Question

Find the solution of $ \dfrac{2}{{x + 3}} - \dfrac{4}{{x - 3}} = \dfrac{{ - 6}}{{x + 3}} $ .

Answer 1

Hint: Various operations such as multiplication, addition subtraction can be used for simplifying an equation. For an equation consisting of terms in the form of fraction, the product of the denominators of all the terms can be multiplied on both sides of the equation, in order to initiate the simplification. Then after all the brackets are to be solved first using the BODMAS method, i.e. first solve brackets, then division, multiplication, addition and subtraction based on the type of question being asked.

Complete step-by-step answer:
The given equation is $ \dfrac{2}{{x + 3}} - \dfrac{4}{{x - 3}} = \dfrac{{ - 6}}{{x + 3}} $ .
To simplify the entire equation, we can multiply the entire equation by $ \left( {x + 3} \right)\left( {x - 3} \right) $ .
On multiplying the entire equation by $ \left( {x + 3} \right)\left( {x - 3} \right) $ , we get, $ \left( {x + 3} \right)\left( {x - 3} \right) \cdot \left( {\dfrac{2}{{x + 3}} - \dfrac{4}{{x - 3}}} \right) = \left( {x + 3} \right)\left( {x - 3} \right) \cdot \left( {\dfrac{{ - 6}}{{x + 3}}} \right) $
After simplifying the equation further, we get,
 $
\Rightarrow \left( {\left( {x + 3} \right)\left( {x - 3} \right)\dfrac{2}{{x + 3}} - \left( {x + 3} \right)\left( {x - 3} \right)\dfrac{4}{{x - 3}}} \right) = \left( {x + 3} \right)\left( {x - 3} \right) \cdot \left( {\dfrac{{ - 6}}{{x + 3}}} \right)\\
2\left( {x - 3} \right) - 4\left( {x + 3} \right) = - 6\left( {x - 3} \right)
 $
On multiplying the constants outside the bracket to the linear equation, we get,
 $
\Rightarrow 2x - 6 - 4x - 12 = - 6x + 18\\
\Rightarrow - 2x - 18 = - 6x + 18
 $
On simplifying the equation to reach the final solution, we get,
 $
\Rightarrow - 2x + 6x = 18 + 18\\
\Rightarrow 4x = 36
 $
Divide the entire equation by 4.
 $
\Rightarrow x = \dfrac{{36}}{4}\\
\Rightarrow x = 9
 $

Therefore, the value of x = 9.

Additional Information:
Solving any equation means to find the solution for the given equation. There are various methods which can be used to find the solution.
One can either simplify the equation or use a hit and trial method.
Simplifying the equation is a better approach as compared to hit and trial.
The time required for solving an equation using hit and trial method cannot be determined. But by using a simplification method, the equation can be solved easily and within the stipulated time.
There are different types of equations such as matrix equations, simple linear equations, differential equations, polynomial equations, system of linear equations, etc.

Note: In order to start the simplification of the equation, apply the same operations on both sides of the equation using the same expression, for instance if you want to divide the left-hand side of equation by 2, then at the same time, it is necessary to keep in mind that the right-side of the equation should also be divided by 2.