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

Full solution

Q. The number

x

is irrational. Which statement about

x - \sqrt{10}

is true?

\newline

Choices:

\newline

(A)

x - \sqrt{10}

is rational.

\newline

(B)

x - \sqrt{10}

is irrational.

\newline

(C)

x - \sqrt{10}

can be rational or irrational, depending on the value of

x

Identify Type of Number: Identify whether $\sqrt{10}$ is a rational or irrational number. $10$ is a non-perfect square, which means that its square root cannot be expressed as a ratio of two integers. Therefore, $\sqrt{10}$ is an irrational number.
Properties of Irrational Numbers: Consider the properties of irrational numbers when combined with arithmetic operations. $\newline$ The sum or difference of an irrational number and a rational number is always irrational. $\newline$ However, the sum or difference of two irrational numbers can be either rational or irrational, depending on the specific numbers.
Apply Properties to Expression: Apply the properties to the given expression $x - \sqrt{10}$ . Since $x$ is irrational and $\sqrt{10}$ is irrational, their difference $x - \sqrt{10}$ can be either rational or irrational. For example, if $x = \sqrt{10}$ , then $x - \sqrt{10} = \sqrt{10} - \sqrt{10} = 0$ , which is rational. If $x$ is any irrational number not related to $\sqrt{10}$ , then $x - \sqrt{10}$ will remain irrational.