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

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Properties of operations on rational and irrational numbers

Full solution

Q. The number

y

is rational. Which statement about

\sqrt{3} + y

is true?

\newline

Choices:

\newline

(A)

\sqrt{3} + y

is rational.

\newline

(B)

\sqrt{3} + y

is irrational.

\newline

(C)

\sqrt{3} + y

can be rational or irrational, depending on the value of

y

.

Identify Type of $\sqrt{3}$ : Identify whether $\sqrt{3}$ is a rational or irrational number. $\newline$ Since $3$ is not a perfect square, $\sqrt{3}$ is an irrational number.
Nature of Sum: Consider the nature of the sum of a rational number and an irrational number. The sum of a rational number and an irrational number is always irrational.
Application to $\sqrt{3} + y$ : Apply the above principle to $\sqrt{3} + y$ . Since $y$ is rational and $\sqrt{3}$ is irrational, $\sqrt{3} + y$ must be irrational.

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