$$\frac{(e^x-2)}{x-1}+\frac{(e^x-1/4)}{x+1}=0$$
The above equation are given, and the question asks to show the equation has a solution in interval $(-1,1)$.
I have no idea at all. I have just studied continuous function and I think it might be done by using intermediate value theorem.
As $x \to 1$ the function on LHS tends to $-\infty$ and $x \to -1$ it tends to $\infty$. Being a continuous function it takes all real values in the interval $(-1,1)$.
Remember that $2 <e <4$.