Answer
See the proof below.
Work Step by Step
Consider the following statement:
The sum of a rational number and an irrational number is irrational.
We will prove by contradiction that the statement is true.
$\textbf{Proof:}$ Suppose there exists a rational number $r$ and an irrational number $x$ such that $r+x$ is rational.
$r$ being a rational number implies that $-r$ is a rational number.
Also, since $r+x$ is a rational number and the rationals are closed under addition, it follows that $\left(-r\right)+\left(r+x\right)$ is a rational number.
Now, observe that
$$\left(-r\right)+\left(r+x\right)=\left(\left(-r\right)+r\right)+x=0+x=x.$$
Thus, $x$ is a rational number and this contradicts our assumption that $x$ is an irrational number.
Hence, the statement is true.
