The number $x$ is irrational. Which statement about $x + \sqrt{50}$ is true? $\newline$ Choices: $\newline$ (A) $x + \sqrt{50}$ is rational. $\newline$ (B) $x + \sqrt{50}$ is irrational. $\newline$ (C) $x + \sqrt{50}$ can be rational or irrational, depending on the value of $x$ .

Full solution

Q. The number

x

is irrational. Which statement about

x + \sqrt{50}

is true?

\newline

Choices:

\newline

(A)

x + \sqrt{50}

is rational.

\newline

(B)

x + \sqrt{50}

is irrational.

\newline

(C)

x + \sqrt{50}

can be rational or irrational, depending on the value of

x

Identify type of number: Identify whether $\sqrt{50}$ is a rational or irrational number. $\sqrt{50}$ can be simplified to $\sqrt{25\times2}$ which is equal to $5\times\sqrt{2}$ . Since $\sqrt{2}$ is an irrational number, $5\times\sqrt{2}$ is also an irrational number.
Simplify $\sqrt{50}$ : Consider the sum of two irrational numbers, $x$ and $5\sqrt{2}$ . The sum of two irrational numbers can be either rational or irrational. It is not guaranteed to be one or the other without additional information about the specific numbers involved.
Consider sum of irrationals: Analyze the given choices in the context of the sum of $x$ and $5\sqrt{2}$ . $\newline$ Choice (A) states that $x + \sqrt{50}$ is rational. This is not necessarily true without knowing the specific value of $x$ . $\newline$ Choice (B) states that $x + \sqrt{50}$ is irrational. This is not necessarily true either, as there could be a specific value of $x$ that makes the sum rational. $\newline$ Choice (C) states that $x + \sqrt{50}$ can be rational or irrational, depending on the value of $x$ . This is the correct choice because if $x$ were chosen to be the negative of $5\sqrt{2}$ , the sum would be $5\sqrt{2}$ $0$ , which is rational. If $x$ is any other irrational number, the sum would be irrational.