To solve the equation \( \frac{3}{2}(4 - 8x) + 5x = 1 - 7x \), we will follow a systematic approach and justify each step using properties of equality.
Step 1: Distribute \( \frac{3}{2} \) into \( (4 - 8x) \).
Using the Distributive Property:
\[ \frac{3}{2} \cdot 4 - \frac{3}{2} \cdot 8x = 6 - 12x \]
So, we can rewrite the equation as:
\[ 6 - 12x + 5x = 1 - 7x \]
Step 2: Combine like terms on the left side.
Combine the \( x \) terms:
\[ 6 - 12x + 5x = 6 - 7x \]
Now the equation is:
\[ 6 - 7x = 1 - 7x \]
Step 3: Add \( 7x \) to both sides to isolate the constants.
Adding \( 7x \) to both sides:
\[ 6 - 7x + 7x = 1 - 7x + 7x \]
This simplifies to:
\[ 6 = 1 \]
Step 4: Analyze the resulting equation.
The equation \( 6 = 1 \) is a contradiction, meaning that there are no values of \( x \) that can satisfy the original equation.
Conclusion
Since we derived a contradiction, the number of solutions to the equation \( \frac{3}{2}(4 - 8x) + 5x = 1 - 7x \) is:
\[ \boxed{0} \]