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Finite range of values of 'x' - please explain

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by GMATinsight » Fri Aug 01, 2014 11:05 am
BShubh wrote:Which of the following inequalities has a finite range of values of 'x' satisfying it?

A. x^2 + 5x + 6 > 0
B. |x + 2| > 4
C. 9x - 7 < 3x + 14
D. x^2 - 4x + 3 < 0
E. 4x + 3 > 2x + 1

Answer: D
Let's Check Options one-by-one

A. x^2 + 5x + 6 > 0
i.e. (x+2) (x+3) > 0
Which is true for any value of x either greater than -2 or lesser than -3
Therefore Infinite Values
INCORRECT OPTION

B. |x + 2| > 4
i.e. x+2 > 4 or x+2 < -4
i.e. either x greater than 2 or lesser than -6
Therefore Infinite Values
INCORRECT OPTION

C. 9x - 7 < 3x + 14
i.e. 9x-3x < 14+7
i.e. 6x < 21
i.e. x < 7/2
Therefore Infinite Values
INCORRECT OPTION

E. 4x + 3 > 2x + 1 [EASIER OPTION to Check than OPTION D]
i.e. 4x - 2x > 1 - 3
i.e. 2x > -2
i.e. x > -1
Therefore Infinite Values
INCORRECT OPTION

Therefore Answer D must be correct

But just to Verify (However not a necessary step in test scenario)

D. x^2 - 4x + 3 < 0
i.e. (x-3) (x-1) < 0
i.e. 1 < x < 3
FINITE VALUES
CORRECT

Answer: Option D
Last edited by GMATinsight on Fri Aug 01, 2014 11:08 am, edited 1 time in total.
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by [email protected] » Fri Aug 01, 2014 11:08 am
Hi Bshubh,

For an inequality to have a finite range of values, there has to be a "boundary" at both ends.

For example: 0 < X < 4 has a finite range of values because there's a lower boundary (0) and an upper boundary (4).

One of the 5 answer choices has a lower and upper boundary; we have to do enough work to figure out which one it is (or we can figure out the 4 that are NOT). Some of the work will be "math" (simplifying the inequality), some of the work will be "conceptual" (recognizing that there is no boundary).

A: X^2 +5X + 6 > 0

Here, not much math is required; as X gets bigger, the sum gets bigger. There's no upper limit to X.

B: |X+2| > 4

Here, starting with any value greater than 2, as X gets bigger, the sum gets bigger. There's no upper limit to X.

C: 9X - 7 < 3X + 14

Here, we can simplify:

6X < 21

This clearly has an upper limit, but no lower limit.

D: Let's leave this one alone for now.

E: 4X + 3 > 2X + 1

Just like in C, we can simplify:

2X > -2

This clearly has a lower limit, but no upper limit.

We're found 4 answers that are "open ended", so they're not finite. There's only one answer left....

Final Answer: D

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by BShubh » Fri Aug 01, 2014 11:16 am
Thanks to both of you! GMATinsight and [email protected]
Your explanations helped.
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by Matt@VeritasPrep » Sat Aug 02, 2014 11:41 am
There is a major issue in the question: NONE of the answers has a finite set of solutions.

Both explanations above correctly note that D implies that 1 < x < 3, but this is NOT a finite set of values! There are an infinite number of real numbers between 1 and 3. (Just to illustrate a few of them, we could have x = 1.2, x = 2.3, x = 2.079, etc.)

The range is finite in the sense that the solutions satisfying the inequality are bounded, but this is a concept from analysis, not the introductory algebra to which the GMAT is typically limited. The distinction between "a finite number of solutions" and "a finite range" doesn't seem to be one that the GMAT would expect you to know.
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by BShubh » Sat Aug 02, 2014 12:42 pm
Matt@VeritasPrep wrote:There is a major issue in the question: NONE of the answers has a finite set of solutions.

Both explanations above correctly note that D implies that 1 < x < 3, but this is NOT a finite set of values! There are an infinite number of real numbers between 1 and 3. (Just to illustrate a few of them, we could have x = 1.2, x = 2.3, x = 2.079, etc.)

The range is finite in the sense that the solutions satisfying the inequality are bounded, but this is a concept from analysis, not the introductory algebra to which the GMAT is typically limited. The distinction between "a finite number of solutions" and "a finite range" doesn't seem to be one that the GMAT would expect you to know.
Thank you for your response Matt@VeritasPrep!
I am a bit confused though.
From what I understood by earlier two explanations (and please correct me, if I am wrong) -
The question asks for finite range, i.e it will have some fix boundary at both ends. Only option D seems to be best answer among the given answer choices.
And so it isn't the same as finite set..which means, a set of finite numbers,right?
I agree with you totally here, when you see it as 'finite set of values', the answer choices clearly are not satisfying.
Please help me understand :(

Also, do you mean such question wont be asked in actual GMAT?


Thank You!
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by Matt@VeritasPrep » Sat Aug 02, 2014 4:07 pm
Hi BShubh!

Yeah, I doubt it would be asked in this form -- "finite range" seems like a dubious term -- but I could be wrong. I remember seeing a similar question to this in the GMATPrep software, and it asked for an answer whose graph "was a single line segment of finite length", which is the same thing we're asking for here, but much more clearly stated.

You're right that if we take a "finite range" to mean a set of numbers bounded above and bounded below, a "finite range" is obviously different from a finite set. My issue is with the term itself: I can't remember hearing it used in proper mathematics (arithmetical, statistical, or otherwise), and it seems to me to be pretty questionable.
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by GMATinsight » Sat Aug 02, 2014 8:41 pm
Hi Bshubh,

I would only comment that excessive thinking usually causes serious doubts on your knowledge. Therefore, take GMAT positively and use your knowledge constructively.

I am sure you are not willing to do "Research on GMAT Questions" and more interested in learning to get the correct answers of the questions probable in GMAT.

You may always come across elements who keep raising questions on every single thing. They are not wrong, It's just their objective is not the same as your.

I hope now you understand the meaning and difference between the Mathematical terms 'Finite Range' and 'finite set of values' and if you do, then you are GOOD!!!
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