jmc

We can write $$j(x)=\frac{5x+3}{x}=5+\frac{3}{x}.$$First, $x$ can take on any non-zero value. Second, as $x$ takes on all non-zero values, $\frac{3}{x}$ also takes on all real values except 0, which means $5+\frac{3}{x}$ takes on all values except 5. Therefore, the range of the function is $\boxed{(-\infty ,5)\cup (5,\infty )}.$