Question

What is the opposite and reciprocal of $-\dfrac{1}{x}$?

Answer 1

Hint: First of all understand the terms ‘opposite’ and ‘reciprocal’ of a number. Assume the opposite of the given number as k. Now, add k and $-\dfrac{1}{x}$ and equate it with 0. Solve for the value of k to get the answer of one part of the question. Further, assume the reciprocal of $-\dfrac{1}{x}$ as m. To find the value of m, multiply $-\dfrac{1}{x}$ and m and equate it with 1. Solve for the value of m to get the answer.

Complete step by step answer:
Here we have been asked to find the opposite and reciprocal of $-\dfrac{1}{x}$. First we need to understand the meaning of the terms ‘opposite’ and ‘reciprocal’. Let us check them one by one.
(1) The term ‘opposite’ of a number is also called its additive inverse. Additive inverse of a number (y) is defined as a number which when added to y gives the sum 0. So let us assume the additive inverse of $-\dfrac{1}{x}$ is k, therefore the sum of k and $-\dfrac{1}{x}$ will be 0.
\[\begin{align}
  & \Rightarrow k+\left( -\dfrac{1}{x} \right)=0 \\
 & \Rightarrow k-\dfrac{1}{x}=0 \\
 & \therefore k=\dfrac{1}{x} \\
\end{align}\]
Hence, the additive inverse (opposite) of $-\dfrac{1}{x}$ is $\dfrac{1}{x}$.
(2) The term ‘reciprocal’ of a number is also called its multiplicative inverse. Multiplicative inverse of a number (y) is defined as a number which when multiplied to k gives the product 1. So let us assume the multiplicative inverse of $-\dfrac{1}{x}$ is m, therefore the product of $-\dfrac{1}{x}$ and m will be 1.
$\begin{align}
  & \Rightarrow m\times \left( -\dfrac{1}{x} \right)=1 \\
 & \Rightarrow m=\dfrac{1}{\left( -\dfrac{1}{x} \right)} \\
 & \therefore m=-x \\
\end{align}$

Hence, the multiplicative inverse (reciprocal) of $-\dfrac{1}{x}$ is $-x$

Note: Remember the definitions of the terms mentioned in the solution otherwise it will be difficult to solve the question. To find the additive inverse of any number just multiply it with -1 and to find the multiplicative inverse of any number divide 1 by that number. These are the simple rules to get the answer. Note that in the above question x cannot be 0 otherwise the fraction $-\dfrac{1}{x}$ will become undefined.