# As x increases from 165 to 166…

## Solution:

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.

**Answer: C**