Question

How do you solve \[\dfrac{3x}{x+1}+\dfrac{6}{2x}=\dfrac{7}{x}\]?

Answer 1

Hint: Simplify the term \[\dfrac{6}{2x}\] by cancelling the common factor. Take the simplified form of this expression to the R.H.S. and simplify further by taking L.C.M. Now, cross – multiply the terms and form a quadratic equation in x. Solve this quadratic equation by using the middle term split method and get the two values of x.

Complete step by step answer:
Here, we have been provided with the equation: - \[\dfrac{3x}{x+1}+\dfrac{6}{2x}=\dfrac{7}{x}\] and we are asked to solve it. That means we have to find the values of x. So, we have,
\[\Rightarrow \dfrac{3x}{x+1}+\dfrac{6}{2x}=\dfrac{7}{x}\]
Simplifying the term \[\dfrac{6}{2x}\] by cancelling the common factor and taking it to the R.H.S., we have,
\[\Rightarrow \dfrac{3x}{x+1}=\dfrac{7}{x}-\dfrac{3}{x}\]
Taking L.C.M. in the R.H.S. and simplifying, we get,
\[\begin{align}
  & \Rightarrow \dfrac{3x}{x+1}=\dfrac{7-3}{x} \\
 & \Rightarrow \dfrac{3x}{x+1}=\dfrac{4}{x} \\
\end{align}\]
By cross – multiplication we get,
\[\begin{align}
  & \Rightarrow 3{{x}^{2}}=4\left( x+1 \right) \\
 & \Rightarrow 3{{x}^{2}}=4x+4 \\
\end{align}\]
Taking all the terms to the L.H.S., we get,
\[\Rightarrow 3{{x}^{2}}-4x-4=0\]
Clearly, we can see that the above equation is a quadratic equation in x, so using the middle term split method, we have,
\[\begin{align}
  & \Rightarrow 3{{x}^{2}}-6x+2x-4=0 \\
 & \Rightarrow 3x\left( x-2 \right)+2\left( x-2 \right)=0 \\
\end{align}\]
Taking \[\left( x-2 \right)\] common we have,
\[\Rightarrow \left( x-2 \right)\left( 3x+2 \right)=0\]
Substituting each term equal to 0, we get,
\[\Rightarrow x-2=0\] or \[3x+2=0\]
\[\Rightarrow x=2\] or \[x=\dfrac{-2}{3}\]
Hence, the solutions of the given equation are: - \[x=2,\dfrac{-2}{3}\].

Note:
One may note that if we would have obtained x = -1 or x = 0 as our solution then we have to reject them. This is because for these values of x the initial equation is undefined. So, we need to check once before writing the answers. Note that we have used the middle terms split method to solve the quadratic equation. You can also apply the discriminant method to get the values of x.