(x+2)/(x2+1)>1/2
⇒ \(\frac{x+2}{x^2+1}\) - \(\frac{1}{2}\) > 0
⇒ \(\frac{2(x+2)-(x^2+1)}{2(x^2+1)}\) > 0
⇒ \(\frac{-x^2+2x+3}{2(x^2+1)}\) > 0
Here,
Denominator i.e., x2 + 1 is always positive and not equal to zero.
So, neglect it.
⇒ - x2 + 2x + 3 > 0
⇒ x2 – 2x – 3 < 0
⇒ (x – 3)(x + 1) < 0
Case I : (x – 3) < 0 and (x + 1) > 0
⇒ x < 3 and x > -1
By takin intersection x ∈ (-1, 3)
Case II : (x – 3) > 0 and (x + 1) < 0
⇒ x > 3 and x < -1
By taking intersection x ∈ ∅.
So, case II is irrelevant.
So, the complete solution is x ∈ (-1, 3)
The integral solution is 0, 1 and 2.
So, number of integral solution is 3.