Prove that the product of a nonzero rational number together with an irrational number is an irrational number.

Each time they assume the sum is rational; however, upon rearranging the terms of their equation, they get a contradiction (that an irrational number is equal to a rational number). Since the assumption that the sum of a rational and irrational number is rationalleads to a contradiction, the sum must be irrational.

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