The number of integers satisfying the inequality `log_(sqrt(0.9))log_(5)(sqrt(x^(2)+5+x))gt 0` is

Options

A. 6
B. 7
C. 8
D. 9

Explanation

To solve the inequality \( \log_{\sqrt{0.9}} \log_{5}(\sqrt{x^2 + 5 + x}) > 0 \), we will follow these steps:

### Step 1: Analyze the Inner Logarithm
We start with the inner logarithm \( \log_{5}(\sqrt{x^2 + 5 + x}) \). For this logarithm to be greater than 0, the argument must be greater than 1:
\[ \sqrt{x^2 + 5 + x} > 1 \]
Squaring both sides, we get:
\[ x^2 + 5 + x > 1 \]
This simplifies to:
\[ x^2 + x + 4 > 0 \]

### Step 2: Analyze the Quadratic Expression
The quadratic \( x^2 + x + 4 \) is always positive because its discriminant is negative:
\[ D = b^2 - 4ac = 1^2 - 4 \cdot 1 \cdot 4 = 1 - 16 = -15 \]
Since the quadratic has no real roots, it does not change sign and is always positive.

### Step 3: Analyze the Outer Logarithm
Next, we consider the outer logarithm \( \log_{\sqrt{0.9}} \). The base \( \sqrt{0.9} \) is less than 1, which means the logarithm is positive when its argument is less than 1:
\[ \log_{5}(\sqrt{x^2 + 5 + x}) < 1 \]
This implies:
\[ \sqrt{x^2 + 5 + x} < 5 \]
Squaring both sides gives:
\[ x^2 + 5 + x < 25 \]
This simplifies to:
\[ x^2 + x - 20 < 0 \]

### Step 4: Factor the Quadratic
Now, we factor the quadratic \( x^2 + x - 20 \):
\[ (x - 4)(x + 5) < 0 \]

### Step 5: Determine the Intervals
To find the intervals where this inequality holds, we analyze the critical points \( x = -5 \) and \( x = 4 \). Testing intervals:
- For \( x < -5 \): Choose \( x = -6 \) → \( (-6 - 4)(-6 + 5) = (-10)(-1) > 0 \)
- For \( -5 < x < 4 \): Choose \( x = 0 \) → \( (0 - 4)(0 + 5) = (-4)(5) < 0 \)
- For \( x > 4 \): Choose \( x = 5 \) → \( (5 - 4)(5 + 5) = (1)(10) > 0 \)

Thus, the solution to the inequality \( (x - 4)(x + 5) < 0 \) is:
\[ -5 < x < 4 \]

### Step 6: Count the Integer Solutions
The integers that satisfy \( -5 < x < 4 \) are:
\[ -4, -3, -2, -1, 0, 1, 2, 3 \]
Counting these gives us a total of 8 integers.

### Final Answer
The number of integers satisfying the inequality is \( \boxed{8} \).