To solve the equation
\[ \frac{3}{2}(4 - 8x) + 5x = 1 - 7x, \]
let's proceed step by step and justify each step using properties of equality.
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Distribute \( \frac{3}{2} \) on the left side: \[ \frac{3}{2} \cdot 4 - \frac{3}{2} \cdot 8x + 5x = 1 - 7x \] This simplifies to: \[ 6 - 12x + 5x = 1 - 7x. \]
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Combine like terms on the left side: \[ 6 - 12x + 5x = 6 - 7x. \] Therefore, we have: \[ 6 - 7x = 1 - 7x. \]
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Add \( 7x \) to both sides (addition property of equality): \[ 6 = 1, \] since the \( -7x \) terms cancel out on both sides.
At this point, we see that we have arrived at a statement \( 6 = 1 \), which is a contradiction.
- Conclusion: Since \( 6 \neq 1 \), the original equation has no solutions. This indicates that the equation is inconsistent.
Thus, the number of solutions to the equation \( \frac{3}{2}(4 - 8x) + 5x = 1 - 7x \) is \( \boxed{0} \).
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