1. Suppose x is a nonzero real number such that both x 5 and 20x + 19 x are rational numbers. Prove that x is a rational number.

Sure, let's break this down:

1. We are given that x is a nonzero real number and that both x/5 and (20x + 19)/x are rational numbers.

2. A rational number is a number that can be expressed as the quotient of two integers. So, we can say that x/5 = a/b and (20x + 19)/x = c/d, where a, b, c, and d are integers.

3. From x/5 = a/b, we can rearrange to find x = 5a/b.

4. We can substitute x = 5a/b into the second equation to get (20*(5a/b) + 19)/(5a/b) = c/d.

5. Simplifying this gives 100a/b + 19b/5a = c/d.

6. This equation tells us that the left-hand side must be a rational number since the right-hand side is a rational number (c/d).

7. Therefore, x must be a rational number because it is the sum of two rational numbers (100a/b and 19b/5a).

So, we have proven that if x is a nonzero real number such that both x/5 and (20x + 19)/x are rational numbers, then x is a rational number.