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The number $u$ is irrational. Which statement about $u - 10$ is true? $\newline$ Choices: $\newline$ (A) $u - 10$ is rational. $\newline$ (B) $u - 10$ is irrational. $\newline$ (C) $u - 10$ can be rational or irrational, depending on the value of $u$ .

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Properties of operations on rational and irrational numbers

Full solution

Q. The number

u

is irrational. Which statement about

u - 10

is true?

\newline

Choices:

\newline

(A)

u - 10

is rational.

\newline

(B)

u - 10

is irrational.

\newline

(C)

u - 10

can be rational or irrational, depending on the value of

u

.

Identify Number Nature: Identify the nature of the number $10$ . $10$ is an integer, and all integers are rational numbers.
Consider Irrational Properties: Consider the properties of irrational numbers. Subtracting a rational number from an irrational number results in an irrational number. This is because if the result were rational, adding the rational number back would yield the original irrational number, which contradicts the definition of irrational numbers.
Apply Property to Problem: Apply the property to the given problem. $\newline$ Since $u$ is irrational and $10$ is rational, $u - 10$ must be irrational.
Determine True Statement: Determine which statement is true based on the previous steps. $\newline$ The correct statement is that $u - 10$ is irrational.

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