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The number $q$ is rational. Which statement about $4 + q$ is true? $\newline$ Choices: $\newline$ (A) $4 + q$ is rational. $\newline$ (B) $4 + q$ is irrational. $\newline$ (C) $4 + q$ can be rational or irrational, depending on the value of $q$ .

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Properties of operations on rational and irrational numbers

Full solution

Q. The number

q

is rational. Which statement about

4 + q

is true?

\newline

Choices:

\newline

(A)

4 + q

is rational.

\newline

(B)

4 + q

is irrational.

\newline

(C)

4 + q

can be rational or irrational, depending on the value of

q

.

Identify Number $4$ : Identify the nature of the number $4$ . $4$ is an integer, and all integers are rational numbers because they can be expressed as a ratio of integers (for example, $4$ can be written as $\frac{4}{1}$ ).
Consider Number $q$ : Consider the nature of the number $q$ . Since $q$ is given as a rational number, it can be expressed as a ratio of two integers, where the denominator is not zero.
Analyze Sum of Rationals: Analyze the sum of two rational numbers. The sum of two rational numbers is always rational. This is because if you have two rational numbers, $\frac{a}{b}$ and $\frac{c}{d}$ , their sum is $\frac{ad + bc}{bd}$ , which is also a ratio of two integers (assuming $b$ and $d$ are not zero).
Apply Property to $4 + q$ : Apply the property to $4 + q$ . $\newline$ Since $4$ (which is rational) and $q$ (which is rational) are being added, their sum $4 + q$ must also be rational according to the property of rational numbers.

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