The complete solution set of inequality `((x-5)^(1005)(x+8)^(1008)(x-1))/(x^(1006)(x-2)^3(x-3)^5(x-6)(x+9)^(1010)) le 0 `

Options

A. `(-oo,-9)cup(-8,0) cup (0,1)cup (2,3)cup [ 5,6]`
B. `(-oo,-9)cup(-9,0)cup(0,1) cup (2,3)cup [5,6)`
C. `(-oo,-9)cup (-9,0)cup (0,1] cup (2,3)cup [ 5,6)`
D. `(-oo,0) cup (0,1] cup (2,3)cup [5,6)]`

Explanation

To solve the inequality

\[ \frac{(x-5)^{1005}(x+8)^{1008}(x-1)}{x^{1006}(x-2)^3(x-3)^5(x-6)(x+9)^{1010}} \leq 0, \]

we will follow these steps:

### Step 1: Identify the critical points
The critical points occur when the numerator or denominator is zero. Set each factor to zero:

- From the numerator:
- \(x - 5 = 0 \Rightarrow x = 5\)
- \(x + 8 = 0 \Rightarrow x = -8\)
- \(x - 1 = 0 \Rightarrow x = 1\)

- From the denominator:
- \(x = 0\) (from \(x^{1006}\))
- \(x - 2 = 0 \Rightarrow x = 2\) (from \((x-2)^3\))
- \(x - 3 = 0 \Rightarrow x = 3\) (from \((x-3)^5\))
- \(x - 6 = 0 \Rightarrow x = 6\)
- \(x + 9 = 0 \Rightarrow x = -9\) (from \((x+9)^{1010}\))

Thus, the critical points are:
\[ -9, -8, 0, 1, 2, 3, 5, 6. \]

### Step 2: Create a number line
We will place the critical points on a number line:

```
---|---|---|---|---|---|---|---|---|---|---
-9 -8 0 1 2 3 5 6
```

### Step 3: Test intervals
We will test the sign of the expression in each interval created by these critical points:

1. **Interval \((-∞, -9)\)**: Choose \(x = -10\)
\[ \text{Sign} = \frac{(-)(+)(-)}{(-)(-)(-)(-)(+)} = + \quad \text{(positive)} \]

2. **Interval \((-9, -8)\)**: Choose \(x = -9.5\)
\[ \text{Sign} = \frac{(-)(+)(-)}{(-)(-)(-)(-)(+)} = + \quad \text{(positive)} \]

3. **Interval \((-8, 0)\)**: Choose \(x = -1\)
\[ \text{Sign} = \frac{(-)(+)(-)}{(+)(-)(-)(-)(+)} = - \quad \text{(negative)} \]

4. **Interval \((0, 1)\)**: Choose \(x = 0.5\)
\[ \text{Sign} = \frac{(-)(+)(-)}{(+)(-)(-)(-)(+)} = - \quad \text{(negative)} \]

5. **Interval \((1, 2)\)**: Choose \(x = 1.5\)
\[ \text{Sign} = \frac{(-)(+)(+)}{(+)(-)(-)(-)(+)} = + \quad \text{(positive)} \]

6. **Interval \((2, 3)\)**: Choose \(x = 2.5\)
\[ \text{Sign} = \frac{(-)(+)(+)}{(+)(+)(-)(-)(+)} = - \quad \text{(negative)} \]

7. **Interval \((3, 5)\)**: Choose \(x = 4\)
\[ \text{Sign} = \frac{(-)(+)(+)}{(+)(+)(+)(-)(+)} = + \quad \text{(positive)} \]

8. **Interval \((5, 6)\)**: Choose \(x = 5.5\)
\[ \text{Sign} = \frac{(+)(+)(+)}{(+)(+)(+)(-)(+)} = - \quad \text{(negative)} \]

9. **Interval \((6, ∞)\)**: Choose \(x = 7\)
\[ \text{Sign} = \frac{(+)(+)(+)}{(+)(+)(+)(+)(+)} = + \quad \text{(positive)} \]

### Step 4: Determine where the expression is less than or equal to zero
From our tests, the expression is negative in the intervals:
- \((-8, 0)\)
- \((0, 1)\)
- \((2, 3)\)
- \((5, 6)\)

### Step 5: Include critical points
The critical points where the expression equals zero are \(x = -8\), \(x = 1\), and \(x = 5\). However, \(x = 0\) is not included since it makes the denominator zero.

### Final solution
The complete solution set is:
\[ (-8, 0) \cup [1, 2) \cup [5, 6). \]