The function f(x) = tan(0.8x) is a discontinuous function that goes from negative infinity to positive infinity in each interval -pi/1.6<x<pi/1.6, pi/1.6<x<3pi/1.6; 3pi/1.6<x<5pi/1.6, etc. Thus there is a solution for tan(0.8x)+x= 0 in each of interval, and hence there are an infinite number of solutions. See the attached plot which shows 5 intervals of the function.