Final answer:
To prove that the sum of a rational number (q) and an irrational number (x) is irrational, we assume the opposite and find a contradiction, thus the sum q + x must be irrational.
Step-by-step explanation:
The subject of the question is to prove that if q is a rational number (Q) and x is an irrational number (not in Q), then the sum q + x is also an irrational number (in R-Q). To show this, we use the assumption that q is rational, which means it can be expressed as a ratio of two integers (q = a/b where a, b are integers and b is not zero). Since x is irrational, it cannot be expressed as a ratio of two integers. If we were to assume that q + x is rational, then x would also have to be rational, which is a contradiction because we know x is irrational. Because this assumption leads to a contradiction, it demonstrates that q + x must be irrational, and thus q + x is in R-Q.