Problem Solving

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

I. 2x-5

II. 1-1/x

III. 1/x squared - x

a) I only
b)III only
c)I and II
d)I and III
e)II and III

shibotta23 wrote:As x increases from 165 to 166, which of the following must increase?

I. 2x-5

II. 1-1/x

III. 1/x squared - x

a) I only
b)III only
c)I and II
d)I and III
e)II and III

Hi!

We're told that x is positive and increasing and we're asked to determine which of 3 expressions will increase along with x.

Here's something to remember: the GMAT is not trying to be your friend, but it is trying to see how clever you are. So, we're always using our cleverness to outwit the test and turn complex questions into simpler ones.

How can we do so on this question? By recognizing that "165 to 166" is an arbitrary range. We can make our lives much easier by saying that x is increasing from 5 to 6 instead, then using those numbers to test the choices.

(I) looks simplest, so let's start there.

If x=5, then 2x-5 = 10-5=5
If x=6, then 2x-5 = 12-5=7

Did (I) increase? YES. Eliminate (b) and (e), since they don't include (I).

(As an aside: the "-5" is irrelevant, since it's a constant and has no impact on whether that expression grows, so we could have just tested "2x" instead.)

Our 3 remaining choices are I only, I and II and I and III. Not much help there, so let's test II next, since it seems simpler than III.

(II) 1 - 1/x

If x = 5, then we get 1 - 1/5 = 4/5
If x = 6, then we get 1 - 1/6 = 5/6

Is 5/6 > 4/5? YES. So, x is increasing once again, which means that (II) is also a winner.

(As another aside, 1 is constant so we could have ignored it and only tested "-1/x"... but you have to remember to include the negative sign, so this one may be easier to test by leaving the 1 in there.)

Only one choice includes both I and II... choose (C)!

Answer is C by substituting values & subtract 2 values.
It answer is positive than second one is greater

It answer is negative than second one is fewer.