You still get a rational number.
The sum/subtraction between rational numbers is defined as follows:
[tex]\dfrac{a}{b}\pm\dfrac{c}{d} = \dfrac{ad\pm bc}{bd}[/tex]
The expression [tex]ad\pm bc[/tex] is the sum/difference between the products of integers, and so is still an integer. This means that
[tex]\dfrac{ad\pm bc}{bd}[/tex]
is still a ratio between integers, and thus a rational.