IB Mathematics Analysis and Approaches Question Type 3: Using Contradiction with Conditions Exercises

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

Prove by contradiction that if $n^2$ is even, then $n$ is even.

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

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

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

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

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

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