2x^(7)=1-x (0,1)????
Are you familiar with the Intermediate Value Theorem?
We have $\displaystyle 2x^7+x-1=0$. Consider the function $\displaystyle f(x)=2x^7+x-1$. We know that $\displaystyle f(0) = -1$ and $\displaystyle f(1) = 2$. Since $\displaystyle f$ is continuous, what does this tell us?