Chapter 1 - Introduction to Algebraic Expressions - 1.7 Multiplication and Division of Real Numbers - 1.7 Exercise Set - Page 59: 127

Yes. The reciprocals are: $\frac{1}{x}$ and $\frac{1}{-x}=-\frac{1}{x}$

The two numbers $x$ and $-x$ are opposite of each other. Their reciprocals are: $\frac{1}{x}$ and $\frac{1}{-x}=-\frac{1}{x}$ They are also opposites of each other.