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The number $x$ is rational. Which statement about $x + \sqrt{18}$ is true? $\newline$ Choices: $\newline$ (A) $x + \sqrt{18}$ is rational. $\newline$ (B) $x + \sqrt{18}$ is irrational. $\newline$ (C) $x + \sqrt{18}$ can be rational or irrational, depending on the value of $x$ .

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Properties of operations on rational and irrational numbers

Full solution

Q. The number

x

is rational. Which statement about

x + \sqrt{18}

is true?

\newline

Choices:

\newline

(A)

x + \sqrt{18}

is rational.

\newline

(B)

x + \sqrt{18}

is irrational.

\newline

(C)

x + \sqrt{18}

can be rational or irrational, depending on the value of

x

.

Identify Type of Number: Identify whether $\sqrt{18}$ is a rational or irrational number. $18$ is a non-perfect square, which means that its square root cannot be expressed as a ratio of two integers. $\sqrt{18}$ is an irrational number.
Nature of Sum: Consider the nature of the sum of a rational number $x$ and an irrational number $\sqrt{18}$ . The sum of a rational number and an irrational number is always irrational. This is because you cannot express the sum as a ratio of two integers when one of the addends is not expressible as such a ratio.
Statement about $x + \sqrt{18}$ : Determine which statement about $x + \sqrt{18}$ is true. $\newline$ Since $x$ is rational and $\sqrt{18}$ is irrational, their sum $x + \sqrt{18}$ must be irrational. $\newline$ This is true regardless of the value of $x$ , as long as $x$ is rational.

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