jmc

The line is of the form $y=mx+5.$ Substituting, we get $$9x^2+16(mx+5{)}^2=144.$$Expanding, we get $$(16m^2+9)x^2+160mx+256=0.$$For the line and ellipse to intersect, this quadratic must have a real root, which means that its discriminant is nonnegative: $$(160m{)}^2-4(16m^2+9)(256)\ge 0.$$This reduces to $m^2\ge 1.$ Thus, the possible slopes are $m\in \boxed{(-\infty ,-1]\cup [1,\infty )}.$