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The number $x$ is rational. Which statement about $7 + x$ is true? $\newline$ Choices: $\newline$ (A) $7 + x$ is rational. $\newline$ (B) $7 + x$ is irrational. $\newline$ (C) $7 + x$ 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

7 + x

is true?

\newline

Choices:

\newline

(A)

7 + x

is rational.

\newline

(B)

7 + x

is irrational.

\newline

(C)

7 + x

can be rational or irrational, depending on the value of

x

.

Identify Number Nature: Identify the nature of the number $7$ . $7$ is an integer, and all integers are rational numbers because they can be expressed as a fraction where the denominator is $1$ ( $\frac{7}{1}$ ).
Sum Property of Rationals: Understand the property of the sum of two rational numbers. The sum of two rational numbers is always a rational number. This is because if you have two fractions $\frac{a}{b}$ and $\frac{c}{d}$ , their sum $\frac{ad + bc}{bd}$ is also a fraction, which means it is a rational number.
Apply Property to Numbers: Apply the property to the given numbers. $\newline$ Since $x$ is given to be a rational number and $7$ is also a rational number, their sum $7 + x$ must also be a rational number according to the property of rational numbers.
Choose Correct Statement: Choose the correct statement from the given choices. $\newline$ Based on the previous steps, the statement that $7 + x$ is rational is true. Therefore, the correct choice is: $\newline$ (A) $7 + x$ is rational.

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