Please help with rational inequalities?

Question

Hi all. I have a problem which I fear may be easy to solve but I’ve spent some time reseaching on it and still can’t do it. The question is:

Solve 1/x^2 + 1 >= 2/x^2 + 3

If anyone can show me how to do it, I’d really, really appreciate it.

Answer 1

1/x² + 1 ≥ 2/x² + 3

=> 1 - 3 ≥ 2/x² - 1/x²

=> - 2 ≥ 1/x²

This is impossibe for real values of x as 1/x² > 0 and hence -2 cannot be ≥ 1/x².

Answer 2

1/x^2 + 1 = 2/x^2 + 3

take x^2 asLCD

1+x^2/x^2=2+3x^2/x^2

i.e 1 + x^2= 2+3x^2

or 2+3x^2 = 1+ x^2

3x^2- x^2 = 1-2

2x^2 = -1

x^2 =-1/2

x= Square root of -1/2

x= i/square root 2