Prove that x+1/x is less than or equal to -2?

Answer 1

x + 1/x < -2

There are 2 possibilities:

Case 1: x > 0

Then, x + 1/x is clearly positive. So, the inequality x + 1/x < -2 has no solution.

Case 2: x < 0

Then, x² + 1 > -2x

x² + 2x + 1 > 0

(x+1)² > 0

This is true for all x ≠ -1.

However, we assumed that x < 0.

Solution set: (-∞, -1) ∪ (-1, 0)

Answer 2

Without knowing what x is, we can't prove anything.......

Perhaps you meant that we should solve for x?