Given that a, b are rational, b ≠ 0, and r is irrational.
Assume a + br is rational
if a + br is rational, then a has a rational additive inverse, then a + br - a is rational, because rational numbers are closed under addition. Which means br is rational, because of closure under addition.
if br is rational, br/b must be rational because b, as a non-zero rational number, has rational multiplicative inverse and rational numbers are closed under multiplication. Thus, once we simplify b/b = 1, r is rational because of closure of rational numbers under multiplication.
But, it was given that r was irrational, therefore the assumption that
a + br is rational leads to a contradiction.
