Question

How do you solve \[\dfrac{{8\left( {x - 1} \right)}}{{{x^2} - 4}} = \dfrac{4}{{x - 2}}\]?

Answer 1

Hint: Here in this question, we have to solve the given equation to the x variable. The given equation is the algebraic equation with one variable x, this can be solve by add or subtract the necessary term from each side of the equation to isolate the term with the variable x, then multiply or divide each side of the equation by the appropriate value, while keeping the equation balanced then solve the resultant balance equation for the x value.

Complete step-by-step solution:
Consider the given equation
\[\dfrac{{8\left( {x - 1} \right)}}{{{x^2} - 4}} = \dfrac{4}{{x - 2}}\]--------(1)
Where x is the variable
Rewrite the equation (1) as
\[\dfrac{{8\left( {x - 1} \right)}}{{{x^2} - {2^2}}} = \dfrac{4}{{x - 2}}\]
Now, we have to solve the above equation for the variable x
By using the algebraic formula \[\left( {{a^2} - {b^2}} \right) = \left( {a + b} \right)\left( {a - b} \right)\] the denominator of LHS can be written as
\[ \Rightarrow \,\,\,\dfrac{{8\left( {x - 1} \right)}}{{\left( {x - 2} \right)\left( {x + 2} \right)}} = \dfrac{4}{{x - 2}}\]
Multiply both side by \[\left( {x - 2} \right)\]
\[ \Rightarrow \,\,\,\dfrac{{8\left( {x - 1} \right)}}{{\left( {x - 2} \right)\left( {x + 2} \right)}}\left( {x - 2} \right) = \dfrac{4}{{x - 2}}\left( {x - 2} \right)\]
On simplification, we get
\[ \Rightarrow \,\,\,\dfrac{{8\left( {x - 1} \right)}}{{\left( {x + 2} \right)}} = 4\]--------(2)
Multiply \[\left( {x + 2} \right)\] on both side, then
\[ \Rightarrow \,\,\,8\left( {x - 1} \right) = 4\left( {x + 2} \right)\]
Using the distributive property
\[ \Rightarrow \,\,\,8x - 8 = 4x + 8\]
To isolate the x variable take all the x variable and its coefficient to the LHS, similarly the remaining term take in to the RHS, i.e.,
\[ \Rightarrow \,\,\,8x - 4x = 8 + 8\]
On simplification, we get
\[ \Rightarrow \,\,\,4x = 16\]
For solving the x, divide both side by 4
\[ \Rightarrow \,\,\,\dfrac{4}{4}x = \dfrac{{16}}{4}\]
\[ \Rightarrow \,\,\,x = 4\]

Hence, the required solution is \[\,x = 4\].

Note: In this question we solve the given equation for the one variable. While shifting the terms we must take care of signs. When the term is moving from one side to another side the sign will change. The term is in the form of fraction so we use the LCM procedure and we obtain the solution for the question.