Find the roots of the given equation:

contexto.140

3 years ago

$\begin{array}{l} \frac{3}{x+1}-\frac{1}{2}=\frac{2}{3 x-1}, x \neq-1, \frac{1}{3} \\ \Rightarrow \frac{3}{x+1}-\frac{2}{3 x-1}=\frac{1}{2} \\ \Rightarrow \frac{9 x-3-2 x-2}{(x+1)(3 x-1)}=\frac{1}{2} \\ \Rightarrow \frac{7 x-5}{3 x^{2}+2 x-1}=\frac{1}{2} \\ \Rightarrow 3 x^{2}+2 x-1=14 x-10 \\ \Rightarrow 3 x^{2}-12 x+9=0 \\ \Rightarrow x^{2}-4 x+3=0 \\ \Rightarrow x^{2}-3 x-x+3=0 \\ \Rightarrow x(x-3)-1(x-3)=0 \\ \Rightarrow(x-3)(x-1)=0 \\ \Rightarrow x-3=0 \text { or } x-1=0 \\ \Rightarrow x=3 \text { or } x=1 \end{array}$

Hence, 1 and 3 are the roots of the given equation.