Justify your steps using the properties of equality and determine the number of solutions to the equation 3/2(4−8x)+5x=1−7x

To solve the equation \( \frac{3}{2}(4 - 8x) + 5x = 1 - 7x \), we will proceed step by step and justify each step using the properties of equality.

1. **Distribute \( \frac{3}{2} \)**:
\[
\frac{3}{2} \cdot 4 - \frac{3}{2} \cdot 8x + 5x = 1 - 7x
\]
This simplifies to:
\[
6 - 12x + 5x = 1 - 7x
\]
(Justification: Distributive Property)

2. **Combine like terms on the left side**:
\[
6 - 7x = 1 - 7x
\]
(Justification: Combining like terms)

3. **Add \( 7x \) to both sides**:
\[
6 - 7x + 7x = 1 - 7x + 7x
\]
This simplifies to:
\[
6 = 1
\]
(Justification: Addition Property of Equality)

4. **Conclude the result**:
The statement \( 6 = 1 \) is a contradiction. This means that there are no 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 **zero**.