Write the solution set of the inequation x + 1/x ≥ 2. ​

Question

Write the solution set of the inequation \(x\) + \(\frac{1}{x}\) ≥ 2.

x + 1/x ≥ 2.

Answer 1

x + 1/x ≥ 2

⇒  \(x\) + \(\frac{1}{x}\) - 2 ≥ 0

⇒ \(\frac{x^2-2x+1}{x}\) ≥ 0 

Case I : x² – 2x + 1 ≥ 0 and x > 0 

(x – 1)² ≥ 0 and x > 0 

So, by taking intersection x > 0 

Case II : x² – 2x + 1 ≤ 0 and x < 0 

(x – 1)² ≤ 0 and x < 0 

Square term is always positive so case II is irrelevant. 

Then, 

The final answer of question is x ∈ (0, ∞).