To solve the equation \(\frac{3}{2}(4 - 8x) + 5x = 1 - 7x\), we will apply the properties of equality step by step:
Step 1: Distribute \(\frac{3}{2}\) into \(4 - 8x\)
\[ \frac{3}{2} \cdot 4 - \frac{3}{2} \cdot 8x = \frac{12}{2} - \frac{24}{2}x = 6 - 12x \]
So, the equation becomes:
\[ 6 - 12x + 5x = 1 - 7x \]
Step 2: Combine like terms on the left side
Combine \(-12x\) and \(5x\):
\[ 6 - 7x = 1 - 7x \]
Step 3: Add \(7x\) to both sides
This helps to eliminate \(-7x\) from both sides:
\[ 6 - 7x + 7x = 1 - 7x + 7x \]
This simplifies to:
\[ 6 = 1 \]
Step 4: Analyze the resulting equation
Since we have the statement \(6 = 1\), which is a contradiction (this statement is false), this indicates that there are no possible values of \(x\) that can satisfy the original equation.
Conclusion:
Thus, the number of solutions to the equation \(\frac{3}{2}(4 - 8x) + 5x = 1 - 7x\) is \(\boxed{0}\).
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