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The number $z$ is irrational. Which statement about $2 + z$ is true? $\newline$ Choices: $\newline$ (A) $2 + z$ is rational. $\newline$ (B) $2 + z$ is irrational. $\newline$ (C) $2 + z$ can be rational or irrational, depending on the value of $z$ .

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Properties of operations on rational and irrational numbers

Full solution

Q. The number

z

is irrational. Which statement about

2 + z

is true?

\newline

Choices:

\newline

(A)

2 + z

is rational.

\newline

(B)

2 + z

is irrational.

\newline

(C)

2 + z

can be rational or irrational, depending on the value of

z

.

Identify Number $2$ : Identify the nature of the number $2$ . $2$ is a rational number because it can be expressed as a fraction of two integers $(2/1)$ .
Consider Number $z$ : Consider the nature of the number $z$ . $z$ is given as an irrational number, which means it cannot be expressed as a fraction of two integers.
Analyze Sum: Analyze the sum of a rational number and an irrational number. $\newline$ The sum of a rational number $2$ and an irrational number $z$ will always be irrational. This is because if it were rational, then subtracting the rational part $2$ from the sum would leave us with a rational number, which contradicts the fact that $z$ is irrational.
Conclude Result: Conclude the nature of $2 + z$ . Since adding a rational number to an irrational number results in an irrational number, the statement that $2 + z$ is irrational is true.

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