Question

The inequality \( x + \frac{1}{x} \leq -2 \) is true, if:

• A. \( x \geq -1 \)
• B. \( x>0 \)
• C. \( x<0 \)
• D. \( x = -1 \text{ only} \)

Hint:

For inequalities involving fractions, multiply by the denominator carefully, considering sign changes.

Answer 1

The Correct Option is D

Given inequality: \[ x + \frac{1}{x} \leq -2 \] Multiply by \( x \) (considering sign changes): \[ x^2 + 1 \leq -2x \] Rearrange: \[ x^2 + 2x + 1 \leq 0 \] \[ (x+1)^2 \leq 0 \] Since a square term is always non-negative, the only solution is: \[ x = -1 \]
Thus, the inequality holds only when \( x = -1 \).