Number of integers, which satisfy the inequality `((16)^(1/ x))/((2^(x+3))) gt1,` is equal to

Options

A. 0
B. 3
C. 4
D. infinite

Answer

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

### Step 1: Rewrite the expression
First, we can rewrite \(16\) 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:
\[ 2^{\frac{4}{x} - (x+3)} > 1 \]
This implies:
\[ \frac{4}{x} - (x + 3) > 0 \]

### Step 3: Simplify the inequality
Rearranging the inequality gives:
\[ \frac{4}{x} - x - 3 > 0 \]
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
Now we factor the quadratic:
\[ (x + 4)(x - 1) < 0 \]

### Step 5: Find the critical points
The critical points are \(x = -4\) and \(x = 1\). We will test the intervals determined by these points: \((-∞, -4)\), \((-4, 1)\), and \((1, ∞)\).

### Step 6: Test the intervals
1. For \(x < -4\) (e.g., \(x = -5\)):
\[ (-5 + 4)(-5 - 1) = (-1)(-6) > 0 \quad \text{(not valid)} \]

2. For \(-4 < x < 1\) (e.g., \(x = 0\)):
\[ (0 + 4)(0 - 1) = (4)(-1) < 0 \quad \text{(valid)} \]

3. For \(x > 1\) (e.g., \(x = 2\)):
\[ (2 + 4)(2 - 1) = (6)(1) > 0 \quad \text{(not valid)} \]

### Step 7: Determine the solution set
The valid interval is \(-4 < x < 1\).

### Step 8: Count the integers
The integers in the interval \((-4, 1)\) are \(-3, -2, -1, 0\). Thus, there are **4 integers** that satisfy the inequality.

### Final Answer:
The number of integers that satisfy the inequality is **4**.

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