Answer
For every nonzero real number $x$, the multiplicative inverse of $x$ is unique.
Work Step by Step
Let $x$ be a nonzero real number and assume $a$ and $b$ are both multiplicative inverses of $x$.
Then $a\cdot x=1$ and $b\cdot x=1$. By substitution, $a\cdot x=b\cdot x$. Right multiplication by the generalized multiplicative inverse $x^{-1}$ gives:
$a\cdot x\cdot x^{-1}=b\cdot x\cdot x^{-1}$
$a\cdot 1=b\cdot 1$. By Theorem 3, 1 is a multiplicative identity of the real numbers.
So $a\cdot 1=a$, $b\cdot 1=b$, and $a=b$. There can only be one multiplicative inverse for each real number.
