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The number $y$ is irrational. Which statement about $y - 4$ is true? $\newline$ Choices: $\newline$ (A) $y - 4$ is rational. $\newline$ (B) $y - 4$ is irrational. $\newline$ (C) $y - 4$ 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 - 4

is true?

\newline

Choices:

\newline

(A)

y - 4

is rational.

\newline

(B)

y - 4

is irrational.

\newline

(C)

y - 4

can be rational or irrational, depending on the value of

y

.

Nature of Number $4$ : Understand the nature of the number $4$ . $4$ is a whole number, and therefore it is a rational number.
Subtraction of Rational and Irrational Numbers: Consider the subtraction of a rational number from an irrational number. Subtracting a rational number from an irrational number will result in an irrational number. This is because the difference between a rational and an irrational number cannot be expressed as a ratio of two integers.
Application to Given Problem: Apply this understanding to the given problem. $\newline$ Since $y$ is irrational and $4$ is rational, $y - 4$ will be irrational.
Checking for Exceptions: Check for any exceptions that could make $y - 4$ rational. $\newline$ There are no values of $y$ that are irrational where subtracting $4$ would result in a rational number. Unlike addition where two irrationals could potentially sum to a rational (as in the case of $r + \sqrt{10}$ ), subtraction of a rational from an irrational will always yield an irrational number.

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