Based on what we established about the classification of x and using the closure of integers, what does the equation tell you about the type of number x must be for the sum to be rational? What conclusion can you now make about the result of adding a rational and an irrational number?
Answer: The product of a rational number with an irrational number is an irrational number. To see this assume that x is a rational number and y an irrational number. Then let us assume that the product xy is rational, which means that there are integers a,b such that xy=a/b. But then we obtain y=(1/x)(a/b) which is also rational since the set of rational numbers is closed under multiplication. But this is a contradiction since y was assumed to be an irrational number.
