`(x+3)/(x ^(2)-x-2) ge (1)/(x-4)` holds for all x satisfying :

Options

A. ` -2 lt x lt 1 or x gt 4`
B. ` -1 lt x lt 2 or x gt 4`
C. `x lt -1 or 2 lt x lt 4`
D. `x gt -1 or 2 lt x lt 4`

Answer

To solve the inequality \(\frac{x+3}{x^2 - x - 2} \geq \frac{1}{x-4}\), we will follow these steps:

### Step 1: Rewrite the inequality
We start by rewriting the inequality:
\[ \frac{x+3}{x^2 - x - 2} - \frac{1}{x-4} \geq 0 \]
This can be rearranged as:
\[ \frac{x+3}{x^2 - x - 2} - \frac{1}{x-4} \geq 0 \]

### Step 2: Find a common denominator
The common denominator for the fractions is \((x^2 - x - 2)(x - 4)\). We can express the inequality as:
\[ \frac{(x+3)(x-4) - (x^2 - x - 2)}{(x^2 - x - 2)(x - 4)} \geq 0 \]

### Step 3: Simplify the numerator
Now, we simplify the numerator:
\[ (x+3)(x-4) = x^2 - 4x + 3x - 12 = x^2 - x - 12 \]
Now substituting back into the inequality:
\[ \frac{x^2 - x - 12 - (x^2 - x - 2)}{(x^2 - x - 2)(x - 4)} \geq 0 \]
This simplifies to:
\[ \frac{-10}{(x^2 - x - 2)(x - 4)} \geq 0 \]

### Step 4: Factor the denominator
Next, we factor the quadratic in the denominator:
\[ x^2 - x - 2 = (x - 2)(x + 1) \]
So the inequality becomes:
\[ \frac{-10}{(x - 2)(x + 1)(x - 4)} \geq 0 \]

### Step 5: Analyze the sign of the expression
Now we need to find the critical points where the expression is undefined or equal to zero:
- The expression is undefined at \(x = 2\), \(x = -1\), and \(x = 4\).
- The expression is negative because of the \(-10\) in the numerator.

### Step 6: Test intervals
We will test the intervals determined by the critical points:
1. \( (-\infty, -1) \)
2. \( (-1, 2) \)
3. \( (2, 4) \)
4. \( (4, \infty) \)

- For \(x < -1\): Choose \(x = -2\)
\[ \frac{-10}{(-2 - 2)(-2 + 1)(-2 - 4)} = \frac{-10}{(-4)(-1)(-6)} < 0 \]
- For \(-1 < x < 2\): Choose \(x = 0\)
\[ \frac{-10}{(0 - 2)(0 + 1)(0 - 4)} = \frac{-10}{(-2)(1)(-4)} > 0 \]
- For \(2 < x < 4\): Choose \(x = 3\)
\[ \frac{-10}{(3 - 2)(3 + 1)(3 - 4)} = \frac{-10}{(1)(4)(-1)} > 0 \]
- For \(x > 4\): Choose \(x = 5\)
\[ \frac{-10}{(5 - 2)(5 + 1)(5 - 4)} = \frac{-10}{(3)(6)(1)} < 0 \]

### Step 7: Conclusion
The inequality \(\frac{-10}{(x - 2)(x + 1)(x - 4)} \geq 0\) holds in the intervals:
- \((-1, 2)\)
- \((2, 4)\)

Thus, the solution set is:
\[ x \in (-\infty, -1) \cup (2, 4) \]

### Final Answer
The inequality holds for all \(x\) satisfying:
\[ x < -1 \quad \text{or} \quad 2 < x < 4 \]