Question

Divide the following and write your answer in lowest terms: \[\dfrac{x}{{x + 1}} \div \dfrac{{{x^2}}}{{{x^2} - 1}}\]
A. \[\dfrac{{x - 1}}{x}\]
B. \[\dfrac{{x + 1}}{x}\]
C. \[\dfrac{{x - 1}}{{{x^2}}}\]
D. None of these

Answer 1

Hint: In this question, first of all inter change the numerator and denominator of the right-hand side part to change the division part to multiplication part. Then simply the obtained expression by using the formula \[{a^2} - {b^2} = \left( {a + b} \right)\left( {a - b} \right)\]. So, use this concept to reach the solution of the given problem.

Complete Step-by-Step solution:
The given expression \[\dfrac{x}{{x + 1}} \div \dfrac{{{x^2}}}{{{x^2} - 1}}\] can be simplified as shown below:
\[ \Rightarrow \dfrac{x}{{x + 1}} \div \dfrac{{{x^2}}}{{{x^2} - 1}}\]
By interchange the numerator and denominator of the right-hand side part, we have
\[
   \Rightarrow \dfrac{x}{{x + 1}} \div \dfrac{{{x^2}}}{{{x^2} - 1}} = \dfrac{x}{{x + 1}} \times \dfrac{{{x^2} - 1}}{{{x^2}}} \\
   \Rightarrow \dfrac{x}{{x + 1}} \div \dfrac{{{x^2}}}{{{x^2} - 1}} = \dfrac{x}{{x + 1}} \times \dfrac{{{x^2} - {1^2}}}{{{x^2}}} \\
\]
We know that the term \[{a^2} - {b^2}\] can be written as \[\left( {a + b} \right)\left( {a - b} \right)\]
\[
   \Rightarrow \dfrac{x}{{x + 1}} \div \dfrac{{{x^2}}}{{{x^2} - 1}} = \dfrac{x}{{x + 1}} \times \dfrac{{\left( {x + 1} \right)\left( {x - 1} \right)}}{{x \times x}}{\text{ }}\left[ {\because {a^2} - {b^2} = \left( {a + b} \right)\left( {a - b} \right)} \right] \\
    \\
\]
Cancelling the common terms, we get
\[
   \Rightarrow \dfrac{x}{{x + 1}} \div \dfrac{{{x^2}}}{{{x^2} - 1}} = \dfrac{1}{1} \times \dfrac{{x - 1}}{x} \\
  \therefore \dfrac{x}{{x + 1}} \div \dfrac{{{x^2}}}{{{x^2} - 1}} = \dfrac{{x - 1}}{x} \\
\]
Hence, the lowest terms of \[\dfrac{x}{{x + 1}} \div \dfrac{{{x^2}}}{{{x^2} - 1}}\] is given by \[\dfrac{{x - 1}}{x}\].
Thus, the correct option is A. \[\dfrac{{x - 1}}{x}\]

Note: In this question, it is very important to know that we can rewrite the given expression by changing the second term. For simplicity, we may even write the divide expression as a division over division or inverse of multiplication.