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

Full solution

Q. The number

c

is irrational. Which statement about

6 - c

is true?

\newline

Choices:

\newline

(A)

6 - c

is rational.

\newline

(B)

6 - c

is irrational.

\newline

(C)

6 - c

can be rational or irrational, depending on the value of

c

Identify Number Nature: Identify the nature of the number $6$ . $6$ is a rational number because it can be expressed as a fraction of two integers ( $\frac{6}{1}$ ).
Consider Number $c$ : Consider the nature of the number $c$ . $c$ is given as an irrational number, which means it cannot be expressed as a fraction of two integers.
Analyze Operation: Analyze the operation $6 - c$ . Subtracting an irrational number $c$ from a rational number $6$ could result in either a rational or an irrational number. However, if $c$ is a specific irrational number such that when subtracted from $6$ results in a rational number, then $6 - c$ can be rational. For example, if $c$ is an irrational number such that $c = 6 - \text{a rational number}$ , then $6 - c$ would be rational. Conversely, if $c$ is not of this form, then $6 - c$ will be irrational.
Determine Possible Outcomes: Determine the possible outcomes for $6 - c$ . If $c$ is not specifically chosen to make $6 - c$ rational, then in general, subtracting an irrational number from a rational number will result in an irrational number. However, since there exists at least one case where $c$ can be chosen such that $6 - c$ is rational, the correct statement must allow for both possibilities.