Find the number of integal values of x satisfying
`sqrt(-x^2+10x-16) lt x -2`

Explanation

To solve the inequality \( \sqrt{-x^2 + 10x - 16} < x - 2 \), we will follow these steps:

### Step 1: Set up the inequality
We start with the given inequality:
\[ \sqrt{-x^2 + 10x - 16} < x - 2 \]

### Step 2: Ensure the expression under the square root is non-negative
For the square root to be defined, we need:
\[ -x^2 + 10x - 16 \geq 0 \]
This can be rewritten as:
\[ x^2 - 10x + 16 \leq 0 \]

### Step 3: Factor the quadratic expression
Next, we factor the quadratic:
\[ x^2 - 10x + 16 = (x - 2)(x - 8) \]
Thus, we need to solve:
\[ (x - 2)(x - 8) \leq 0 \]

### Step 4: Determine the intervals
To find the intervals where this inequality holds, we can analyze the sign of the product:
- The roots are \( x = 2 \) and \( x = 8 \).
- Testing intervals:
- For \( x < 2 \) (e.g., \( x = 0 \)): \( (0 - 2)(0 - 8) > 0 \) (positive)
- For \( 2 < x < 8 \) (e.g., \( x = 5 \)): \( (5 - 2)(5 - 8) < 0 \) (negative)
- For \( x > 8 \) (e.g., \( x = 9 \)): \( (9 - 2)(9 - 8) > 0 \) (positive)

Thus, the solution to the inequality is:
\[ 2 \leq x \leq 8 \]

### Step 5: Solve the second part of the inequality
Now we square both sides of the original inequality (noting that \( x - 2 > 0 \) in the interval \( x > 2 \)):
\[ -x^2 + 10x - 16 < (x - 2)^2 \]
Expanding the right side:
\[ -x^2 + 10x - 16 < x^2 - 4x + 4 \]

### Step 6: Rearranging the inequality
Bringing all terms to one side gives:
\[ -x^2 - x^2 + 10x + 4x - 16 - 4 > 0 \]
This simplifies to:
\[ -2x^2 + 14x - 20 > 0 \]
Dividing by -2 (which reverses the inequality):
\[ x^2 - 7x + 10 < 0 \]

### Step 7: Factor the quadratic again
Factoring gives:
\[ (x - 2)(x - 5) < 0 \]

### Step 8: Determine the intervals for this inequality
The roots are \( x = 2 \) and \( x = 5 \). Testing intervals:
- For \( x < 2 \): positive
- For \( 2 < x < 5 \): negative
- For \( x > 5 \): positive

Thus, the solution is:
\[ 2 < x < 5 \]

### Step 9: Find the intersection of the intervals
The two intervals we have are:
1. \( 2 \leq x \leq 8 \)
2. \( 2 < x < 5 \)

The intersection of these intervals is:
\[ 2 < x < 5 \]

### Step 10: Identify integral values
The integral values of \( x \) in the interval \( (2, 5) \) are:
- \( 3, 4 \)

Thus, there are **2 integral values** of \( x \) that satisfy the original inequality.

### Final Answer
The number of integral values of \( x \) satisfying the inequality is **2**.

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