The number w is rational. Which statement about w - 6 is true? Choices: (A)w - 6 is rational. (B)w - 6 is irrational. (C)w - 6 can be rational or irrational, depending on the value of w.

Definition of Rational Number: We know that w is a rational number. By definition, a rational number is any number that can be expressed as the quotient or fraction p/q of two integers, with the denominator q not equal to zero. Subtracting 6 from w: Since 6 is an integer, and integers are also rational numbers (because they can be expressed as a fraction with 1 as the denominator), subtracting 6 from w will involve an operation between two rational numbers.Difference of Rational Numbers: The difference of two rational numbers is always rational. This is because if you have two rational numbers, a/b and c/d, their difference (a/b) - (c/d) can be expressed as a single fraction, which is (ad - bc) / (bd), and since integers are closed under multiplication and subtraction, ad - bc and bd are still integers, and bd is not zero if b and d are not zero.Conclusion: Therefore, since w is rational and 6 is rational, their difference w - 6 must also be rational.

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The number $w$ is rational. Which statement about $w - 6$ is true? $\newline$ Choices: $\newline$ (A) $w - 6$ is rational. $\newline$ (B) $w - 6$ is irrational. $\newline$ (C) $w - 6$ can be rational or irrational, depending on the value of $w$ .

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Q. The number

w

is rational. Which statement about

w - 6

is true?

\newline

Choices:

\newline

(A)

w - 6

is rational.

\newline

(B)

w - 6

is irrational.

\newline

(C)

w - 6

can be rational or irrational, depending on the value of

w

Definition of Rational Number: We know that $w$ is a rational number. By definition, a rational number is any number that can be expressed as the quotient or fraction $\frac{p}{q}$ of two integers, with the denominator $q$ not equal to zero.
Subtracting $6$ from $w$ : Since $6$ is an integer, and integers are also rational numbers (because they can be expressed as a fraction with $1$ as the denominator), subtracting $6$ from $w$ will involve an operation between two rational numbers.
Difference of Rational Numbers: The difference of two rational numbers is always rational. This is because if you have two rational numbers, $\frac{a}{b}$ and $\frac{c}{d}$ , their difference $(\frac{a}{b}) - (\frac{c}{d})$ can be expressed as a single fraction, which is $\frac{ad - bc}{bd}$ , and since integers are closed under multiplication and subtraction, $ad - bc$ and $bd$ are still integers, and $bd$ is not zero if $b$ and $d$ are not zero.
Conclusion: Therefore, since $w$ is rational and $6$ is rational, their difference $w - 6$ must also be rational.