Show that if 𝑥 is rational, then 3𝑥 − 1 is rational

Sure, here is the step by step proof:

1. Let's start by defining what a rational number is. A rational number is any number that can be expressed as the quotient or fraction p/q of two integers, with the denominator q not equal to zero.

2. Since x is given as rational, we can express it as x = p/q, where p and q are integers and q ≠ 0.

3. Now, let's consider the expression 3x - 1. Substituting x = p/q into this expression, we get 3(p/q) - 1.

4. Simplifying this expression, we get (3p - q) / q.

5. Since the numerator (3p - q) and the denominator q are both integers (because p and q are integers and the operations addition, subtraction, and multiplication of integers always result in integers), this means that (3p - q) / q is a rational number.

6. Therefore, if x is rational, then 3x - 1 is also rational. This completes the proof.