Question

Write the solution set of the inequation 

\[x + \frac{1}{x} \geq 2\] 

Answer 1

\[\text{ We have }, \]
\[x + \frac{1}{x} \geq 2\]
\[ \Rightarrow \frac{x^2 + 1}{x} \geq 2\]
\[ \Rightarrow \frac{x^2 + 1}{x} - 2 \geq 0\]
\[ \Rightarrow \frac{x^2 - 2x + 1}{x} \geq 0\]
\[ \Rightarrow \frac{(x - 1 )^2}{x} \geq 0\]
\[ \Rightarrow \text{ Either } (x - 1 )^2 \geq 0 \text{ and } x > 0 or (x - 1 )^2 < 0 \text{ and } x < 0 . \]
\[\text{ But }, (x - 1 )^2 \text{ is always greater than zero } . \]
\[ \therefore (x - 1 )^2 \geq 0 \text{ and } x > 0\]
\[ \Rightarrow x > 0\]
\[ \Rightarrow x \in \left( 0, \infty \right)\]