https://file.aiquickdraw.com/mathbot/file/1731069048127hji9ga8k.jpg

In the image, the inequalities presented are:

1. \( 3 \leq a \leq 6 \)
2. \( 2 \leq b \leq 3 \)
3. \( ? \leq \frac{a \cdot b}{a + b} \leq ? \)

We need to find the bounds of the expression \(\frac{a \cdot b}{a + b}\), given the ranges for \(a\) and \(b\).

### Step 1: Exploring the Expression \(\frac{a \cdot b}{a + b}\)
This expression is often used in calculations involving harmonic means. To determine its range, let’s calculate its value for the boundary values of \(a\) and \(b\) based on the given intervals.

### Step 2: Calculating for Boundary Values
Using the boundary values for \(a\) and \(b\):

- **When \(a = 3\) and \(b = 2\):**
  \[
  \frac{a \cdot b}{a + b} = \frac{3 \cdot 2}{3 + 2} = \frac{6}{5} = 1.2
  \]

- **When \(a = 3\) and \(b = 3\):**
  \[
  \frac{a \cdot b}{a + b} = \frac{3 \cdot 3}{3 + 3} = \frac{9}{6} = 1.5
  \]

- **When \(a = 6\) and \(b = 2\):**
  \[
  \frac{a \cdot b}{a + b} = \frac{6 \cdot 2}{6 + 2} = \frac{12}{8} = 1.5
  \]

- **When \(a = 6\) and \(b = 3\):**
  \[
  \frac{a \cdot b}{a + b} = \frac{6 \cdot 3}{6 + 3} = \frac{18}{9} = 2
  \]

### Step 3: Conclusion
From these calculations, the minimum value of \(\frac{a \cdot b}{a + b}\) is \(1.2\), and the maximum value is \(2\).

Thus, the inequality for \(\frac{a \cdot b}{a + b}\) is:
\[
1.2 \leq \frac{a \cdot b}{a + b} \leq 2
\]

### Final Answer
So, the missing values are:
\[
1.2 \leq \frac{a \cdot b}{a + b} \leq 2
\]

Would you like a deeper explanation or have any questions about this solution?

Here are some related questions:
1. How does the harmonic mean differ from the arithmetic mean in terms of bounds?
2. What would the result be if the ranges of \(a\) and \(b\) were different?
3. How can we interpret inequalities geometrically?
4. What are other ways to find bounds for rational expressions?
5. How does this relate to optimization problems in calculus?

**Tip:** When dealing with inequalities, always check boundary values carefully to understand the range of an expression.

Given the inequalities 3 ≤ a ≤ 6 and 2 ≤ b ≤ 3, determine the range for the expression \( \frac{a \cdot b}{a + b} \).

Explore how to find the range of the expression \( \frac{a \cdot b}{a + b} \) given the inequalities 3 ≤ a ≤ 6 and 2 ≤ b ≤ 3. This solution involves evaluating boundary values to determine the minimum and maximum values for the expression. Ideal for students in Grades 10-12.

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