The solution set of the equation is $(-2-x)=\sqrt[5]{(x-7)}$

We have, $(-2-x)=\sqrt[5]{(x-7)}$ 
The LHS of the equation is defined for $x\le -2$ and its value is zero at $x=-2$ and positive $x<-2$ whereas RHS of the equation is defined for all real values and its value is zero at $x=7,$ positive at $x>7$ and negative at $x<7.$ So, the equation has no real roots.