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Must be true question

Problem Solving — algebra and arithmetic (GMAT Focus Edition)
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by rahulvsd » Sun May 20, 2012 8:04 pm
If a^5 ≤ a, which of the following must be true?
I -1 ≤ a ≤ 0
II a=0
III 0 ≤ a ≤ 1


A. None of the above
B. I only
C. II only
D. III only
E. I and III only
[spoiler]
OA: A. Source: MasterGMAT[/spoiler]
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by ankita1709 » Mon May 21, 2012 7:53 am
By seeing the question we can easily make out that
0<=a<=1 and a<=-1

which implies
I -1 ≤ a ≤ 0 --> Can't be true
II a=0 --> Not the only possible value for a. Hence not true
III 0 ≤ a ≤ 1 --> this is true but again it doesn't include the whole range

Try and focus on the word must be true
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by tutorphd » Sun Jun 24, 2012 5:42 pm
First interpret "MUST BE TRUE" as "given the condition IS ALWAYS TRUE".
Compare to "COULD BE TRUE" questions which are interpreted as "given the condition IS SOMETIMES TRUE".

If you can't factorize the given inequality as in the previous solution, you can resort to picking numbers. The usual numbers to pick are -2, -1, -1/2, 0, 1/2, 1, 2. Test these and you will see that only -2, -1, 0, 1/2, 1 satisfy a^5 <= a.

The numbers -2, -1, 0, 1/2, 1 that satisfy a^5<=a MUST also satisfy any statement that is a TRUE consequence of a^5<=a:

I. is not true because it doesn't include -2, 1/2, 1
II. is not ture because it doesn't include -2, -1, 1/2, 1
III. is not true because it doesn't include -2, -1
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by KapTeacherEli » Tue Jun 26, 2012 8:31 am
tutorphd wrote:First interpret "MUST BE TRUE" as "given the condition IS ALWAYS TRUE".
Compare to "COULD BE TRUE" questions which are interpreted as "given the condition IS SOMETIMES TRUE".

If you can't factorize the given inequality as in the previous solution, you can resort to picking numbers. The usual numbers to pick are -2, -1, -1/2, 0, 1/2, 1, 2. Test these and you will see that only -2, -1, 0, 1/2, 1 satisfy a^5 <= a.

The numbers -2, -1, 0, 1/2, 1 that satisfy a^5<=a MUST also satisfy any statement that is a TRUE consequence of a^5<=a:

I. is not true because it doesn't include -2, 1/2, 1
II. is not ture because it doesn't include -2, -1, 1/2, 1
III. is not true because it doesn't include -2, -1
Hi tutor PHD,

Great suggestion of picking numbers! Problems like these are classic opportunities to earn points by plugging in values. The key is that on a "must be true" problem, a single counterexample is all you need to rule a choice out!

However, I do want to point something out. We're dealing with a^5--that's an odd power. Odd powers and even powers are very common food for GMAT problems. Odd powers preserve the base, so a negative power will stay negative, and will get more negative as long as it's not a fraction.

Recognizing this pattern, we can plug -2 into the problem. And once we do, we're done! -2 rules out all three statements. We don't need to try a single other number, saving us lots of time!

Regards.
Eli Meyer
Kaplan GMAT Teacher
Cambridge, MA
www.kaptest.com/gmat

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by tutorphd » Tue Jun 26, 2012 9:16 am
Yes, it's a good observation that only plugging -2 works to reject all statements, in this problem.

However, I wanted to point out the general approach that Princeton Review suggests for testing inequalities by using the universal numbers -2, -1, -1/2, 0, 1/2, 1, 2.

Inequalities of one varialbe on GMAT amost always factorize in x, x-1, and x+1 factors. The above test numbers are universal for such inequalities because they test exactly the roots of those factors and in between. In such inequalities, the above numbers can be used not only to reject a statement in a MUST BE TRUE question but also to accept it.

Of course, if GMAT puts in an inequality that has another type of factor like x-3, in the question stem or in the must be true statements, the choice of testing numbers should be different. But for the majority of questions, that doesn't happen. Seems that GMAT is testing more number properties than general factorization of inequalities. Nevertheless, I always teach solving inequalities by factorization to my students aiming at 700+ scores, because it allows solving more complicated types of questions mechanically.
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