The number $u$ is irrational. Which statement about $\sqrt{3} - u$ is true? $\newline$ Choices: $\newline$ (A) $\sqrt{3} - u$ is rational. $\newline$ (B) $\sqrt{3} - u$ is irrational. $\newline$ (C) $\sqrt{3} - u$ can be rational or irrational, depending on the value of $u$ .

Full solution

Q. The number

u

is irrational. Which statement about

\sqrt{3} - u

is true?

\newline

Choices:

\newline

(A)

\sqrt{3} - u

is rational.

\newline

(B)

\sqrt{3} - u

is irrational.

\newline

(C)

\sqrt{3} - u

can be rational or irrational, depending on the value of

u

Identify Type of Number: Identify whether $\sqrt{3}$ is a rational or irrational number. $\newline$ Since $3$ is not a perfect square, the square root of $3$ is an irrational number.
Properties of Irrational Numbers: Consider the properties of irrational numbers. The difference between two irrational numbers can be rational or irrational. It depends on the specific numbers involved.
Analyze Outcomes for $\sqrt{3} - u$ : Analyze the possible outcomes for $\sqrt{3} - u$ . If $u$ is an irrational number that is not related to $\sqrt{3}$ , then $\sqrt{3} - u$ is likely to be irrational. However, if $u$ is some irrational number that, when subtracted from $\sqrt{3}$ , results in a rational number (for example, if $u = \sqrt{3} +$ a rational number), then $\sqrt{3} - u$ could be rational.
Determine Correct Choice: Determine the correct choice based on the analysis. $\newline$ Since there are scenarios where $\sqrt{3} - u$ could be rational and others where it could be irrational, the correct statement is that $\sqrt{3} - u$ can be rational or irrational, depending on the value of $u$ .