Vincen wrote: ↑

Sun May 23, 2021 12:32 pm

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

I. \(2x - 5\)

II. \(1 - \dfrac1{x}\)

III. \(\dfrac1{x^2 - x}\)

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

Answer: C

Source: Official Guide

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