The number \( 0.\overline{9} \) (which means 0.99999...) is a rational number. Using algebra (let x = \( 0.\overline{9} \) , multiply by 10, and subtract), explain why \( 0.\overline{9} \) is exactly equal to 1.

Let \(x = 0.\overline{9} = 0.999... \) ---(1)
Multiplying equation (1) by 10, we get
\(10x = 10(0.999...)\)
⇒ \(10x = 9.999...)\) ---(2)
Subtracting equation (1) from (2), we get
\(10x − x = 9.999... − 0.999...\)
⇒ \(9x = 9\)
⇒ \(\frac{9x}{9} = \frac{9}{9}\) [Dividing both sides by 9]
⇒ \(x = 1\)
So, \(x = 1\)
Hence we prove that \( 0.\overline{9} = 1\).