Question 2

Step 1 · Assume $r + x$ is Rational and Isolate $x$

Assume on the contrary that $r + x$ is a rational number.

Then, there exist integers $a$ and $b$ ($b \neq 0$) such that: $r + x = \dfrac{a}{b}$

Since $r$ is given to be a rational number, let $r = \dfrac{c}{d}$, where $c$ and $d$ are integers and $d \neq 0$.

Substitute $r = \dfrac{c}{d}$ into the equation: $\dfrac{c}{d} + x = \dfrac{a}{b}$

Isolating $x$:

Step 2 · Analyze $x$ and Reach a Contradiction

Since $a, b, c, d$ are integers and $b, d \neq 0$:

• $ad - cb$ is an integer (integers are closed under multiplication and subtraction).
• $bd$ is a non-zero integer.

Therefore, $\dfrac{ad - cb}{bd}$ is a rational number, which implies: $x \text{ is a rational number}$

This contradicts the given statement that $x$ is an irrational number.

Hence, our assumption that $r + x$ is rational is false.