Question

Write the solution set of the inequation $x+\frac{1}{x}\ge 2$.

Answer 1

$$\begin{gathered} \mathrm{We}\mathrm{have}, \\ x+\frac{1}{x}\ge 2 \\ \Rightarrow \frac{x^2+1}{x}\ge 2 \\ \Rightarrow \frac{x^2+1}{x}-2\ge 0 \\ \Rightarrow \frac{x^2-2x+1}{x}\ge 0 \\ \Rightarrow \frac{(x-1{)}^2}{x}\ge 0 \\ \Rightarrow \mathrm{Either}(x-1{)}^2\ge 0\mathrm{and}x>0\mathrm{or}(x-1{)}^2<0\mathrm{and}x<0. \\ \mathrm{But},(x-1{)}^2\mathrm{is}\mathrm{always}\mathrm{greater}\mathrm{than}\mathrm{zero}. \\ \therefore (x-1{)}^2\ge 0\mathrm{and}x>0 \\ \Rightarrow x>0 \\ \Rightarrow x\in \left(0,\infty \right) \end{gathered}$$