The number $c$ is irrational. Which statement about $5 - c$ is true? $\newline$ Choices: $\newline$ (A) $5 - c$ is rational. $\newline$ (B) $5 - c$ is irrational. $\newline$ (C) $5 - c$ can be rational or irrational, depending on the value of $c$ .

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Math Problems

Algebra 1

Properties of operations on rational and irrational numbers

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Q. The number

c

is irrational. Which statement about

5 - c

is true?

\newline

Choices:

\newline

(A)

5 - c

is rational.

\newline

(B)

5 - c

is irrational.

\newline

(C)

5 - c

can be rational or irrational, depending on the value of

c

Identify Number $5$ : Identify the nature of the number $5$ . $5$ is a rational number because it can be expressed as a fraction ( $\frac{5}{1}$ ).
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 between a rational and an irrational number. $\newline$ Subtracting an irrational number from a rational number typically results in an irrational number. This is because the difference between a rational and an irrational number cannot be expressed as a fraction of two integers.
Exceptions to Typical Case: Consider exceptions to the typical case. $\newline$ If $c$ were a specific irrational number such that when subtracted from $5$ , the result is a rational number, then $5 - c$ could be rational. However, since $c$ is not specified and can be any irrational number, there is no guarantee that $5 - c$ will be rational.
Values of $c$ : Determine if there are any values of $c$ that would make $5 - c$ rational. $\newline$ If $c$ were $5$ , then $5 - c$ would be $0$ , which is rational. However, since $c$ is irrational, $c$ cannot be $5$ . Therefore, there are no values of $c$ that would make $5 - c$ rational.
Conclude Statement: Conclude the statement about $5 - c$ . Since there are no specific values of $c$ given that would make $5 - c$ rational, and the typical operation between a rational and an irrational number results in an irrational number, the statement that $5 - c$ is irrational is true.