Use Simplify, Collect together all the j, and then push things not containing j across the inequality.

Collect[Simplify[j-((β-1)(β^2 j λ+β^2(-j)-β j λ+β j-β^2 j λ q+β^2 j q+β^2 q x z-
  β^2 q z+β q z-β^2 λ w+β λ w+β^2 λ w x-β^2 x z+β x z+β^2 z-2 β z+z))/
  ((β-1)(-β-β^2 λ q+β q+β^2 q x+β^2 λ x-β^2 x+β x+1))]>0,j]/.p1_+p2_>0->p1>-p2

which results in

(j*(1+β*(-1+λ+q))*(1+β*(-1+x)))/(1+β*(-1+q+x)+β^2*((-1+q)*x+λ*(-q+x))) >
-(((1+β*(-1+x))*(-z+β*(-(λ*w)+z-q*z)))/(1+β*(-1+q+x)+β^2*((-1+q)*x+λ*(-q+x))))

Notice that contains a single j on the left hand side and the left and right hand side denominators are the same and the left and right hand side numerators have a common factor.

IF you can justify to yourself that you can multiply and divide both sides without changing the inequality to eliminate those common factors then the condition for j is

j(1+β(-1+λ+q)) > -(-z+β(λ w+z-q z))

IF you can further justify one more division without changing the inequality then the condition for j is

j > -(-z+β(λ w+z-q z))/(1+β(-1+λ+q))

Please check every step of this very carefully to convince yourself that I have made no mistakes.