To solve the equation \( \frac{1}{2} + \left( \frac{5}{6} + \frac{11}{12} + \frac{19}{20} + \frac{29}{30} + \frac{41}{42} + \frac{55}{56} + \frac{71}{72} + \frac{89}{90} \right) x = \frac{81}{10} \), we need to follow these steps:

1. Simplify the sum inside the parentheses.
2. Combine the simplified sum with \(\frac{1}{2}\).
3. Solve for \(x\).

### Step 1: Simplify the sum inside the parentheses

We need to find a common denominator to add these fractions:

\[
\frac{5}{6}, \frac{11}{12}, \frac{19}{20}, \frac{29}{30}, \frac{41}{42}, \frac{55}{56}, \frac{71}{72}, \frac{89}{90}
\]

The least common multiple (LCM) of the denominators (6, 12, 20, 30, 42, 56, 72, 90) is 2520. However, for simplicity, let's add them step by step:

\[
\frac{5}{6} + \frac{11}{12} = \frac{10}{12} + \frac{11}{12} = \frac{21}{12} = \frac{7}{4}
\]

\[
\frac{7}{4} + \frac{19}{20} = \frac{35}{20} + \frac{19}{20} = \frac{54}{20} = \frac{27}{10}
\]

\[
\frac{27}{10} + \frac{29}{30} = \frac{81}{30} + \frac{29}{30} = \frac{110}{30} = \frac{11}{3}
\]

\[
\frac{11}{3} + \frac{41}{42} = \frac{154}{42} + \frac{41}{42} = \frac{195}{42} = \frac{65}{14}
\]

\[
\frac{65}{14} + \frac{55}{56} = \frac{260}{56} + \frac{55}{56} = \frac{315}{56} = \frac{45}{8}
\]

\[
\frac{45}{8} + \frac{71}{72} = \frac{405}{72} + \frac{71}{72} = \frac{476}{72} = \frac{119}{18}
\]

\[
\frac{119}{18} + \frac{89}{90} = \frac{595}{90} + \frac{89}{90} = \frac{684}{90} = \frac{38}{5}
\]

So, the sum inside the parentheses is \(\frac{38}{5}\).

### Step 2: Combine the simplified sum with \(\frac{1}{2}\)

The equation now becomes:

\[
\frac{1}{2} + \left( \frac{38}{5} \right) x = \frac{81}{10}
\]

### Step 3: Solve for \(x\)

First, isolate the term with \(x\):

\[
\left( \frac{38}{5} \right) x = \frac{81}{10} - \frac{1}{2}
\]

Convert \(\frac{1}{2}\) to a fraction with a common denominator of 10:

\[
\frac{1}{2} = \frac{5}{10}
\]

So,

\[
\frac{81}{10} - \frac{5}{10} = \frac{76}{10} = \frac{38}{5}
\]

Now, we have:

\[
\left( \frac{38}{5} \right) x = \frac{38}{5}
\]

Divide both sides by \(\frac{38}{5}\):

\[
x = 1
\]

So, the solution is:

\[
x = 1
\]