To solve the equation
\[ 4 \frac{3}{10} - \left(2 \frac{2}{5}x + 5 \frac{1}{2}\right) = \frac{1}{2} \left(-3 \frac{3}{5}x + 1 \frac{1}{5}\right), \]
let's first convert all mixed numbers to improper fractions.
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Convert \(4 \frac{3}{10}\): \[ 4 \frac{3}{10} = \frac{40}{10} + \frac{3}{10} = \frac{43}{10}. \]
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Convert \(2 \frac{2}{5}\): \[ 2 \frac{2}{5} = \frac{10}{5} + \frac{2}{5} = \frac{12}{5}. \]
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Convert \(5 \frac{1}{2}\): \[ 5 \frac{1}{2} = \frac{10}{2} + \frac{1}{2} = \frac{11}{2}. \]
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Convert \(-3 \frac{3}{5}\): \[ -3 \frac{3}{5} = -\left(\frac{15}{5} + \frac{3}{5}\right) = -\frac{18}{5}. \]
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Convert \(1 \frac{1}{5}\): \[ 1 \frac{1}{5} = \frac{5}{5} + \frac{1}{5} = \frac{6}{5}. \]
Now replace the values in the equation:
\[ \frac{43}{10} - \left(\frac{12}{5}x + \frac{11}{2}\right) = \frac{1}{2} \left(-\frac{18}{5}x + \frac{6}{5}\right). \]
Start by simplifying the left side:
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To combine fractions, find a common denominator for the terms: The common denominator for \(10\), \(5\), and \(2\) is \(10\).
Rewrite \(\frac{12}{5}x\) and \(\frac{11}{2}\): \[ \frac{12}{5}x = \frac{24}{10}x, \quad \frac{11}{2} = \frac{55}{10}. \]
The left side becomes: \[ \frac{43}{10} - \left(\frac{24}{10}x + \frac{55}{10}\right) = \frac{43}{10} - \frac{24}{10}x - \frac{55}{10}. \]
Combine the constants: \[ \frac{43 - 55}{10} = \frac{-12}{10} = -\frac{6}{5}. \]
Thus, the left side simplifies to: \[ -\frac{6}{5} - \frac{24}{10}x = -\frac{6}{5} - \frac{12}{5}x. \]
Now simplify the right side:
\[ \frac{1}{2} \left(-\frac{18}{5}x + \frac{6}{5}\right) = -\frac{9}{5}x + \frac{3}{5}. \]
Now the equation is:
\[ -\frac{6}{5} - \frac{12}{5}x = -\frac{9}{5}x + \frac{3}{5}. \]
To eliminate the fractions, multiply through by \(5\):
\[ -6 - 12x = -9x + 3. \]
Now isolate \(x\):
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Add \(9x\) to both sides: \[ -6 - 12x + 9x = 3 \implies -6 - 3x = 3. \]
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Add \(6\) to both sides: \[ -3x = 9. \]
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Divide by \(-3\): \[ x = -3. \]
Thus, the value of \(x\) is \(\boxed{-3}\).
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