Which of the following is not the solution of `(16^(1//x))/(2^(x+3))gt 1` ?

Options

A. `(-oo,-4)`
B. (0,1)
C. `(0, oo)`
D. none of these

Explanation

To solve the inequality \( \frac{16^{\frac{1}{x}}}{2^{x+3}} > 1 \), we will follow these steps:

### Step 1: Rewrite the inequality
We start by rewriting \( 16 \) as \( 2^4 \):
\[ \frac{(2^4)^{\frac{1}{x}}}{2^{x+3}} > 1 \]
This simplifies to:
\[ \frac{2^{\frac{4}{x}}}{2^{x+3}} > 1 \]

### Step 2: Combine the exponents
Using the properties of exponents, we can combine the fractions:
\[ 2^{\frac{4}{x} - (x + 3)} > 1 \]
Since \( 1 = 2^0 \), we can rewrite the inequality as:
\[ \frac{4}{x} - (x + 3) > 0 \]

### Step 3: Simplify the inequality
Now, we simplify the left-hand side:
\[ \frac{4}{x} - x - 3 > 0 \]
Multiplying through by \( x \) (noting that we need to consider the sign of \( x \)):
\[ 4 - x^2 - 3x > 0 \]
This simplifies to:
\[ -x^2 - 3x + 4 > 0 \]
or
\[ x^2 + 3x - 4 < 0 \]

### Step 4: Factor the quadratic
Next, we factor the quadratic:
\[ (x + 4)(x - 1) < 0 \]

### Step 5: Find critical points
The critical points from the factors are:
\[ x = -4 \quad \text{and} \quad x = 1 \]

### Step 6: Test intervals
We will test intervals defined by the critical points:
1. \( (-\infty, -4) \)
2. \( (-4, 1) \)
3. \( (1, \infty) \)

- **Interval \( (-\infty, -4) \)**: Choose \( x = -5 \)
\[ (-5 + 4)(-5 - 1) = (-1)(-6) > 0 \quad \text{(not a solution)} \]

- **Interval \( (-4, 1) \)**: Choose \( x = 0 \)
\[ (0 + 4)(0 - 1) = (4)(-1) < 0 \quad \text{(solution)} \]

- **Interval \( (1, \infty) \)**: Choose \( x = 2 \)
\[ (2 + 4)(2 - 1) = (6)(1) > 0 \quad \text{(not a solution)} \]

### Step 7: Conclusion
The solution to the inequality is:
\[ x \in (-4, 1) \]
Thus, the values that are not solutions include \( (-\infty, -4) \) and \( (1, \infty) \).

### Final Answer
Among the options given, the one that is **not** a solution is:
\[ \text{Option C: } (0, \infty) \]