Tim x biết : \(\left( \frac{1}{2} - 1 \right) \cdot \left( \frac{1}{3} - 1 \right) \cdots \left( \frac{1}{9} - 1 \right) \cdot (2x - 9) = \frac{2}{8} + \frac{3}{28} + \frac{4}{77} + \frac{7}{198}\)

To solve the equation:

\[
\left( \frac{1}{2} - 1 \right) \cdot \left( \frac{1}{3} - 1 \right) \cdots \left( \frac{1}{9} - 1 \right) \cdot (2x - 9) = \frac{2}{8} + \frac{3}{28} + \frac{4}{77} + \frac{7}{198}
\]

let’s break it down step by step.

### Step 1: Simplify the left side

The left side can be simplified as follows:

\[
\left( \frac{1}{2} - 1 \right) = -\frac{1}{2}
\]
\[
\left( \frac{1}{3} - 1 \right) = -\frac{2}{3}
\]
\[
\left( \frac{1}{4} - 1 \right) = -\frac{3}{4}
\]
\[
\left( \frac{1}{5} - 1 \right) = -\frac{4}{5}
\]
\[
\left( \frac{1}{6} - 1 \right) = -\frac{5}{6}
\]
\[
\left( \frac{1}{7} - 1 \right) = -\frac{6}{7}
\]
\[
\left( \frac{1}{8} - 1 \right) = -\frac{7}{8}
\]
\[
\left( \frac{1}{9} - 1 \right) = -\frac{8}{9}
\]

Now multiply these fractions together:

\[
-\frac{1}{2} \cdot -\frac{2}{3} \cdot -\frac{3}{4} \cdot -\frac{4}{5} \cdot -\frac{5}{6} \cdot -\frac{6}{7} \cdot -\frac{7}{8} \cdot -\frac{8}{9}
\]
This gives us an overall sign of \((-1)^8 = 1\), and multiplying the numeric parts:

\[
\frac{1 \cdot 2 \cdot 3 \cdot 4 \cdot 5 \cdot 6 \cdot 7 \cdot 8}{2 \cdot 3 \cdot 4 \cdot 5 \cdot 6 \cdot 7 \cdot 8 \cdot 9} = \frac{8!}{2 \cdot 9!} = \frac{1}{9}
\]

Thus, we have:

\[
\frac{1}{9} \cdot (2x - 9)
\]

### Step 2: Simplify the right side

Next, let's simplify the right side:

\[
\frac{2}{8} + \frac{3}{28} + \frac{4}{77} + \frac{7}{198}
\]

Finding a common denominator would be complicated, so we can convert each fraction:

\[
\frac{2}{8} = \frac{1}{4} = \frac{21}{84}
\]
\[
\frac{3}{28} = \frac{9}{84}
\]
\[
\frac{4}{77} = \frac{48}{924} \text{ (common denominator with others)}
\]
\[
\frac{7}{198} = \frac{84}{924}
\]
Combining all terms gives:

\[
\frac{21 + 9 + 48 + 84}{924} = \frac{162}{924} = \frac{27}{154}
\]

### Step 3: Set the two sides equal to each other

Now we equate both sides:

\[
\frac{1}{9}(2x - 9) = \frac{27}{154}
\]

### Step 4: Solve for \(x\)

Multiply both sides by 9:

\[
2x - 9 = \frac{243}{154}
\]

Now, add 9 (or \(\frac{1386}{154}\)) to both sides:

\[
2x = \frac{243 + 1386}{154}
\]
\[
2x = \frac{1629}{154}
\]

Dividing both sides by 2:

\[
x = \frac{1629}{308} = \frac{543}{102.67} \approx 5.28
\]

Thus, the value of \(x\) is approximately 5.28.