## GRE Math Tutorial

*Problem Solving*

#### Intro

A sample question similar to some that might show up on the GRE Math portion.

#### Sample Problem

If x is an integer such that x > 0, and given the following statements:

I. x – x^{2} > 0

II. 2x – x^{3} < 0

III. x^{-7} > 0

which of the statements are always true?

#### Solution

Let’s go through each option, remembering that x is always a positive integer. That means x is a value from 1 to infinity.

I. Will never be true because x^{2} is always greater than x with the exception of x = 1. Even in the case of x = 1, the result of (I) would be 0, and the inequality is exclusive.

II. x^{3} will always be bigger than 2x when x is always a positive integer EXCEPT when x = 1 as in this case. While any integer value from 2 onward will make the comparison true, x = 1 will result in:

2(1) – (1)^{3} =

2 – 1 = 1

1 > 0

That means II is false for one case.

III. x^{-7} is the same as 1/x^{7}. While you could make the argument that, as x approaches infinity, the value of x^{-7} is practically 0, it is not actually 0. The result is just a very small positive fraction that is still bigger than 0, and so III only is the correct answer.

# About The Author

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