Simple act practice test problem

Question

I am going through an act practice test and I came to a problem that said.

Which of the following is equal to the product of x and the square of its reciprocal for all x < 0.

My first step would be to set up the problem. Which would result in

x * (1/x)^2. I could simplify this here and change the negatives to 0 then, or I could substitute the -x in now.

I decided to substitute now

This would equate to: -x * (1/-x)^2 which would be simplified as: -x * 1/(-x^2) = -x * 1/(x^2) = -x/(x^2) = -1/x

Or you could simplify first and get 1/x and then substitute the negative to get the same answer.

However the book says the answer is 1/x. Its justification for this is it says that x being negative doesn't change anything and "the fact that x is negative will automatically make the whole thing negative -- there's no need to add an extra negative sign. "

I don't see there logic. Am I or the book wrong?

Answer 1

Take an example value say, $x=-4$, that would give you $$(-4)\times\frac{1}{(-4)^2}=(-4)\times\frac{1}{16}=-\frac{1}{4}\tag{1}$$ So if you took the answer to be $-\frac{1}{x}$ then you would get:$$-\frac{1}{(-4)}=\frac{1}{4}\tag{2}$$ (2) does not match (1) which should indicate to you that the book is indeed correct.

Another way to look at this is to modify the question slightly to:

Which of the following is equal to the product of x and the square of its reciprocal for all $x\ne0$.

What would you then say is the correct answer?

In both cases the answer is $\frac{1}{x}$. The answer does not depend on the actual value of $x$.