The number $q$ is irrational. Which statement about $q + \sqrt{33}$ is true? $\newline$ Choices: $\newline$ (A) $q + \sqrt{33}$ is rational. $\newline$ (B) $q + \sqrt{33}$ is irrational. $\newline$ (C) $q + \sqrt{33}$ can be rational or irrational, depending on the value of $q$ .

Full solution

Q. The number

q

is irrational. Which statement about

q + \sqrt{33}

is true?

\newline

Choices:

\newline

(A)

q + \sqrt{33}

is rational.

\newline

(B)

q + \sqrt{33}

is irrational.

\newline

(C)

q + \sqrt{33}

can be rational or irrational, depending on the value of

q

Identify Type of Number: Identify whether $\sqrt{33}$ is a rational or irrational number. $33$ is a non-perfect square, which means that its square root cannot be expressed as a ratio of two integers. Therefore, $\sqrt{33}$ is an irrational number.
Sum of Irrational Numbers: Consider the sum of two irrational numbers, $q$ and $\sqrt{33}$ . If $q$ is any irrational number that is not the additive inverse of $\sqrt{33}$ , then their sum is also irrational. This is because the sum of an irrational number and a rational number is irrational, and the sum of two irrational numbers is generally irrational unless they are additive inverses.
Special Cases Consideration: Examine the special cases where $q$ could be the additive inverse of $\sqrt{33}$ . If $q = -\sqrt{33}$ , then $q + \sqrt{33} = -\sqrt{33} + \sqrt{33} = 0$ , which is a rational number. This shows that there is at least one value of $q$ for which $q + \sqrt{33}$ is rational.
Final Conclusion: Determine the correct statement based on the previous steps. $\newline$ Since $q + \sqrt{33}$ can be rational if $q$ is the additive inverse of $\sqrt{33}$ , but is generally irrational for other values of $q$ , the correct statement is that $q + \sqrt{33}$ can be rational or irrational, depending on the value of $q$ .