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

Question

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

JeffTargetTestPrep · Accepted Answer

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.

The answer is C.

Bunuel · Answer

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.

Answer: C.

selfishmofo · Answer

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.

GMATSPARTAN · Answer

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

appleid · Answer

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?

Bunuel · Answer

x^2 - x is an equation of upward parabola, intercepting the x axis at 0 and 1. 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: as-x-increases-from-209-to-210-which-of-the-following-must-128561.html 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.

EMPOWERgmatRichC · Answer

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 
Eliminate Answers B and E.

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 
Eliminate Answers A and D.

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

Final Answer:  C

GMAT assassins aren't born, they're made, 
Rich

Kinshook · Answer

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

TheUltimateWinner · Answer

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. 
It'll save your time...
Thanks__

BrentGMATPrepNow · Answer

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 Answer: C

Rainman91 · Answer

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.

Adarsh_24 · Answer

For 2nd option,

Old value 164/165(less than 1)
New value 165/166 = 164+1/165+1
When you add positive k to numerator and denominator, it moves towards to 1.
For numbers less than 1 it increases and greater than 1 it decreases.

Ayeka · Answer

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

Take 2, 3 & plug in them to find the answer

Posted from my mobile device

AdarshSambare · Answer

Whether we use 165 and 166 does not matter. To make the calculations easier, we use the smaller numbers 2 and 3. The question becomes:

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

I. 2x – 5
x = 2 → 2(2) – 5 = -1
x = 3 → 2(3) – 5 = 1
This increases → Correct

II. 1 – 1/x
x = 2 → 1 – 1/2 = 1/2
x = 3 → 1 – 1/3 = 2/3
This increases → Correct

III. 1/(x2 – x)
x = 2 → 1/(4 – 2) = 1/2
x = 3 → 1/(9 – 3) = 1/6
This decreases → Not correct

Thus, only I and II are correct.

Answer: C

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

Prep Toolkit

Top 5 GMAT Debrief Videos

615 to 715 in 15 Days (Ria's Strategies)

More than Quant/Verbal Abilities: Speed vs. Accuracy

745 with Only Self-Prep and GMAT Club

From 555 to 765: 210-point Improvement

Perfect 805 Score Debrief by Julia

My Rewards

GMAT Problem Solving (PS) Questions

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

Prep Toolkit

615 to 715 in 15 Days (Ria's Strategies)

More than Quant/Verbal Abilities: Speed vs. Accuracy

745 with Only Self-Prep and GMAT Club

From 555 to 765: 210-point Improvement

Perfect 805 Score Debrief by Julia

My Rewards