`log_(1/4)((35-x^2)/x)geq-1/2`

Answer

To solve the inequality \( \log_{1/4}\left(\frac{35 - x^2}{x}\right) \geq -\frac{1}{2} \), we will follow these steps:

### Step 1: Rewrite the logarithmic inequality
Using the property of logarithms, we can rewrite the inequality:
\[ \log_{1/4}\left(\frac{35 - x^2}{x}\right) \geq -\frac{1}{2} \implies \frac{35 - x^2}{x} \geq (1/4)^{-\frac{1}{2}} \]
Calculating \( (1/4)^{-\frac{1}{2}} \):
\[ (1/4)^{-\frac{1}{2}} = 4^{\frac{1}{2}} = 2 \]
Thus, we have:
\[ \frac{35 - x^2}{x} \geq 2 \]

### Step 2: Clear the fraction
To eliminate the fraction, multiply both sides by \( x \) (keeping in mind that \( x \) must be positive):
\[ 35 - x^2 \geq 2x \]
Rearranging gives:
\[ x^2 + 2x - 35 \leq 0 \]

### Step 3: Factor the quadratic
Next, we factor the quadratic expression:
\[ x^2 + 2x - 35 = (x - 5)(x + 7) \leq 0 \]

### Step 4: Determine the critical points
The critical points from the factors are \( x = 5 \) and \( x = -7 \).

### Step 5: Analyze the intervals
We will use a sign chart or the wavy curve method to determine where the product is less than or equal to zero:
- Test intervals: \( (-\infty, -7) \), \( (-7, 5) \), and \( (5, \infty) \).
- Choose test points:
- For \( x = -8 \): \( (-)(-) = + \) (positive)
- For \( x = 0 \): \( (+)(-) = - \) (negative)
- For \( x = 6 \): \( (+)(+) = + \) (positive)

Thus, the solution to \( (x - 5)(x + 7) \leq 0 \) is:
\[ x \in [-7, 5] \]

### Step 6: Check the conditions for the logarithm
We also need to ensure that \( \frac{35 - x^2}{x} > 0 \):
1. \( 35 - x^2 > 0 \) implies \( x^2 < 35 \) or \( -\sqrt{35} < x < \sqrt{35} \).
2. \( x > 0 \) implies \( x \in (0, \sqrt{35}) \).

### Step 7: Combine the intervals
Now we combine the intervals:
- From the quadratic inequality: \( [-7, 5] \)
- From the logarithmic condition: \( (0, \sqrt{35}) \)

The common solution is:
\[ x \in (0, 5] \]

### Final Answer
The solution to the inequality \( \log_{1/4}\left(\frac{35 - x^2}{x}\right) \geq -\frac{1}{2} \) is:
\[ x \in (0, 5] \]