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

Identify Number 4: Identify the nature of the number 4. 4 is a rational number because it can be expressed as a fraction of two integers (4/1).Consider Number w: Consider the nature of the number w. w is given as an irrational number, which means it cannot be expressed as a fraction of two integers.Analyze Sum: Analyze the sum of a rational and an irrational number. The sum of a rational number and an irrational number is always irrational. This is because if it were rational, then subtracting the rational number from both sides would leave an irrational number equal to a rational number, which is a contradiction.Apply Principle: Apply this principle to the given numbers. Since 4 is rational and w is irrational, their sum 4 + w must be irrational.

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

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Math Problems

Algebra 1

Properties of operations on rational and irrational numbers

Full solution

Q. The number

w

is irrational. Which statement about

4 + w

is true?

\newline

Choices:

\newline

(A)

4 + w

is rational.

\newline

(B)

4 + w

is irrational.

\newline

(C)

4 + w

can be rational or irrational, depending on the value of

w

Identify Number $4$ : Identify the nature of the number $4$ . $4$ is a rational number because it can be expressed as a fraction of two integers ( $\frac{4}{1}$ ).
Consider Number $w$ : Consider the nature of the number $w$ . $w$ is given as an irrational number, which means it cannot be expressed as a fraction of two integers.
Analyze Sum: Analyze the sum of a rational and an irrational number. $\newline$ The sum of a rational number and an irrational number is always irrational. This is because if it were rational, then subtracting the rational number from both sides would leave an irrational number equal to a rational number, which is a contradiction.
Apply Principle: Apply this principle to the given numbers. $\newline$ Since $4$ is rational and $w$ is irrational, their sum $4 + w$ must be irrational.