Which of the Following is True About \(x = 0.\overline{3}\)?
Options:
- (a) x is a rational number, because \(10x = 3 + x\)
- (b) x is a rational number, because \(10x = 3 – x\)
- (c) x is an irrational number, because \(10x = 3 + x\)
- (d) x is an irrational number, because \(10x = 3 – x\)
Solution:
Let \(x = 0.\overline{3}\)
\[ 10x = 3.\overline{3} \]
\[ 10x – x = 3 \]
\[ 9x = 3 \Rightarrow x = \frac{1}{3} \]
Thus, \(x\) is rational and satisfies: \[ 10x = 3 + x \]
Correct Answer:
(a)