Answer: The answer is \(\dfrac{5}{8},~\dfrac{3}{2}\) and no extraneous solution.
Step-by-step explanation: The given equation is
\(2|7-7x|=2x+4.\)
We are given to solve this equation and also find its extraneous solution.
The solution is as follows:
\(2|7-7x|=2x+4\\\\\Rightarrow |7-7x|=x+2\\\\\Rightarrow 7-7x=x+2,~~~~~-7+7x=x+2\\\\\Rightarrow 8x=5,~~~~~~~~~~~~~~~~\Rightarrow 6x=9\\\\\\\Rightarrow x=\dfrac{5}{8},~~~~~~~~~~~~~~~~~\Rightarrow x=\dfrac{3}{2}.\)
now, for \(x=\dfrac{5}{8}\), we have
\(L.H.S.=2|7-7\times \dfrac{5}{8}|=\dfrac{21}{4},\\\\R.H.S.=2\times\dfrac{5}{8}+4=\dfrac{21}{4}.\)
So, this solution is not extraneous.
For \(x=\dfrac{3}{2}\), we have
\(L.H.S.=2|7-7\times \dfrac{3}{2}|=7,\\\\R.H.S.=2\times\dfrac{3}{2}+4=7.\)
So, this solution is also not extraneous.
Hence, the solution is [tex]\dfrac{5}{8},~\dfrac{3}{2}[tex] and there is no extraneous solution.
