Solve for `x:log_(2)(sin""(x)/(2))lt-1`

Answer

To solve the inequality \( \log_{2}\left(\frac{\sin(x)}{2}\right) < -1 \), we can follow these steps:

### Step 1: Rewrite the logarithmic inequality
Using the property of logarithms, we can rewrite the inequality:
\[ \log_{2}\left(\frac{\sin(x)}{2}\right) < -1 \implies \frac{\sin(x)}{2} < 2^{-1} \]
This simplifies to:
\[ \frac{\sin(x)}{2} < \frac{1}{2} \]

### Step 2: Simplify the inequality
Multiplying both sides of the inequality by 2 (since 2 is positive, the direction of the inequality remains unchanged):
\[ \sin(x) < 1 \]
Since the sine function is always less than or equal to 1, we need to consider the condition from the logarithm that requires \( \frac{\sin(x)}{2} > 0 \):
\[ \sin(x) > 0 \]

### Step 3: Determine the intervals for \( x \)
The sine function is positive in the intervals:
\[ (0, \pi) \quad \text{and} \quad (2n\pi, 2n\pi + \pi) \quad \text{for any integer } n. \]
Now we need to find where \( \sin(x) < 1 \) in these intervals.

### Step 4: Find critical points
The sine function equals \( \frac{1}{2} \) at:
\[ x = \frac{\pi}{6} \quad \text{and} \quad x = \frac{5\pi}{6} \quad \text{in the interval } (0, \pi). \]
Thus, we need to check the intervals:
1. \( (0, \frac{\pi}{6}) \)
2. \( (\frac{\pi}{6}, \frac{5\pi}{6}) \)
3. \( (\frac{5\pi}{6}, \pi) \)

### Step 5: Analyze the intervals
- In \( (0, \frac{\pi}{6}) \), \( \sin(x) < \frac{1}{2} \).
- In \( (\frac{\pi}{6}, \frac{5\pi}{6}) \), \( \sin(x) \) is increasing and reaches \( \frac{1}{2} \) at \( \frac{\pi}{6} \) and decreases back to \( 1 \) at \( \frac{5\pi}{6} \), so \( \sin(x) > \frac{1}{2} \).
- In \( (\frac{5\pi}{6}, \pi) \), \( \sin(x) < \frac{1}{2} \).

### Step 6: Combine the intervals
Thus, the solution in the interval \( (0, \pi) \) is:
\[ x \in (0, \frac{\pi}{6}) \cup (\frac{5\pi}{6}, \pi) \]

### Step 7: General solution
To find the general solution, we add \( 2n\pi \) for any integer \( n \):
\[ x \in (2n\pi, 2n\pi + \frac{\pi}{6}) \cup (2n\pi + \frac{5\pi}{6}, 2n\pi + \pi) \]

### Final Answer
The complete solution for \( x \) is:
\[ x \in (2n\pi, 2n\pi + \frac{\pi}{6}) \cup (2n\pi + \frac{5\pi}{6}, 2n\pi + \pi) \quad \text{for any integer } n. \]