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

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Manager
Joined: 02 Dec 2012
Posts: 174
As x increases from 165 to 166, which of the following must  [#permalink]

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26 Dec 2012, 06:13
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25% (medium)

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74% (01:17) correct 26% (01:30) wrong based on 1158 sessions

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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
Intern
Joined: 11 Aug 2013
Posts: 29
Re: As x increases from 165 to 166, which of the following must  [#permalink]

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12 Aug 2013, 19:31
5
1
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.
##### General Discussion
Math Expert
Joined: 02 Sep 2009
Posts: 52971
Re: As x increases from 165 to 166, which of the following must  [#permalink]

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26 Dec 2012, 06:16
2
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]

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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]

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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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Joined: 22 Oct 2014
Posts: 27
As x increases from 165 to 166, which of the following must  [#permalink]

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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?
Math Expert
Joined: 02 Sep 2009
Posts: 52971
Re: As x increases from 165 to 166, which of the following must  [#permalink]

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22 Oct 2014, 06:49
1
1
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 14025 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.

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Hope it helps.
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Posts: 27
As x increases from 165 to 166, which of the following must  [#permalink]

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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?
Manager
Joined: 10 May 2014
Posts: 138
Re: As x increases from 165 to 166, which of the following must  [#permalink]

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25 Nov 2015, 16:10
Bunuel wrote:
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.

Hi Bunuel,
I was thinking that plugging in 2 instead of 165 and 3 instead of 166 and then solve could be an efficient approach for this particular question.
Do you agree?
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Re: As x increases from 165 to 166, which of the following must  [#permalink]

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28 Nov 2015, 20:05
3
Hi All,

Roman Numeral questions are relatively rare on Test Day (you'll likely see just 1 in the Quant section), but they are almost always designed around Number Properties and logic 'shortcuts'...

Here, we're told that X increase from 165 to 166. We're asked which of the following MUST increase. While this prompt might appear 'calculation heavy', you can actually solve it WITHOUT doing calculations (and paying attention to the answer choices).

I. 2X - 5

The "-5" has the same 'math effect', regardless of what X is: the '-5' reduces the value by 5. So as X goes from 165 to 166, 2X gets BIGGER. Since the '-5' has the same effect, 2X-5 definitely gets bigger as X increases from 165 to 166.
Roman Numeral 1 INCREASES

II. 1 - 1/X

As the denominator of a fraction gets BIGGER, the fraction gets SMALLER. Thus, 1/165 > 1/166. The "1" in this calculation is constant, so subtracting a SMALLER fraction from it will lead to a BIGGER result. As X increases from 165 to 166, 1/X gets SMALLER, so 1-(1/X) definitely gets bigger.
Roman Numeral 2 INCREASES

There's only one answer remaining (and we don't even have to deal with Roman Numeral 3)...

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Re: As x increases from 165 to 166, which of the following must  [#permalink]

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07 Jul 2016, 09:18
3
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

Before jumping into this problem using the numbers provided, we must analyze what is really being tested here. We are really being asked, what happens when we increase a positive integer by a value of 1. Whether we use 165 and 166 does not matter. So to make the calculations easier, we will use the smaller numbers 2 and 3. We can now rephrase the question:

As x increases from 2 to 3, which of the following must increase?

Let’s test out the new numbers for each Roman numeral.

I. 2x – 5

When x = 2:

2(2) – 5 = -1

When x = 3:

2(3) – 5 = 1

This DOES increase. Roman numeral I is correct.

II. 1 - 1/x

When x = 2:

1 – 1/2 = 1/2

When x = 3:

1 – 1/3 = 2/3

This DOES increase. Roman numeral II is also correct.

III. 1/(x^2 - x)

When x = 2:

1/(2^2 – 2)

1/(4 – 2) = 1/2

When x = 3:

1/(3^2 – 3)

1/(9 – 3) = 1/6

This DOES NOT increase. Roman numeral III is not correct.

Thus, only I and II are correct.

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Re: As x increases from 165 to 166, which of the following must  [#permalink]

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02 Dec 2018, 06:40
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Re: As x increases from 165 to 166, which of the following must   [#permalink] 02 Dec 2018, 06:40
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