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The number $y$ is irrational. Which statement about $y - \sqrt{38}$ is true? $\newline$ Choices: $\newline$ (A) $y - \sqrt{38}$ is rational. $\newline$ (B) $y - \sqrt{38}$ is irrational. $\newline$ (C) $y - \sqrt{38}$ 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 irrational. Which statement about

y - \sqrt{38}

is true?

\newline

Choices:

\newline

(A)

y - \sqrt{38}

is rational.

\newline

(B)

y - \sqrt{38}

is irrational.

\newline

(C)

y - \sqrt{38}

can be rational or irrational, depending on the value of

y

.

Identify Type of Number: Identify whether $\sqrt{38}$ is a rational or irrational number. $38$ is a non-perfect square. $\sqrt{38}$ 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.
Specific Case Analysis: Analyze the specific case where $y = \sqrt{38}$ . If $y = \sqrt{38}$ , then $y - \sqrt{38} = \sqrt{38} - \sqrt{38} = 0$ , which is a rational number.
Analysis of Different Cases: Analyze the specific case where $y$ is any irrational number not equal to $\sqrt{38}$ . If $y$ is not equal to $\sqrt{38}$ , then $y - \sqrt{38}$ is the difference of two distinct irrational numbers, which is generally irrational.
Conclusion: Conclude that $y - \sqrt{38}$ can be rational or irrational, depending on the value of $y$ . If $y$ is chosen specifically to be $\sqrt{38}$ , the result is rational. Otherwise, it is typically irrational.

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