If for `x in R, (1)/(3) lt (x^(2) - 2x + 4)/(x^(2) + 2x + 4) lt 3`, then `(9.3^(2x) + 6.3^(x) + 4)/(9.3^(2x) - 6.3^(x) + 4)` lies between

Options

A. `(1)/(2)` and 2
B. `(1)/(3)` and 3
C. 0 and 2
D. None of these

Answer

To solve the inequality \( \frac{1}{3} < \frac{x^2 - 2x + 4}{x^2 + 2x + 4} < 3 \), we will follow these steps:

### Step 1: Analyze the Inequality
We start with the given inequality:
\[ \frac{1}{3} < \frac{x^2 - 2x + 4}{x^2 + 2x + 4} < 3 \]

### Step 2: Break it into Two Parts
We can break this into two separate inequalities:
1. \( \frac{1}{3} < \frac{x^2 - 2x + 4}{x^2 + 2x + 4} \)
2. \( \frac{x^2 - 2x + 4}{x^2 + 2x + 4} < 3 \)

### Step 3: Solve the First Inequality
For the first inequality:
\[ \frac{1}{3} < \frac{x^2 - 2x + 4}{x^2 + 2x + 4} \]
Cross-multiplying gives:
\[ 1 \cdot (x^2 + 2x + 4) < 3(x^2 - 2x + 4) \]
This simplifies to:
\[ x^2 + 2x + 4 < 3x^2 - 6x + 12 \]
Rearranging terms leads to:
\[ 0 < 2x^2 - 8x + 8 \]
Factoring out a 2:
\[ 0 < 2(x^2 - 4x + 4) \]
This simplifies to:
\[ 0 < 2(x - 2)^2 \]
Since \( (x - 2)^2 \) is always non-negative, the inequality holds for all \( x \) except \( x = 2 \).

### Step 4: Solve the Second Inequality
Now, for the second inequality:
\[ \frac{x^2 - 2x + 4}{x^2 + 2x + 4} < 3 \]
Cross-multiplying gives:
\[ x^2 - 2x + 4 < 3(x^2 + 2x + 4) \]
This simplifies to:
\[ x^2 - 2x + 4 < 3x^2 + 6x + 12 \]
Rearranging terms leads to:
\[ 0 < 2x^2 + 8x + 8 \]
Factoring out a 2:
\[ 0 < 2(x^2 + 4x + 4) \]
This simplifies to:
\[ 0 < 2(x + 2)^2 \]
Again, since \( (x + 2)^2 \) is always non-negative, this inequality holds for all \( x \) except \( x = -2 \).

### Step 5: Combine Results
From both inequalities, we find that:
- The first inequality holds for all \( x \) except \( x = 2 \).
- The second inequality holds for all \( x \) except \( x = -2 \).

### Step 6: Find the Range for the Expression
Now we need to evaluate:
\[ \frac{9 \cdot 3^{2x} + 6 \cdot 3^x + 4}{9 \cdot 3^{2x} - 6 \cdot 3^x + 4} \]
Let \( y = 3^x \). Then the expression becomes:
\[ \frac{9y^2 + 6y + 4}{9y^2 - 6y + 4} \]

### Step 7: Analyze the New Expression
We can analyze the new expression:
1. The numerator \( 9y^2 + 6y + 4 \) is always positive.
2. The denominator \( 9y^2 - 6y + 4 \) is also always positive since its discriminant \( (-6)^2 - 4 \cdot 9 \cdot 4 < 0 \).

### Step 8: Find the Range
Now we need to find the range of:
\[ \frac{9y^2 + 6y + 4}{9y^2 - 6y + 4} \]
Using the results from the inequalities, we find that this expression lies between \( \frac{1}{3} \) and \( 3 \).

### Final Conclusion
Thus, the expression \( \frac{9 \cdot 3^{2x} + 6 \cdot 3^x + 4}{9 \cdot 3^{2x} - 6 \cdot 3^x + 4} \) lies between \( \frac{1}{3} \) and \( 3 \).