As x increases from 165 to 166,

Cydney Seigerman, Magoosh Tutor

In the explanation video, Mike's analysis is based on general observations that apply to any two positive integers, where the first one is larger than the second one and for which the 3 statements are defined. Let's look at the statements more closely to see why.

I) 2x - 5
In this case, we are subtracting a constant from 2x. If we increase x, then 2x will be larger. Since we will still be subtracting the same value (5), the answer will be bigger when 2x is bigger. So, for any two positive integers, the difference is larger when x is the larger integer.

II) 1 - 1/x
In this expression, 1/x is being subtracted from a constant. For two values of x, when 1/x is smaller, 1 - 1/x will be larger. At the same time, for two fractions with the same numerator, the fraction with the smaller denominator will be the larger quantity (e.g. 1/2 > 1/3; 2 < 3). Together, this means that when x is greater, the fraction being subtracted will be smaller, so 1 - 1/x will be greater.

III) 1/ (x^2 - x)
Now, we know that when the denominator is greater, the fraction will be smaller. So, the question becomes, will x^2 -x be smaller when x is larger or smaller. For x>1, x^2 > x, so for all values of x larger than 1, the difference x^2 - x will be larger for larger values of x. Since the denominator is larger for larger values of x, the expression will be smaller for the greater value of x. Therefore this expression will decrease as x increases.

With that in mind, we can also solve the problem by substituting values in for x to analyze the statements, as you've suggested. If we plug in values for x, it is definitely ok to use quantities that are easier to work with. However, you need to be careful that the expression is defined for the values of x you choose. We cannot use 1, for example, since (III) 1/(x^2 -x) = 1/(1-1) = 1/0 is undefined. However, we could use x=2 and x=3, for which all three statements are defined, in order to analyze the statements in this problem.

Hope this helps :)

Dec 8, 2015 • Reply