Question Type 3: Using contradiction with conditions Bootcamps

Let $x \notin \mathbb{Q}$. Prove by contradiction that $|x| \notin \mathbb{Q}$.

Given $x \notin \mathbb{Q}$, prove by contradiction that $2x+1 \notin \mathbb{Q}$.

Given $x \notin \mathbb{Q}$, prove by contradiction that $3 - 5x \notin \mathbb{Q}$.

Let $x \ne 0$ be irrational. Prove by contradiction that $\dfrac{1}{x}$ is irrational.

Let $n \in \mathbb{Z}$. Prove by contradiction that if $n^2$ is even, then $n$ is even.

Let $x \notin \mathbb{Q}$ and $r \in \mathbb{Q}\setminus\{0\}$. Prove by contradiction that $\dfrac{r}{x} + 5$ is irrational.

Let $r \in \mathbb{Q}\setminus\{0\}$. If $x \notin \mathbb{Q}$, prove by contradiction that $\dfrac{x}{r} \notin \mathbb{Q}$.

Let $a,b \in \mathbb{Q}$ with $a \ne 0$. Given $x \notin \mathbb{Q}$, prove by contradiction that $ax + b \notin \mathbb{Q}$.