Question

How do you solve $ \dfrac{7}{{x + 5}} + \dfrac{3}{{x - 5}} = \dfrac{{30}}{{{x^2} - 25}} $ and check for extraneous solutions?

Answer 1

Hint: Here first of we will take the given expression and apply Least common multiple concepts to simplify the denominators and then will accordingly find the value for the variable “x”.

Complete step-by-step answer:
Take the given expression –
 $ \dfrac{7}{{x + 5}} + \dfrac{3}{{x - 5}} = \dfrac{{30}}{{{x^2} - 25}} $
Take LCM (Least common multiple) for the terms on the left side of the given expression.
 $ \Rightarrow \dfrac{{7(x - 5)}}{{(x + 5)(x - 5)}} + \dfrac{{3(x + 5)}}{{(x - 5)(x + 5)}} = \dfrac{{30}}{{{x^2} - 25}} $
Simplify the above equation –
 $ \Rightarrow \dfrac{{7(x - 5)}}{{{x^2} - 25}} + \dfrac{{3(x + 5)}}{{{x^2} - 25}} = \dfrac{{30}}{{{x^2} - 25}} $
Since, the denominators on the left-hand side of the equation are the same, we can write under both the terms together considering the given signs in between them.
\[ \Rightarrow \dfrac{{7(x - 5) + 3(x + 5)}}{{{x^2} - 25}} = \dfrac{{30}}{{{x^2} - 25}}\]
Since denominators on both sides of the equation are the same, they cancel each other.
\[ \Rightarrow 7(x - 5) + 3(x + 5) = 30\]
Open the brackets and simplify the above equation –
\[ \Rightarrow 7x - 35 + 3x + 15 = 30\]
Make the pair of like terms together.
\[ \Rightarrow \underline {7x + 3x} - \underline {35 + 15} = 30\]
Simplify the above equation, remember when you simplify between one positive term and negative term you have to subtract and sign a bigger number.
 $ \Rightarrow 10x - 20 = 30 $
Take constant on one side of the equation. Remember when you move any term from one side to another, sign also changes. Positive terms become negative and vice-versa.
 $ \Rightarrow 10x = 30 + 20 $
Simplify the above equation –
 $ \Rightarrow 10x = 50 $
When any term multiplicative on one side is moved to the opposite side, then it goes to the denominator of the opposite side.
 $ \Rightarrow x = \dfrac{{50}}{{10}} $
Find the factors of the term on the numerator of the above expression.
 $ \Rightarrow x = \dfrac{{5 \times 10}}{{10}} $
Common factors from the numerator and the denominator cancel each other. Therefore, remove from the numerator and the denominator.
 $ \Rightarrow x = 5 $
This is the required solution.
So, the correct answer is “ $ x = 5 $ ”.

Note: Be careful about the sign while doing simplification remember the golden rules-
Addition of two positive terms gives the positive term
Addition of one negative and positive term, you have to do subtraction and give signs of bigger numbers, whether positive or negative.
Addition of two negative numbers gives a negative number but in actual you have to add both the numbers and give a negative sign to the resultant answer.