jmc

We know that the asymptotes of the hyperbola are given by the two equations $$\frac{x-3}{5}=\pm \frac{y+1}{4}.$$We see that choosing the $+$ sign will give an asymptote with positive slope: $$\frac{x-3}{5}=\frac{y+1}{4}.$$To compute the $x-$intercept of this line, we set $y=0,$ giving $$\frac{x-3}{5}=\frac{1}{4}.$$Then $x-3=\frac{5}{4},$ so $x=3+\frac{5}{4}=\frac{17}{4}.$ Thus, the $x-$intercept is $(x,y)=\boxed{\left(\frac{17}{4},0\right)}.$