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# As x increases from 165 to 166, which of the following must

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As x increases from 165 to 166, which of the following must [#permalink]  26 Dec 2012, 06:13
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As x increases from 165 to 166, which of the following must increase?

I. 2x - 5
II. 1 - 1/x
III. 1/(x^2 - x)

(A) I only
(B) III only
(C) I and II
(D) I and III
(E) II and III
[Reveal] Spoiler: OA
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Posts: 28747
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Kudos [?]: 47217 [1] , given: 7115

Re: As x increases from 165 to 166, which of the following must [#permalink]  26 Dec 2012, 06:16
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Expert's post
As x increases from 165 to 166, which of the following must increase?

I. 2x - 5
II. 1 - 1/x
III. 1/(x^2 - x)

(A) I only
(B) III only
(C) I and II
(D) I and III
(E) II and III

I. 2x - 5 --> x increases from 165 to 166 --> 2x increases --> 2x - 5 increases. Correct.

II. 1 - 1/x --> x increases from 165 to 166 --> 1/x decreases --> 1 -1/x increases. Correct.

III. 1/(x^2 - x) --> x increases from 165 to 166 --> x^2-x increases --> 1/(x^2 - x) decreases.

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Re: As x increases from 165 to 166, which of the following must [#permalink]  12 Aug 2013, 19:31
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For the first part, I simply plugged in a positive whole number for x, then I retested with a higher number than the number I chose originally for x,

X=5

2x-5=
2*5-5
10-5=5

X=6

2*6-5
12-5=7

X increases, and so does 2x-5

For the second problem, I used the same method, third problem, decreases by theory, no need to solve.
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Kudos [?]: 186 [0], given: 102

Re: As x increases from 165 to 166, which of the following must [#permalink]  13 Aug 2013, 07:12
As x increases from 165 to 166, which of the following must increase?

I. 2x - 5
II. 1 - 1/x
III. 1/(x^2 - x)

(A) I only
(B) III only
(C) I and II
(D) I and III
(E) II and III

...
I and II can be counted easily, both increases.
And III is 1/x(x-1)
Now, 1/(165×164) > 1/(166×165) , so it decreases as 165 turns to 166 .
so I and II = C (Answer)
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Re: As x increases from 165 to 166, which of the following must [#permalink]  13 Aug 2013, 09:40
As x increases from 165 to 166, which of the following must increase?

I. 2x - 5
II. 1 - 1/x
III. 1/(x^2 - x)

(A) I only
(B) III only
(C) I and II
(D) I and III
(E) II and III

The max difference b/w 165 and 166 is '1' and the min value can be 0.000000000000000000001 but for our convenience we shall choose 0.1
Equation 1 ==> for min value of 0.1, the value is 2(0.1) - 5 = -4.8

for max value of 1, 2(1)-5=-3
so the values are increasing from -4.8 to -3

similary submit the values 0.1 and 1 and we notice that the value increases from -9 to 0 ==> equation 2 is also increasing

for double confirmation substitute the values of 0.1 and 1 in equation 3.
for min value of x=0.1, the value is $$1/(0.01-0.1)=-1/0.09$$
for max value of 1, the value is 'infinite' ... hence we dont count this.

So equations 1 & 2 are increasing. This is the answer
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As x increases from 165 to 166, which of the following must [#permalink]  22 Oct 2014, 06:26
selfishmofo wrote:
For the first part, I simply plugged in a positive whole number for x, then I retested with a higher number than the number I chose originally for x,
.....
......
For the second problem, I used the same method, third problem, decreases by theory, no need to solve.

I have done the first two in the same way you did, but third one took time. Can you please connect or explain the theory that helped you?
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Posts: 28747
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Kudos [?]: 47217 [1] , given: 7115

Re: As x increases from 165 to 166, which of the following must [#permalink]  22 Oct 2014, 06:49
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appleid wrote:
selfishmofo wrote:
For the first part, I simply plugged in a positive whole number for x, then I retested with a higher number than the number I chose originally for x,
.....
......
For the second problem, I used the same method, third problem, decreases by theory, no need to solve.

I have done the first two in the same way you did, but third one took time. Can you please connect or explain the theory that helped you?

x^2 - x is an equation of upward parabola, intercepting the x axis at 0 and 1.
Attachment:

graph.png [ 7.08 KiB | Viewed 2440 times ]
So, far away from the roots, the values of the function (x^2 - x) are increasing as x increases. Therefore 1/(x^2 - x) decreases.

Similar questions to practice:
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as-j-increases-from-135-to-137010.html
as-y-increases-from-y-247-to-y-248-which-of-the-follow-161679.html

Hope it helps.
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As x increases from 165 to 166, which of the following must [#permalink]  24 Oct 2014, 03:11
Bunuel wrote:
appleid wrote:
selfishmofo wrote:
For the first part, I simply plugged in a positive whole number for x, then I retested with a higher number than the number I chose originally for x,
.....
......
For the second problem, I used the same method, third problem, decreases by theory, no need to solve.

I have done the first two in the same way you did, but third one took time. Can you please connect or explain the theory that helped you?

x^2 - x is an equation of upward parabola, intercepting the x axis at 0 and 1.
Attachment:
graph.png
So, far away from the roots, the values of the function (x^2 - x) are increasing as x increases. Therefore 1/(x^2 - x) decreases.

Similar questions to practice:
....
Hope it helps.

Thank you! However, I did understand the value of x^2-x will always goes up, but I am not getting the basic theory (or relevant foundation/topic) behind it. Could you please suggest me any topic which will teach me this from ground level?
As x increases from 165 to 166, which of the following must   [#permalink] 24 Oct 2014, 03:11
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