = 0 - 1 = -1 Eliminate Answer C.

For the remaining two answers, since you're dealing with exponents, you should consider fractions....

Answer E: A = 1/2, B = 1/2 ... |(1/2)^3 + (1/2)^3| - 1/2 - 1/2 = 2/8 - 1 = -6/8 Eliminate Answer E.

Final Answer: D

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by Matt@VeritasPrep » Fri Aug 18, 2017 1:12 pm
If you try numbers, remember that you don't have to try each answer every time! Once you get a negative result for an answer, that answer is eliminated, and you only have to doubletest the answers that showed positive.
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by Matt@VeritasPrep » Fri Aug 18, 2017 1:13 pm
Let me add a (semi-)algebraic approach too.

Suppose that each answer is negative. For instance, for answer B that would give us

|a + b| - |a| < 0, or

|a + b| < |a|

This is obviously possible, if a = 3 and b = -1, so a negative result could work here.

Continuing this way through the other answers, you might find plugging in faster and simpler.
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by Matt@VeritasPrep » Fri Aug 18, 2017 1:18 pm
We could take a conceptual approach too. Remembering that |x - y| = the distance from x to y on the number line, we could say

A::
|a + b| - |a - b| < 0
|a + b| < |a - b|
the distance from a to -b < the distance from a to b

This is definitely possible.

B::

|a + b| < |a|
the distance from a to -b < the distance from a to 0

Also totally possible!

C::

|2a + b| < |a + b|
the distance from 2a to -b < the distance from a to -b

Yup, works!

D::

a² - 2|ab| + b² < 0

(|a| - |b|)² < 0

Aha! No such thing as a negative square - at least not in GMAT math - so we've found our answer.
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