The number of integral solution of the equation `(sgn((x-1)(x-5)))^(x)=1` lying in the interval `[-10, 10]` is not equal to :

Options

A. `11`
B. `16`
C. `17`
D. `18`

Explanation

To solve the equation \((\text{sgn}((x-1)(x-5)))^x = 1\) and find the number of integral solutions lying in the interval \([-10, 10]\), we can break down the problem into several cases based on the properties of the signum function.

### Step 1: Understand the Signum Function
The signum function, \(\text{sgn}(y)\), is defined as:
- \(\text{sgn}(y) = 1\) if \(y > 0\)
- \(\text{sgn}(y) = 0\) if \(y = 0\)
- \(\text{sgn}(y) = -1\) if \(y < 0\)

Thus, we need to analyze the expression \((x-1)(x-5)\).

### Step 2: Determine the Intervals
The expression \((x-1)(x-5)\) changes sign at \(x = 1\) and \(x = 5\). We can evaluate the sign in the following intervals:
- For \(x < 1\): \((x-1)(x-5) > 0\) (positive)
- For \(1 < x < 5\): \((x-1)(x-5) < 0\) (negative)
- For \(x > 5\): \((x-1)(x-5) > 0\) (positive)

### Step 3: Case Analysis
We have three cases to consider based on the value of \((\text{sgn}((x-1)(x-5)))^x\):

#### Case 1: \(\text{sgn}((x-1)(x-5)) = 1\)
This occurs when \((x-1)(x-5) > 0\). Thus, \(x < 1\) or \(x > 5\).
- In the interval \([-10, 10]\), the integers satisfying this condition are:
\[-10, -9, -8, -7, -6, -5, -4, -3, -2, -1, 0, 6, 7, 8, 9, 10\]
- This gives us a total of 16 integral solutions.

#### Case 2: \(\text{sgn}((x-1)(x-5)) = 0\)
This occurs when \((x-1)(x-5) = 0\). Thus, \(x = 1\) or \(x = 5\).
- Both values are included in the interval, giving us 2 additional solutions.

#### Case 3: \(\text{sgn}((x-1)(x-5)) = -1\)
This occurs when \((x-1)(x-5) < 0\). Thus, \(1 < x < 5\).
- The integers in this interval are:
\[2, 3, 4\]
- This gives us 3 integral solutions.

### Step 4: Combine the Solutions
Now we combine the solutions from all cases:
- From Case 1: 16 solutions
- From Case 2: 2 solutions
- From Case 3: 3 solutions

Total integral solutions = \(16 + 2 + 3 = 21\).

### Final Step: Identify the Answer
The question asks for the number of integral solutions that is **not equal to** a certain value. Since we have found 21 integral solutions, we can conclude that the number of integral solutions of the equation in the interval \([-10, 10]\) is not equal to 18.

### Summary
The number of integral solutions of the equation \((\text{sgn}((x-1)(x-5)))^x = 1\) lying in the interval \([-10, 10]\) is **not equal to 18**.