Let's start by rephrasing the question. We could solve algebraically, but I think it's much better to think conceptually. Where is the absolute value of a number less than 1? For positive and negative fractions. In other words:
Is -1 < n < 1 ?
Let's start with statement 2 here, which is much easier.
(2) x^(-1) = -2
This introduces a new variable, and tells us absolutely nothing about n. INSUFFICIENT.
(1) n^x -n <0
Let's rearrange this algebraically by moving the -n to the other side:
n^x < n
So again, let's think conceptually. A number to some exponent is less than that number. What does this tell us? Well, maybe n is a positive integer like 2, and x is a negative, like -2. The statement would be true, and the answer to the question would be "no."
But what if n is a fraction like 1/2, and x is a positive integer, like 2? The statement would be true, but we'd get a "yes" answer to the question. We need more information about x before we know anything about n. INSUFFICIENT
(1) & (2) If x^(-1) = -2, then (1/x) = -2, so x = (-1/2)
Plug that in for x in statement 1:
n^(-1/2) < n
Instead of a negative exponent, take the reciprocal. So, 1/(n^(1/2)) < n
Here's where things get really tricky. We have a square root of n in the denominator. So, we know that n can't be a negative, or that square root wouldn't be a real number. n > 0, but that doesn't answer our question yet.
So let's test values - one positive fraction, and one positive integer, to see if this is sufficient.
If n = 2, then 1/(√2) < 2
We know that √2 is approximately 1.4, so 1/1.4 is less than 2. This fits the statement, and gives us a "no" answer to our question.
If n = 1/2, then 1/(√(1/2)) < 1/2 ? The root of 1/2 will be less than 1, so 1/(√(1/2) will be greater than one. This doesn't fit the statement. If fact, no number less than 1 will fit the statement, so n must be greater than 1. The answer is no, |n| is not less than 1. SUFFICIENT
Answer: C
Ceilidh Erickson
EdM in Mind, Brain, and Education
Harvard Graduate School of Education
