Question

How do you write $\dfrac{5}{{x - 3}} - \dfrac{4}{{x + 3}}$ as a single fraction in its simplest form?

Answer 1

Hint: We have to cross-multiply the two fractions and add the results together to get the numerator of the answer. To get the numerator of the answer, cross-multiply. In other words, multiply the numerator of each fraction by the denominator of the other. Then, add the results to get the numerator of the answer. Next, multiply the two denominators together to get the denominator of the answer. Finally, write the answer as a fraction.

Complete step by step solution:
We have to simplify $\dfrac{5}{{x - 3}} - \dfrac{4}{{x + 3}}$.
We have to cross-multiply the two fractions and add the results together to get the numerator of the answer.
We want to add the fractions $\dfrac{5}{{x - 3}}$ and $\dfrac{{ - 4}}{{x + 3}}$. To get the numerator of the answer, cross-multiply. In other words, multiply the numerator of each fraction by the denominator of the other:
$\dfrac{5}{{x - 3}} - \dfrac{4}{{x + 3}}$
$5\left( {x + 3} \right) = 5x + 15$
$ - 4\left( {x - 3} \right) = - 4x + 12$
Add the results to get the numerator of the answer:
$5x + 15 - 4x + 12 = x + 27$
Multiply the two denominators together to get the denominator of the answer.
To get the denominator, just multiply the denominators of the two fractions:
$\left( {x - 3} \right)\left( {x + 3} \right)$
The denominator of the answer is $\left( {x - 3} \right)\left( {x + 3} \right)$.
Write the answer as a fraction.
$\dfrac{5}{{x - 3}} - \dfrac{4}{{x + 3}} = \dfrac{{x + 27}}{{\left( {x - 3} \right)\left( {x + 3} \right)}}$
This fraction can’t be reduced further, $\dfrac{{x + 27}}{{\left( {x - 3} \right)\left( {x + 3} \right)}}$ is the final answer.

Hence, simplified version of $\dfrac{5}{{x - 3}} - \dfrac{4}{{x + 3}}$ is $\dfrac{{x + 27}}{{\left( {x - 3} \right)\left( {x + 3} \right)}}$.

Note: By simplifying a fraction, we mean to express the fraction as a ratio of prime numbers or we can say that both the numerator and denominator should be prime numbers, that is, they should be divisible by only $1$ and itself. For simplifying a fraction, we write it as a product of prime factors, and then divide both of them with the common factors. In this question both the numerator and denominator are already prime numbers and thus the fraction cannot be simplified further.