To solve the equation
\[ \frac{4}{7} \left(\frac{21}{8} x + \frac{1}{2}\right) = -2 \left(\frac{1}{7} - \frac{5}{28} x\right), \]
let's start by simplifying both sides step by step.
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Left Side:
The left side is
\[ \frac{4}{7} \left(\frac{21}{8} x + \frac{1}{2}\right). \]
Distributing \(\frac{4}{7}\):
\[ \frac{4}{7} \cdot \frac{21}{8} x + \frac{4}{7} \cdot \frac{1}{2}. \]
Let's calculate each term separately:
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For the \(x\) term: \[ \frac{4 \cdot 21}{7 \cdot 8} x = \frac{84}{56} x = \frac{3}{2} x. \]
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For the constant term: \[ \frac{4 \cdot 1}{7 \cdot 2} = \frac{4}{14} = \frac{2}{7}. \]
Thus, the left side simplifies to:
\[ \frac{3}{2} x + \frac{2}{7}. \]
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Right Side:
The right side is
\[ -2 \left(\frac{1}{7} - \frac{5}{28} x\right). \]
Distributing \(-2\):
\[ -2 \cdot \frac{1}{7} + 2 \cdot \frac{5}{28} x. \]
Calculating each term:
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For the constant term: \[ -\frac{2}{7}. \]
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For the \(x\) term: \[ \frac{10}{28} x = \frac{5}{14} x. \]
Thus, the right side simplifies to:
\[ -\frac{2}{7} + \frac{5}{14} x. \]
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Setting the two sides equal:
Now we have the equation:
\[ \frac{3}{2} x + \frac{2}{7} = -\frac{2}{7} + \frac{5}{14} x. \]
First, eliminate the fractions by finding a common denominator, which is 14. Multiply each term by 14:
\[ 14 \left(\frac{3}{2} x\right) + 14 \left(\frac{2}{7}\right) = 14 \left(-\frac{2}{7}\right) + 14 \left(\frac{5}{14} x\right). \]
Simplifying each term results in:
\[ 21x + 4 = -4 + 5x. \]
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Rearranging the equation:
Now, let's move all \(x\) terms to one side and constants to the other:
\[ 21x - 5x = -4 - 4, \] leading to
\[ 16x = -8. \]
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Solving for \(x\):
Now divide both sides by 16:
\[ x = -\frac{8}{16} = -\frac{1}{2}. \]
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Writing in decimal form:
Converting \(-\frac{1}{2}\) to decimal gives:
\[ x = -0.5. \]
Thus, the final answer is
\[ \boxed{-0.5}. \]