To solve for \( x \) in the equation
\[ \frac{1}{5}(x + 0.5) + 5.24 = \frac{3}{2} x + \frac{7}{10}(x + 2.2), \]
we will simplify and combine like terms step-by-step.
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Start with the left side: \[ \frac{1}{5}(x + 0.5) + 5.24 = \frac{1}{5}x + \frac{1}{10} + 5.24. \] We convert \( 5.24 \) into a fraction: \[ 5.24 = \frac{524}{100} = \frac{131}{25}. \] So, \[ \frac{1}{5}(x + 0.5) + 5.24 = \frac{1}{5}x + \frac{1}{10} + \frac{131}{25}. \] Finding a common denominator (which is 50): \[ \frac{1}{10} = \frac{5}{50}, \quad \frac{131}{25} = \frac{262}{50}. \] Thus, \[ \frac{1}{5}x + \frac{1}{10} + \frac{131}{25} = \frac{1}{5}x + \frac{5}{50} + \frac{262}{50} = \frac{1}{5}x + \frac{267}{50}. \]
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Now simplify the right side: \[ \frac{7}{10}(x + 2.2) = \frac{7}{10}x + \frac{7 \cdot 2.2}{10} = \frac{7}{10}x + \frac{15.4}{10} = \frac{7}{10}x + \frac{77}{100}. \] We need a common denominator of 100: \[ \frac{3}{2}x = \frac{150}{100}x, \quad \frac{7}{10}x = \frac{70}{100}x. \] Therefore: \[ \frac{3}{2} x + \frac{7}{10}(x + 2.2) = \frac{150}{100}x + \frac{70}{100}x + \frac{77}{100} = \frac{220}{100}x + \frac{77}{100}. \]
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Now we have: \[ \frac{1}{5}x + \frac{267}{50} = \frac{220}{100}x + \frac{77}{100}. \]
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Next, we'll convert everything to have a common denominator of 100: \[ \frac{1}{5}x = \frac{20}{100}x, \quad \frac{267}{50} = \frac{534}{100}. \] Thus the left side simplifies to: \[ \frac{20}{100}x + \frac{534}{100}. \]
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Equating both sides: \[ \frac{20}{100}x + \frac{534}{100} = \frac{220}{100}x + \frac{77}{100}. \] To simplify: \[ \frac{534 - 77}{100} = \frac{220 - 20}{100}x \implies \frac{457}{100} = \frac{200}{100}x \implies 457 = 200x. \] We can isolate \( x \): \[ x = \frac{457}{200}. \]
So the value of \( x \) is
\[ \frac{457}{200}. \]
To express it more simply, we can write it as 2.285. Hence, in the specified format:
The value of x is \( 2.285 \) or \( \frac{457}{200} \).
Using the final answer in the asked format would give:
The value of x is \( 2.285 \).