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

Options

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

Answer

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

### Step 1: Determine the domain of the square root
The expression inside the square root must be non-negative:
\[ -x^2 + 10x - 16 \geq 0 \]
Rearranging gives:
\[ x^2 - 10x + 16 \leq 0 \]

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

### Step 3: Find the critical points
The critical points are \( x = 2 \) and \( x = 8 \). We will test intervals around these points to determine where the product is non-positive.

### Step 4: Test intervals
1. For \( x < 2 \) (e.g., \( x = 0 \)):
\[ (0 - 2)(0 - 8) = 16 > 0 \]
2. For \( 2 < x < 8 \) (e.g., \( x = 4 \)):
\[ (4 - 2)(4 - 8) = -8 < 0 \]
3. For \( x > 8 \) (e.g., \( x = 9 \)):
\[ (9 - 2)(9 - 8) = 7 > 0 \]

### Step 5: Conclusion from the test
The inequality \( (x - 2)(x - 8) \leq 0 \) holds for:
\[ x \in [2, 8] \]

### Step 6: Solve the original inequality
Now we need to solve:
\[ \sqrt{-x^2 + 10x - 16} < x - 2 \]
Squaring both sides (valid since both sides are non-negative in the interval):
\[ -x^2 + 10x - 16 < (x - 2)^2 \]
Expanding the right side:
\[ -x^2 + 10x - 16 < x^2 - 4x + 4 \]
Rearranging gives:
\[ -x^2 - x^2 + 10x + 4x - 16 - 4 < 0 \]
This simplifies to:
\[ -2x^2 + 14x - 20 < 0 \]
Dividing by -2 (and flipping the inequality):
\[ x^2 - 7x + 10 > 0 \]

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

### Step 8: Test intervals for the new inequality
1. For \( x < 2 \) (e.g., \( x = 0 \)):
\[ (0 - 2)(0 - 5) = 10 > 0 \]
2. For \( 2 < x < 5 \) (e.g., \( x = 3 \)):
\[ (3 - 2)(3 - 5) = -2 < 0 \]
3. For \( x > 5 \) (e.g., \( x = 6 \)):
\[ (6 - 2)(6 - 5) = 4 > 0 \]

### Step 9: Conclusion from the test
The inequality \( (x - 2)(x - 5) > 0 \) holds for:
\[ x \in (-\infty, 2) \cup (5, \infty) \]

### Step 10: Find the intersection with the domain
The intersection of \( [2, 8] \) and \( (-\infty, 2) \cup (5, \infty) \) gives:
\[ x \in (5, 8] \]

### Step 11: Find integer solutions
The integer values in the interval \( (5, 8] \) are \( 6, 7, 8 \).

### Final Answer
Thus, the number of integer values of \( x \) satisfying the inequality is **3**.

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