Write the number of integral solutions of \(\frac{x+2}{x^2+1}\) > \

1.	Write the number of integral solutions of \(\frac{x+2}{x^2+1}\) > \(\frac{1}{2}\)(x+2)/(x2+1)>1/2
Answer» (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.

Write the number of integral solutions of \(\frac{x+2}{x^2+1}\) > \(\frac{1}{2}\)(x+2)/(x2+1)>1/2

Answer»

(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.