Question

How do you solve \[\dfrac{7}{{x - 4}} = 1 + \dfrac{9}{{x + 4}}\]?

Answer 1

Hint: In the given question, we have been given an equation. It involves two distinct terms – in each of the terms, a constant is in the numerator and a linear monomial is in the denominator. We have to solve for the value of the variable. To solve that, we are going to multiply both sides of equality by the product of the two denominators. Then we are going to simplify it, perform the operations, and solve for the required value.

Complete step by step answer:
We have to solve,
\[\dfrac{7}{{x - 4}} = 1 + \dfrac{9}{{x + 4}}\]
First, we are going to remove the denominators from both of the terms by multiplying the both sides with the product of the two monomials,
Product, \[P = \left( {x - 4} \right)\left( {x + 4} \right)\]
Multiplying,
\[\dfrac{7}{{x - 4}} \times \left( {x - 4} \right)\left( {x + 4} \right) = \left( {x - 4} \right)\left( {x + 4} \right) + \dfrac{9}{{x + 4}}\left( {x - 4} \right)\left( {x + 4} \right)\]
Simplifying,
$\Rightarrow$ \[7\left( {x + 4} \right) = {x^2} - 16 + 9\left( {x - 4} \right)\]
$\Rightarrow$ \[7x + 28 = {x^2} - 16 + 9x - 36\]
Taking all the things to one side,
$\Rightarrow$ \[{x^2} + 2x - 24 = 0\]
Solving by factoring using splitting the middle term,
$\Rightarrow$ \[{x^2} + 6x - 4x - 24 = 0\]
Taking commons,
$\Rightarrow$ \[x\left( {x + 6} \right) - 4\left( {x + 6} \right) = 0\]
\[\left( {x - 4} \right)\left( {x + 6} \right) = 0\]

Hence, we have,
\[x = - 6,4\]

Note: In the given question, we had to solve an equation in which two constants were divided by two distinct monomials. We solved it by removing the denominators, simplifying them, and finding the value of the unknown. So, it is really important that we know the formulae and where, when and how to use them so that we can get the correct result.