Let's determine whether adding 0.4 to each of the given numbers produces an irrational number.
### Given numbers:
A. [tex]\( 3\pi \)[/tex]
B. [tex]\( 0.444 \ldots \)[/tex] (a repeating decimal)
C. [tex]\( \frac{2}{7} \)[/tex] (a rational number)
D. [tex]\( \sqrt{9} \)[/tex] (which is a rational number, specifically 3)
Let's analyze each in turn.
1. [tex]\( 3\pi \)[/tex]:
- [tex]\(\pi\)[/tex] (pi) is an irrational number. Multiplying an irrational number by a non-zero rational number (3 in this case) results in another irrational number.
- Adding a rational number (0.4) to an irrational number maintains the irrational nature of the result.
- Thus, [tex]\( 3\pi + 0.4 \)[/tex] is irrational.
2. [tex]\( 0.444 \ldots \)[/tex] (repeating decimal):
- A repeating decimal represents a rational number.
- Adding two rational numbers results in a rational number.
- Therefore, [tex]\( 0.444 \ldots + 0.4 \)[/tex] is rational.
3. [tex]\( \frac{2}{7} \)[/tex]:
- [tex]\( \frac{2}{7} \)[/tex] is already a rational number.
- Adding two rational numbers results in a rational number.
- Hence, [tex]\( \frac{2}{7} + 0.4 \)[/tex] is rational.
4. [tex]\( \sqrt{9} \)[/tex]:
- The square root of 9 is 3, which is a rational number.
- Adding two rational numbers results in a rational number.
- Therefore, [tex]\( 3 + 0.4 \)[/tex] is rational.
### Conclusion:
The only case where adding 0.4 results in an irrational number is with [tex]\( 3\pi \)[/tex]. Thus, the answer is:
A. [tex]\( 3\pi \)[/tex]
