https://r2.erweima.ai/mathbot/image/172444501539009eflqah.jpg

The inequality provided in the image is:

\[
\log_{\frac{1}{2}} x + 2 \log_{\frac{1}{2}} \frac{1}{2} + 3 \geq 0
\]

### Step 1: Simplify the expression

1. **Evaluate \(\log_{\frac{1}{2}} \frac{1}{2}\)**: 
   \[
   \log_{\frac{1}{2}} \frac{1}{2} = 1
   \]
   Since \(\log_b b = 1\) for any base \(b\).

2. **Substitute the value into the inequality**:
   \[
   \log_{\frac{1}{2}} x + 2(1) + 3 \geq 0
   \]
   Simplify further:
   \[
   \log_{\frac{1}{2}} x + 2 + 3 \geq 0
   \]
   \[
   \log_{\frac{1}{2}} x + 5 \geq 0
   \]

### Step 2: Solve the inequality

To isolate \(\log_{\frac{1}{2}} x\), subtract 5 from both sides:
\[
\log_{\frac{1}{2}} x \geq -5
\]

### Step 3: Convert the logarithmic inequality to exponential form

The inequality \(\log_{\frac{1}{2}} x \geq -5\) can be rewritten in exponential form as:
\[
x \leq \left(\frac{1}{2}\right)^{-5}
\]

### Step 4: Simplify the exponent

Since \(\left(\frac{1}{2}\right)^{-5} = 2^5 = 32\), we have:
\[
x \leq 32
\]

### Step 5: Consider the domain of the logarithmic function

The logarithm \(\log_{\frac{1}{2}} x\) is defined only for \(x > 0\). Therefore, the solution to the inequality must also satisfy \(x > 0\).

### Final Solution

The solution to the inequality is:
\[
0 < x \leq 32
\]

Would you like further details or have any questions?

Here are some related questions you might find useful:
1. How do you solve inequalities involving logarithms with different bases?
2. What happens if the logarithmic base is greater than 1 instead of less than 1?
3. How do you handle inequalities with multiple logarithmic terms?
4. What is the importance of the domain in logarithmic inequalities?
5. How can the properties of logarithms be used to simplify more complex expressions?

**Tip:** Always ensure that the argument of a logarithm is positive when solving inequalities involving logarithms.

Learn how to solve the logarithmic inequality log(1/2) x + 2 log(1/2) 1/2 + 3 >= 0 step-by-step. This problem involves logarithmic properties and inequalities. Suitable for advanced high school students (Grades 10-12).

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