Question

Find the product of additive inverse and multiplicative inverse of $\dfrac{{x - 2}}{{{x^2} - 4}}$
A. $x + 2$
B. $x - 2$
C. 1
D. -1

Answer 1

Hint: The additive inverse of a number a is given as -a and the multiplicative inverse of a number a is given as $\dfrac{1}{a}$. Using this we can find the additive inverse and multiplicative inverse of the given expression and their product gives the required answer.

Step by step solution :
We are given a expression $\dfrac{{x - 2}}{{{x^2} - 4}}$
We know that the additive inverse of a number a is given as – a
Hence the additive inverse of the given expression is $ - \left( {\dfrac{{x - 2}}{{{x^2} - 4}}} \right)$
Same way the multiplicative inverse of a number a is given as $\dfrac{1}{a}$
Hence the multiplicative inverse of the given expression is $\dfrac{1}{{\dfrac{{x - 2}}{{{x^2} - 4}}}} = \dfrac{{{x^2} - 4}}{{x - 2}}$
We are asked the product of the additive inverse and the multiplicative inverse
Hence we get
$
   \Rightarrow - \left( {\dfrac{{x - 2}}{{{x^2} - 4}}} \right)\times \dfrac{{{x^2} - 4}}{{x - 2}} \\
   \Rightarrow - 1 \\
 $
Therefore the product of the additive inverse and multiplicative inverse of the given expression is -1.

Therefore the correct answer is option D.

Note :
1) The additive inverse of a number a is the number that, when added to a, yields zero. This number is also known as the opposite (number), sign change, and negation. For a real number, it reverses its sign: the opposite to a positive number is negative, and the opposite to a negative number is positive.
2) A multiplicative inverse or reciprocal for a number x, denoted by $\dfrac{1}{x}{\text{ }}or{\text{ }}{x^{ - 1}}$ , is a number which when multiplied by x yields the multiplicative identity, 1. For the multiplicative inverse of a real number, divide 1 by the number.