Chapter 5.2, Problem 43WE

To determine

To find: By counterexample that the given expressions are not equivalent.

Explanation of Solution

Given information:

  ${\left(x+y\right)}^{-1}$ and $x^{-1}+y^{-1}$

  ${\left(x+y\right)}^{-1}$ and $x^{-1}+y^{-1}$

Taking $x=1$ and $y=2$

We get

  $\begin{array}{l}{\left(x+y\right)}^{-1}={\left(1+2\right)}^{-1}\\ \Rightarrow {\left(x+y\right)}^{-1}={\left(3\right)}^{-1}\\ \Rightarrow {\left(x+y\right)}^{-1}=\frac{1}{3}\\ \Rightarrow {\left(x+y\right)}^{-1}=0.33\end{array}$

  $\begin{array}{l}{x}^{-1}+y^{-1}=1^{-1}+2^{-1}\\ \Rightarrow x^{-1}+y^{-1}=\frac{1}{1}+\frac{1}{2}\\ \Rightarrow x^{-1}+y^{-1}=1+0.5\\ \Rightarrow x^{-1}+y^{-1}=1.5\end{array}$

Therefore,

  ${\left(x+y\right)}^{-1}=0.33$ and $x^{-1}+y^{-1}=1.5$

Which are not equivalent.