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

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

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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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]
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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Re: As x increases from 165 to 166, which of the following must [#permalink]
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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 43281 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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Re: As x increases from 165 to 166, which of the following must [#permalink]
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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]
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

Given: As x increases from 165 to 166

Asked: Which of the following must increase?

I. 2x - 5
As x increases, 2x-5 increases

II. 1 - 1/x
As x increases; 1/x decreases; 1-1/x increases

III. 1/(x^2 - x)
As x increases; x^2 - x increases; 1/ (x^2 - x) decreases

IMO C
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As x increases from 165 to 166, which of the following must [#permalink]
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. $$2x - 5$$-->increase (cancel choice B and E)
II. $$1 - \frac{1}{x}$$-->increase (cancel choice A and D)
III. $$\frac{1}{(x^2- x)}$$-->We already got that I and II increase. There is no answer option like I, II and III in the out of 5 options. So, we should not test/verify III. So, the correct choice is C.
Thanks__
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Re: As x increases from 165 to 166, which of the following must [#permalink]
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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

For each statement, let's evaluate the expression for x = 165, and then x = 166

I. 2x - 5
If x = 165, we get 2(165) - 5 = 325
If x = 166, we get 2(166) - 5 = 327
So, the value of the expression increases as x increases from 165 to 166

Check the answer choices..... we can eliminate B and E, since they say that the value of the expression does NOT increase as x increases from 165 to 166

II. 1 - 1/x
If x = 165, we get 1 - 1/165
If x = 166, we get 1 - 1/166
NOTE: since 165 < 166, we know that 1/165 > 1/166
So, with 1 - 1/165, we are subtracting a larger number from 1 than with 1/166, which means (1 - 1/166) > (1 - 1/165)
So, the value of the expression increases as x increases from 165 to 166
Check the answer choices..... we can eliminate A , since it says the value of the expression does NOT increase as x increases from 165 to 166

III. 1/(x^2 - x)
First recognize that x^2 - x = x(x - 1)
If x = 165, x(x - 1) = 165(165 - 1) = 165(154)
If x = 166, x(x - 1) = 166(166 - 1) = 166(165)
Here it is clear that 165(154) < 166(165), which means 1/165(154) > 1/166(165)
So, the value of the expression DOES NOT increase as x increases from 165 to 166

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Re: As x increases from 165 to 166, which of the following must [#permalink]
I did not use easy numbers such as 3 or 2. But, I looked at the answer choices. There is neither an option that says 1,2, AND 3 are all applicable.

So when I proved that 2 is applicable (165/166 - 164/165, they both have a difference of one in the numerator and denominator, so the number with the higher denominator will be bigger)

I didn't waste any more time calculating squares and chose C.
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Re: As x increases from 165 to 166, which of the following must [#permalink]
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