The number of integral solutions of `(x+2)/(x^2+1)>1/2` is

Options

A. 0
B. 1
C. 2
D. 3

Answer

To solve the inequality \(\frac{x+2}{x^2+1} > \frac{1}{2}\), we can follow these steps:

### Step 1: Rearranging the Inequality
We start by subtracting \(\frac{1}{2}\) from both sides:
\[ \frac{x+2}{x^2+1} - \frac{1}{2} > 0 \]
To combine the fractions, we find a common denominator:
\[ \frac{x+2}{x^2+1} - \frac{1}{2} = \frac{2(x+2) - (x^2+1)}{2(x^2+1)} > 0 \]
This simplifies to:
\[ \frac{2x + 4 - x^2 - 1}{2(x^2 + 1)} > 0 \]
which further simplifies to:
\[ \frac{-x^2 + 2x + 3}{2(x^2 + 1)} > 0 \]

### Step 2: Simplifying the Numerator
Now, we focus on the numerator:
\[ -x^2 + 2x + 3 > 0 \]
Multiplying through by -1 (and reversing the inequality sign):
\[ x^2 - 2x - 3 < 0 \]

### Step 3: Factoring the Quadratic
Next, we factor the quadratic:
\[ x^2 - 2x - 3 = (x - 3)(x + 1) \]
Thus, we have:
\[ (x - 3)(x + 1) < 0 \]

### Step 4: Finding the Critical Points
The critical points are \(x = 3\) and \(x = -1\). We will test the intervals determined by these points:
1. \( (-\infty, -1) \)
2. \( (-1, 3) \)
3. \( (3, \infty) \)

### Step 5: Testing the Intervals
- For \(x < -1\) (e.g., \(x = -2\)):
\((x - 3)(x + 1) = (-2 - 3)(-2 + 1) = (-5)(-1) > 0\) (not a solution)

- For \(-1 < x < 3\) (e.g., \(x = 0\)):
\((x - 3)(x + 1) = (0 - 3)(0 + 1) = (-3)(1) < 0\) (solution)

- For \(x > 3\) (e.g., \(x = 4\)):
\((x - 3)(x + 1) = (4 - 3)(4 + 1) = (1)(5) > 0\) (not a solution)

### Step 6: Conclusion on the Solution Set
The solution to the inequality \((x - 3)(x + 1) < 0\) is:
\[ -1 < x < 3 \]
This means the integral solutions are \(0, 1, 2\).

### Step 7: Counting the Integral Solutions
The integral solutions in the interval \((-1, 3)\) are:
- \(0\)
- \(1\)
- \(2\)

Thus, the number of integral solutions is **3**.